Conformal Geometric Algebra - User contributions [en]

Dot product

2024-12-25T01:17:21Z

Eric Lengyel: Redirected page to Dot products

#REDIRECT [[Dot products]]

Dilation

2024-12-22T09:04:23Z

Eric Lengyel:

__NOTOC__
A ''dilation'' is a conformal transformation of Euclidean space performed by the operator

:$$\mathbf D = \dfrac{1 - \sigma}{2} (m_x \mathbf e_{235} + m_y \mathbf e_{315} + m_z \mathbf e_{125} - \mathbf e_{321}) + \dfrac{1 + \sigma}{2} {\large\unicode{x1d7d9}}$$ .

This operator scales an object $$\mathbf u$$ by the factor $$\sigma$$ about the center point $$\mathbf m = (m_x, m_y, m_z)$$ when used with the sandwich antiproduct $$\mathbf D \mathbin{\unicode{x27C7}} \mathbf u \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{D}}}$$. This dilation operator is scaled so that the round weight of $$\mathbf u$$ remains the same after the dilation is applied.

== Exponential Form ==

A dilation by a scale factor $$\sigma$$ about the center of a unitized, positively oriented [[sphere]] $$\mathbf s$$ can be expressed as an exponential of the sphere's [[attitude]] as

:$$\mathbf D = \exp_\unicode{x27C7}\left(-\dfrac{1}{2} \delta \operatorname{att}(\mathbf s)\right) = -\operatorname{att}(\mathbf s) \sinh \dfrac{\delta}{2} + {\large\unicode{x1d7d9}} \cosh \dfrac{\delta}{2}$$ ,

where $$\delta = \log \sigma$$. Expanding the $$\sinh$$ and $$\cosh$$ functions, we can rewrite this as

:$$\mathbf D = \dfrac{e^{-\delta/2} - e^{\delta/2}}{2} \operatorname{att}(\mathbf s) + \dfrac{e^{\delta/2} + e^{-\delta/2}}{2} {\large\unicode{x1d7d9}}$$ .

Homogeneous multiplication by $$e^{\delta/2}$$ gives us

:$$\mathbf D = \dfrac{1 - e^\delta}{2} \operatorname{att}(\mathbf s) + \dfrac{e^\delta + 1}{2} {\large\unicode{x1d7d9}}$$ ,

and replacing $$e^\delta$$ with $$\sigma$$ produces

:$$\mathbf D = \dfrac{1 - \sigma}{2} \operatorname{att}(\mathbf s) + \dfrac{1 + \sigma}{2} {\large\unicode{x1d7d9}}$$ .

== Matrix Form ==

When a dilation $$\mathbf D$$ is applied to a [[round point]], it is equivalent to premultiplying the point by the $$5 \times 5$$ matrix

:$$\begin{bmatrix}
\sigma & 0 & 0 & m_x (1 - \sigma) & 0 \\
0 & \sigma & 0 & m_y (1 - \sigma) & 0 \\
0 & 0 & \sigma & m_z (1 - \sigma) & 0 \\
0 & 0 & 0 & 1 & 0 \\
m_x \sigma (1 - \sigma) & m_y \sigma (1 - \sigma) & m_z \sigma (1 - \sigma) & \dfrac{\mathbf m^2 (1 - \sigma)^2}{2} & \sigma^2
\end{bmatrix}$$ .

== See Also ==

* [[Translation]]
* [[Rotation]]

Dot products

2024-08-28T01:57:55Z

Eric Lengyel:

The dot products between two unitized objects of the same type are listed in the following table. The vector $$\mathbf v$$ is the difference between their center positions, and the scalars $$r_1$$ and $$r_2$$ are their radii. In the case of dipoles and circles, the vectors $$\mathbf n_1$$ and $$\mathbf n_2$$ correspond to the directions of the carrier lines or the normals of the carrier planes.

{| class="wikitable"
! Type || Dot Product
|-
| style="padding: 12px;" | Round points $$\mathbf a_1$$ and $$\mathbf a_2$$
| style="padding: 12px;" | $$\mathbf a_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf a_2 = -\dfrac{1}{2}(\mathbf v^2 + r_1^2 + r_2^2)$$
|-
| style="padding: 12px;" | Dipoles $$\mathbf d_1$$ and $$\mathbf d_2$$
| style="padding: 12px;" | $$\mathbf d_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf d_2 = -\dfrac{1}{2}(\mathbf n_1 \cdot \mathbf n_2)(\mathbf v^2 + r_1^2 + r_2^2) + (\mathbf n_1 \cdot \mathbf v)(\mathbf n_2 \cdot \mathbf v)$$
|-
| style="padding: 12px;" | Circles $$\mathbf c_1$$ and $$\mathbf c_2$$
| style="padding: 12px;" | $$\mathbf c_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf c_2 = +\dfrac{1}{2}(\mathbf n_1 \cdot \mathbf n_2)(\mathbf v^2 - r_1^2 - r_2^2) - (\mathbf n_1 \cdot \mathbf v)(\mathbf n_2 \cdot \mathbf v)$$
|-
| style="padding: 12px;" | Spheres $$\mathbf s_1$$ and $$\mathbf s_2$$
| style="padding: 12px;" | $$\mathbf s_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf s_2 = +\dfrac{1}{2}(\mathbf v^2 - r_1^2 - r_2^2)$$
|}

In the case of two spheres $$\mathbf s_1$$ and $$\mathbf s_2$$, the dot product gives the product of the radii $$r_1$$ and $$r_2$$ times the cosine of the angle $$\phi$$ between the tangent planes where they intersect, as shown in the following figure. This can be demonstrating by considering the law of cosines for the angle $$\gamma$$, which states

:$$\mathbf v^2 = r_1^2 + r_2^2 - 2r_1r_2 \cos \gamma$$ .

Plugging this into the formula for the dot product yields $$\mathbf s_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf s_2 = r_1r_2 \cos \phi$$.

[[Image:sphere-dot-sphere.svg|512px]]

== See Also ==

* [[Geometric products]]
* [[Exterior products]]

File:Sphere-dot-sphere.svg

2024-08-28T01:55:17Z

Eric Lengyel:

Dot products

2024-08-28T01:50:58Z

Eric Lengyel: Created page with "The dot products between two unitized objects of the same type are listed in the following table. The vector $$\mathbf v$$ is the difference between their center positions, and the scalars $$r_1$$ and $$r_2$$ are their radii. In the case of dipoles and circles, the vectors $$\mathbf n_1$$ and $$\mathbf n_2$$ correspond to the directions of the carrier lines or the normals of the carrier planes. {| class="wikitable" ! Type || Dot Product |- | style="padding: 12px;" | Rou..."

The dot products between two unitized objects of the same type are listed in the following table. The vector $$\mathbf v$$ is the difference between their center positions, and the scalars $$r_1$$ and $$r_2$$ are their radii. In the case of dipoles and circles, the vectors $$\mathbf n_1$$ and $$\mathbf n_2$$ correspond to the directions of the carrier lines or the normals of the carrier planes.

{| class="wikitable"
! Type || Dot Product
|-
| style="padding: 12px;" | Round points $$\mathbf a_1$$ and $$\mathbf a_2$$
| style="padding: 12px;" | $$\mathbf a_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf a_2 = -\dfrac{1}{2}(\mathbf v^2 + r_1^2 + r_2^2)$$
|-
| style="padding: 12px;" | Dipoles $$\mathbf d_1$$ and $$\mathbf d_2$$
| style="padding: 12px;" | $$\mathbf d_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf d_2 = -\dfrac{1}{2}(\mathbf n_1 \cdot \mathbf n_2)(\mathbf v^2 + r_1^2 + r_2^2) + (\mathbf n_1 \cdot \mathbf v)(\mathbf n_2 \cdot \mathbf v)$$
|-
| style="padding: 12px;" | Circles $$\mathbf c_1$$ and $$\mathbf c_2$$
| style="padding: 12px;" | $$\mathbf c_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf c_2 = +\dfrac{1}{2}(\mathbf n_1 \cdot \mathbf n_2)(\mathbf v^2 - r_1^2 - r_2^2) - (\mathbf n_1 \cdot \mathbf v)(\mathbf n_2 \cdot \mathbf v)$$
|-
| style="padding: 12px;" | Spheres $$\mathbf s_1$$ and $$\mathbf s_2$$
| style="padding: 12px;" | $$\mathbf s_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf s_2 = +\dfrac{1}{2}(\mathbf v^2 - r_1^2 - r_2^2)$$
|}

== See Also ==

* [[Geometric products]]
* [[Exterior products]]

Main Page

2024-08-28T01:40:06Z

Eric Lengyel: /* Products and other binary operations */

__NOTOC__
== Conformal Geometric Algebra ==

This wiki is a repository of information about Conformal Geometric Algebra, and specifically the five-dimensional Clifford algebra $$\mathcal G_{4,1}$$. This wiki is associated with the following websites:

* [http://projectivegeometricalgebra.org Projective Geometric Algebra overview site]
* [http://rigidgeometricalgebra.org/wiki/index.php?title=Main_Page Rigid Geometric Algebra companion site]

Conformal geometric algebra is an area of active research, and new information is frequently being added to this wiki.

'''If you are experiencing problems with the LaTeX on this site, please clear the cookies for conformalgeometricalgebra.org and reload.'''

== Introduction ==

Conformal geometric algebra is constructed by adding two projective basis vectors called $$\mathbf e_-$$ and $$\mathbf e_+$$ to the set of ordinary basis vectors $$\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$$ of ''n''-dimensional Euclidean space. The new vectors are named this way because their squares under the dot product are

:$$\mathbf e_- \cdot \mathbf e_- = -1 \qquad \mathrm{and} \qquad \mathbf e_+ \cdot \mathbf e_+ = +1$$ .

This by itself is enough to build the entire algebra and observe all of its emergent properties. What follows is the standard way to interpret what's going on when we talk about specific vectors, bivectors, etc., and start multiplying them together with the [[exterior product]] and [[geometric product]]. Using 3D Euclidean space as an example, we begin by considering a homogeneous point

:$$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + w \mathbf e_-$$ .

As in [http://rigidgeometricalgebra.org/wiki/index.php?title=Main_Page rigid geometric algebra], any nonzero scalar multiple of this point belongs to the same equivalence class, so we can just assume $$w = 1$$. Now we perform a stereographic projection onto the four-dimensional unit hypersphere centered at $$\mathbf e_-$$ and extending into the $${\mathbf e_1, \mathbf e_2, \mathbf e_3, \mathbf e_+}$$ subspace. The north pole of this hypersphere toward which points are projected is $$\mathbf e_- + \mathbf e_+$$. This projection transforms the point $$\mathbf p$$ into

:$$\mathbf p = \dfrac{2x}{x^2 + y^2 + z^2 + 1} \mathbf e_1 + \dfrac{2y}{x^2 + y^2 + z^2 + 1} \mathbf e_2 + \dfrac{2z}{x^2 + y^2 + z^2 + 1} \mathbf e_3 + \mathbf e_- + \dfrac{x^2 + y^2 + z^2 - 1}{x^2 + y^2 + z^2 + 1} \mathbf e_+$$ .

We can homogeneously multiply this by $$(x^2 + y^2 + z^2 + 1)/2$$ without changing the meaning of $$\mathbf p$$ to obtain

:$$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \dfrac{1}{2}(x^2 + y^2 + z^2 + 1) \mathbf e_- + \dfrac{1}{2}(x^2 + y^2 + z^2 - 1) \mathbf e_+$$ .

Regrouping the coefficients of the $$\mathbf e_-$$ and $$\mathbf e_+$$ terms lets us write

:$$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \dfrac{1}{2}(\mathbf e_- - \mathbf e_+) + \dfrac{1}{2}(x^2 + y^2 + z^2)(\mathbf e_- + \mathbf e_+)$$ .

It is convenient to define two new vectors $$\mathbf e_4$$ and $$\mathbf e_5$$ as

:$$\mathbf e_4 = \dfrac{1}{2}(\mathbf e_- - \mathbf e_+) \qquad \mathrm{and} \qquad \mathbf e_5 = \mathbf e_- + \mathbf e_+$$ .

Using these two vectors, the point $$\mathbf p$$ is finally expressed as

:$$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \mathbf e_4 + \dfrac{1}{2}(x^2 + y^2 + z^2) \mathbf e_5$$ .

Numerous alternate definitions for $$\mathbf e_4$$ and $$\mathbf e_5$$ in which $$\mathbf e_4 = a(\mathbf e_- - \mathbf e_+)$$ and $$\mathbf e_5 = b(\mathbf e_- + \mathbf e_+)$$ are possible, but the definitions above where $$a = 1/2$$ and $$b = 1$$ tend to produce the cleanest formulation of the entire algebra. In particular, any definitions in which $$ab = 1/2$$ make the bivectors $$\mathbf e_- \wedge \mathbf e_+$$ and $$\mathbf e_4 \wedge \mathbf e_5$$ equal to each other.

When we consider the case that $$(x, y, z) = (0, 0, 0)$$, we can clearly identify the vector $$\mathbf e_4$$ as the origin. If we homogeneously rescale $$\mathbf p$$ as

:$$\mathbf p = \dfrac{2x}{x^2 + y^2 + z^2} \mathbf e_1 + \dfrac{2y}{x^2 + y^2 + z^2} \mathbf e_2 + \dfrac{2z}{x^2 + y^2 + z^2} \mathbf e_3 + \dfrac{2}{x^2 + y^2 + z^2} \mathbf e_4 + \mathbf e_5$$ ,

and allow the magnitude of $$(x, y, z)$$ to become arbitrarily large, then it's apparent that $$\mathbf e_5$$ represents the point at infinity.

The following diagram illustrates the image of the ''x'' axis under the homogeneous stereographic projection that we defined $$\mathbf p$$ with. Euclidean space becomes a parabolic surface called the ''horosphere''. The ''x''-''y'' plane is mapped to a paraboloid, and the full three-dimensional Euclidean space is mapped to a higher-dimensional parabolic volume.

[[Image:horosphere.svg|512px]]

== Pages ==

=== The seven types of geometric objects ===

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Circle]]
* [[Sphere]]

=== Various properties and unary operations ===

* [[Metrics]]
* [[Duals]]
* [[Carriers]]
* [[Attitude]]
* [[Centers]]
* [[Containers]]
* [[Partners]]

=== Products and other binary operations ===

* [[Geometric products]]
* [[Exterior products]]
* [[Dot products]]
* [[Join and meet]]
* [[Expansion]]
* [[Projections]]

=== Conformal Transformations ===

* [[Translation]]
* [[Rotation]]
* [[Dilation]]
* [[Reflection]]
* [[Inversion]]
* [[Transversion]]

Sphere

2024-07-31T04:31:32Z

Eric Lengyel:

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''sphere'' $$\mathbf s$$ is a quadrivector having the general form

:$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ .

If the $$s_u$$ component is zero, then the sphere contains the [[point at infinity]], and it is thus a flat [[plane]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a sphere can be formulated as

:$$\mathbf s = -\mathbf e_{1234} + p_x \mathbf e_{4235} + p_y \mathbf e_{4315} + p_z \mathbf e_{4125} - \dfrac{\mathbf p^2 - r^2}{2} \mathbf e_{3215}$$ .

The various properties of a sphere are summarized in the following table.

[[Image:sphere.svg|512px]]

== Center and Container ==

The round [[center]] of a sphere $$\mathbf s$$ is the [[round point]] having the same center and radius as $$\mathbf s$$, and it is given by

:$$\operatorname{cen}(\mathbf s) = \operatorname{ccr}(\mathbf s) \vee \mathbf s = -s_xs_u \mathbf e_1 - s_ys_u \mathbf e_2 - s_zs_u \mathbf e_3 + s_u^2 \mathbf e_4 + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\mathbf e_5$$ .

The [[container]] of a sphere $$\mathbf s$$ is the sphere itself with a different weight:

:$$\operatorname{con}(\mathbf s) = \mathbf s \wedge \operatorname{car}(\mathbf s)^\unicode["segoe ui symbol"]{x2606} = s_u^2 \mathbf e_{1234} + s_xs_u \mathbf e_{4235} + s_ys_u \mathbf e_{4315} + s_zs_u \mathbf e_{4125} + s_ws_u \mathbf e_{3215}$$ .

== Norms ==

The radius of a sphere $$\mathbf s$$ is given by

:$$\operatorname{rad}(\mathbf s) = \dfrac{\left\Vert\mathbf s\right\Vert_R}{\left\Vert\mathbf s\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 - 2s_ws_u}{s_u^2}}$$ .

== Contained Points ==

The [[attitude]] of a sphere $$\mathbf s$$ is given by

:$$\operatorname{att}(\mathbf s) = \mathbf s \vee \overline{\mathbf e_4} = s_u \mathbf e_{321} + s_x \mathbf e_{235} + s_t \mathbf e_{315} + s_z \mathbf e_{125}$$ .

The set of round points contained by a sphere $$\mathbf s$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + \operatorname{att}(\mathbf s) \vee (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^\unicode["segoe ui symbol"]{x2605}$$ .

That is, $$\mathbf s \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all bivectors $$\boldsymbol \alpha$$. In particular, points on the surface of a sphere are given when $$\boldsymbol \alpha$$ has magnitude $$\left\Vert\mathbf s\right\Vert_R$$ (the weighted radius), and this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Circle]]

Circle

2024-07-31T04:30:43Z

Eric Lengyel:

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form

:$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .

If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the [[point at infinity]], and it is thus a straight [[line]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{\mathbf p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

The various properties of a circle are summarized in the following table.

[[Image:circle.svg|800px]]

== Constraints ==

A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.

:$$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
:$$\mathbf g \cdot \mathbf v = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

== Carrier and Cocarrier ==

The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]

:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

The [[cocarrier]] of a circle $$\mathbf c$$ is the [[line]]

:$$\operatorname{ccr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

The carrier and cocarrier meet at the flat center of the circle, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf c) \vee \operatorname{ccr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

:$$\operatorname{cen}(\mathbf c) = \operatorname{ccr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

:$$\operatorname{con}(\mathbf c) = \mathbf c \wedge \operatorname{car}(\mathbf c)^\unicode["segoe ui symbol"]{x2606} = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .

== Norms ==

The radius of a circle $$\mathbf c$$ is given by

:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a circle $$\mathbf c$$ is given by

:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \overline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{x2605}$$ .

That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Sphere]]

Dipole

2024-07-31T04:30:04Z

Eric Lengyel:

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form

:$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ .

If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ components are all zero, then the dipole contains the [[point at infinity]], and it is thus a [[flat point]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as

:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{\mathbf p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .

The various properties of a dipole are summarized in the following table.

[[Image:dipole.svg|800px]]

== Constraints ==

A valid dipole $$\mathbf d$$ must satisfy the following constraints, where $$\mathbf v = (d_{vx}, d_{vy}, d_{vz})$$, $$\mathbf m = (d_{mx}, d_{my}, d_{mz})$$, and $$\mathbf p = (d_{px}, d_{py}, d_{pz})$$ are treated as ordinary 3D vectors.

:$$\mathbf p \times \mathbf v - d_{pw}\mathbf m = \mathbf 0$$
:$$\mathbf p \cdot \mathbf m = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.

== Carrier and Cocarrier ==

The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]

:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .

The [[cocarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

:$$\operatorname{ccr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .

The carrier and cocarrier meet at the flat center of the dipole, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf d) \vee \operatorname{ccr}(\mathbf d) = (d_{vz} d_{my} - d_{vx} d_{pw} - d_{vy} d_{mz})\mathbf e_{15} + (d_{vx} d_{mz} - d_{vy} d_{pw} - d_{vz} d_{mx})\mathbf e_{25} + (d_{vy} d_{mx} - d_{vz} d_{pw} - d_{vx} d_{my})\mathbf e_{35} - (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{cen}(\mathbf d) = \operatorname{ccr}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{con}(\mathbf d) = \mathbf d \wedge \operatorname{car}(\mathbf d)^\unicode["segoe ui symbol"]{x2606} = (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\mathbf e_{4235} + (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\mathbf e_{4315} + (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\mathbf e_{4125} + (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .

A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf d = \operatorname{car}(\mathbf d) \vee \operatorname{con}(\mathbf d)$$ .

== Norms ==

The radius of a dipole $$\mathbf d$$ is given by

:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a dipole $$\mathbf d$$ is given by

:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \overline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .

The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\alpha) = \operatorname{cen}(\mathbf d) + \alpha \operatorname{att}(\mathbf d)$$ .

That is, $$\mathbf d \wedge \mathbf p(\alpha) = 0$$ for all real numbers $$\alpha$$. In particular, the two points on the surface of a dipole are given by the parameter $$\alpha_R = \pm \left\Vert\mathbf d\right\Vert_R$$ (the weighted radius). When $$\mathbf d$$ is a real dipole, this is precisely where the radius of $$\mathbf p(\alpha)$$ is zero. For smaller absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is real, and for larger absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is imaginary. When $$\mathbf d$$ is an imaginary dipole, the round point $$\mathbf p(\alpha)$$ is always imaginary, and it has an absolute radius at least as large as the dipole itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Circle]]
* [[Sphere]]

Round point

2024-07-31T04:29:17Z

Eric Lengyel:

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''round point'' $$\mathbf a$$ is a vector having the general form

:$$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ .

Given a position $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a round point can be formulated as

:$$\mathbf a = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + \mathbf e_4 + \dfrac{\mathbf p^2 + r^2}{2} \mathbf e_5$$ .

The various properties of a sphere are summarized in the following table.

[[Image:round.svg|512px]]

== Center and Container ==

The round [[center]] of a round point $$\mathbf a$$ is the round point itself with a different weight:

:$$\operatorname{cen}(\mathbf a) = \operatorname{ccr}(\mathbf a) \vee \mathbf a = a_xa_w \mathbf e_1 + a_ya_w \mathbf e_2 + a_za_w \mathbf e_3 + a_w^2 \mathbf e_4 + a_wa_u \mathbf e_5$$ .

The [[container]] of a round point $$\mathbf a$$ is the sphere having the same center and radius as $$\mathbf a$$, and it is given by

:$$\operatorname{con}(\mathbf a) = \mathbf a \wedge \operatorname{car}(\mathbf a)^\unicode["segoe ui symbol"]{x2606} = -a_w^2 \mathbf e_{1234} + a_xa_w \mathbf e_{4235} + a_ya_w \mathbf e_{4315} + a_za_w \mathbf e_{4125} + (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}$$ .

== Norms ==

The radius of a round point $$\mathbf a$$ is given by

:$$\operatorname{rad}(\mathbf a) = \dfrac{\left\Vert\mathbf a\right\Vert_R}{\left\Vert\mathbf a\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{2a_wa_u - a_x^2 - a_y^2 - a_z^2}{a_w^2}}$$ .

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Dipole]]
* [[Circle]]
* [[Sphere]]

Circle

2024-04-21T20:46:34Z

Eric Lengyel: /* Contained Points */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form

:$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .

If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the [[point at infinity]], and it is thus a straight [[line]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

The various properties of a circle are summarized in the following table.

[[Image:circle.svg|800px]]

== Constraints ==

A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.

:$$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
:$$\mathbf g \cdot \mathbf v = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

== Carrier and Cocarrier ==

The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]

:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

The [[cocarrier]] of a circle $$\mathbf c$$ is the [[line]]

:$$\operatorname{ccr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

The carrier and cocarrier meet at the flat center of the circle, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf c) \vee \operatorname{ccr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

:$$\operatorname{cen}(\mathbf c) = \operatorname{ccr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

:$$\operatorname{con}(\mathbf c) = \mathbf c \wedge \operatorname{car}(\mathbf c)^\unicode["segoe ui symbol"]{x2606} = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .

== Norms ==

The radius of a circle $$\mathbf c$$ is given by

:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a circle $$\mathbf c$$ is given by

:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \overline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{x2605}$$ .

That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Sphere]]

Sphere

2024-04-21T20:45:52Z

Eric Lengyel: /* Contained Points */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''sphere'' $$\mathbf s$$ is a quadrivector having the general form

:$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ .

If the $$s_u$$ component is zero, then the sphere contains the [[point at infinity]], and it is thus a flat [[plane]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a sphere can be formulated as

:$$\mathbf s = -\mathbf e_{1234} + p_x \mathbf e_{4235} + p_y \mathbf e_{4315} + p_z \mathbf e_{4125} - \dfrac{p^2 - r^2}{2} \mathbf e_{3215}$$ .

The various properties of a sphere are summarized in the following table.

[[Image:sphere.svg|512px]]

== Center and Container ==

The round [[center]] of a sphere $$\mathbf s$$ is the [[round point]] having the same center and radius as $$\mathbf s$$, and it is given by

:$$\operatorname{cen}(\mathbf s) = \operatorname{ccr}(\mathbf s) \vee \mathbf s = -s_xs_u \mathbf e_1 - s_ys_u \mathbf e_2 - s_zs_u \mathbf e_3 + s_u^2 \mathbf e_4 + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\mathbf e_5$$ .

The [[container]] of a sphere $$\mathbf s$$ is the sphere itself with a different weight:

:$$\operatorname{con}(\mathbf s) = \mathbf s \wedge \operatorname{car}(\mathbf s)^\unicode["segoe ui symbol"]{x2606} = s_u^2 \mathbf e_{1234} + s_xs_u \mathbf e_{4235} + s_ys_u \mathbf e_{4315} + s_zs_u \mathbf e_{4125} + s_ws_u \mathbf e_{3215}$$ .

== Norms ==

The radius of a sphere $$\mathbf s$$ is given by

:$$\operatorname{rad}(\mathbf s) = \dfrac{\left\Vert\mathbf s\right\Vert_R}{\left\Vert\mathbf s\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 - 2s_ws_u}{s_u^2}}$$ .

== Contained Points ==

The [[attitude]] of a sphere $$\mathbf s$$ is given by

:$$\operatorname{att}(\mathbf s) = \mathbf s \vee \overline{\mathbf e_4} = s_u \mathbf e_{321} + s_x \mathbf e_{235} + s_t \mathbf e_{315} + s_z \mathbf e_{125}$$ .

The set of round points contained by a sphere $$\mathbf s$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + \operatorname{att}(\mathbf s) \vee (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^\unicode["segoe ui symbol"]{x2605}$$ .

That is, $$\mathbf s \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all bivectors $$\boldsymbol \alpha$$. In particular, points on the surface of a sphere are given when $$\boldsymbol \alpha$$ has magnitude $$\left\Vert\mathbf s\right\Vert_R$$ (the weighted radius), and this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Circle]]

Circle

2024-04-21T20:45:31Z

Eric Lengyel: /* Contained Points */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form

:$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .

If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the [[point at infinity]], and it is thus a straight [[line]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

The various properties of a circle are summarized in the following table.

[[Image:circle.svg|800px]]

== Constraints ==

A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.

:$$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
:$$\mathbf g \cdot \mathbf v = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

== Carrier and Cocarrier ==

The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]

:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

The [[cocarrier]] of a circle $$\mathbf c$$ is the [[line]]

:$$\operatorname{ccr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

The carrier and cocarrier meet at the flat center of the circle, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf c) \vee \operatorname{ccr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

:$$\operatorname{cen}(\mathbf c) = \operatorname{ccr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

:$$\operatorname{con}(\mathbf c) = \mathbf c \wedge \operatorname{car}(\mathbf c)^\unicode["segoe ui symbol"]{x2606} = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .

== Norms ==

The radius of a circle $$\mathbf c$$ is given by

:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a circle $$\mathbf c$$ is given by

:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \overline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{x2606}$$ .

That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Sphere]]

Dipole

2024-04-21T20:45:05Z

Eric Lengyel: /* Contained Points */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form

:$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ .

If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ components are all zero, then the dipole contains the [[point at infinity]], and it is thus a [[flat point]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as

:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .

The various properties of a dipole are summarized in the following table.

[[Image:dipole.svg|800px]]

== Constraints ==

A valid dipole $$\mathbf d$$ must satisfy the following constraints, where $$\mathbf v = (d_{vx}, d_{vy}, d_{vz})$$, $$\mathbf m = (d_{mx}, d_{my}, d_{mz})$$, and $$\mathbf p = (d_{px}, d_{py}, d_{pz})$$ are treated as ordinary 3D vectors.

:$$\mathbf p \times \mathbf v - d_{pw}\mathbf m = \mathbf 0$$
:$$\mathbf p \cdot \mathbf m = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.

== Carrier and Cocarrier ==

The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]

:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .

The [[cocarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

:$$\operatorname{ccr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .

The carrier and cocarrier meet at the flat center of the dipole, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf d) \vee \operatorname{ccr}(\mathbf d) = (d_{vz} d_{my} - d_{vx} d_{pw} - d_{vy} d_{mz})\mathbf e_{15} + (d_{vx} d_{mz} - d_{vy} d_{pw} - d_{vz} d_{mx})\mathbf e_{25} + (d_{vy} d_{mx} - d_{vz} d_{pw} - d_{vx} d_{my})\mathbf e_{35} - (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{cen}(\mathbf d) = \operatorname{ccr}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{con}(\mathbf d) = \mathbf d \wedge \operatorname{car}(\mathbf d)^\unicode["segoe ui symbol"]{x2606} = (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\mathbf e_{4235} + (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\mathbf e_{4315} + (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\mathbf e_{4125} + (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .

A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf d = \operatorname{car}(\mathbf d) \vee \operatorname{con}(\mathbf d)$$ .

== Norms ==

The radius of a dipole $$\mathbf d$$ is given by

:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a dipole $$\mathbf d$$ is given by

:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \overline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .

The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\alpha) = \operatorname{cen}(\mathbf d) + \alpha \operatorname{att}(\mathbf d)$$ .

That is, $$\mathbf d \wedge \mathbf p(\alpha) = 0$$ for all real numbers $$\alpha$$. In particular, the two points on the surface of a dipole are given by the parameter $$\alpha_R = \pm \left\Vert\mathbf d\right\Vert_R$$ (the weighted radius). When $$\mathbf d$$ is a real dipole, this is precisely where the radius of $$\mathbf p(\alpha)$$ is zero. For smaller absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is real, and for larger absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is imaginary. When $$\mathbf d$$ is an imaginary dipole, the round point $$\mathbf p(\alpha)$$ is always imaginary, and it has an absolute radius at least as large as the dipole itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Circle]]
* [[Sphere]]

Metrics

2024-04-13T02:01:40Z

Eric Lengyel: Created page with "The ''metric'' used in the 5D conformal geometric algebra over 3D Euclidean space is the $$5 \times 5$$ matrix $$\mathfrak g$$ given by :$$\mathfrak g = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & -1 & 0\\\end{bmatrix}$$ . The ''metric exomorphism matrix'' $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$32 \times 32$$ matrix shown below. Im..."

The ''metric'' used in the 5D conformal geometric algebra over 3D Euclidean space is the $$5 \times 5$$ matrix $$\mathfrak g$$ given by

:$$\mathfrak g = \begin{bmatrix}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & -1 \\
0 & 0 & 0 & -1 & 0\\\end{bmatrix}$$ .

The ''metric exomorphism matrix'' $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$32 \times 32$$ matrix shown below.

[[Image:metric-cga-3d.svg|800px]]

The ''metric antiexomorphism matrix'' $$\mathbb G$$, often called the "antimetric", corresponding to the metric $$\mathfrak g$$ is the negation of the metric exomorphism matrix $$\mathbf G$$.

The metric and antimetric determine [[duals]], [[dot products]], and [[geometric products]].

== See Also ==

* [[Duals]]
* [[Dot products]]

File:Metric-cga-3d.svg

2024-04-13T02:00:03Z

Eric Lengyel:

Main Page

2024-04-13T01:57:37Z

Eric Lengyel: /* Various properties and unary operations */

__NOTOC__
== Conformal Geometric Algebra ==

This wiki is a repository of information about Conformal Geometric Algebra, and specifically the five-dimensional Clifford algebra $$\mathcal G_{4,1}$$. This wiki is associated with the following websites:

* [http://projectivegeometricalgebra.org Projective Geometric Algebra overview site]
* [http://rigidgeometricalgebra.org/wiki/index.php?title=Main_Page Rigid Geometric Algebra companion site]

Conformal geometric algebra is an area of active research, and new information is frequently being added to this wiki.

'''If you are experiencing problems with the LaTeX on this site, please clear the cookies for conformalgeometricalgebra.org and reload.'''

== Introduction ==

Conformal geometric algebra is constructed by adding two projective basis vectors called $$\mathbf e_-$$ and $$\mathbf e_+$$ to the set of ordinary basis vectors $$\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n$$ of ''n''-dimensional Euclidean space. The new vectors are named this way because their squares under the dot product are

:$$\mathbf e_- \cdot \mathbf e_- = -1 \qquad \mathrm{and} \qquad \mathbf e_+ \cdot \mathbf e_+ = +1$$ .

This by itself is enough to build the entire algebra and observe all of its emergent properties. What follows is the standard way to interpret what's going on when we talk about specific vectors, bivectors, etc., and start multiplying them together with the [[exterior product]] and [[geometric product]]. Using 3D Euclidean space as an example, we begin by considering a homogeneous point

:$$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + w \mathbf e_-$$ .

As in [http://rigidgeometricalgebra.org/wiki/index.php?title=Main_Page rigid geometric algebra], any nonzero scalar multiple of this point belongs to the same equivalence class, so we can just assume $$w = 1$$. Now we perform a stereographic projection onto the four-dimensional unit hypersphere centered at $$\mathbf e_-$$ and extending into the $${\mathbf e_1, \mathbf e_2, \mathbf e_3, \mathbf e_+}$$ subspace. The north pole of this hypersphere toward which points are projected is $$\mathbf e_- + \mathbf e_+$$. This projection transforms the point $$\mathbf p$$ into

:$$\mathbf p = \dfrac{2x}{x^2 + y^2 + z^2 + 1} \mathbf e_1 + \dfrac{2y}{x^2 + y^2 + z^2 + 1} \mathbf e_2 + \dfrac{2z}{x^2 + y^2 + z^2 + 1} \mathbf e_3 + \mathbf e_- + \dfrac{x^2 + y^2 + z^2 - 1}{x^2 + y^2 + z^2 + 1} \mathbf e_+$$ .

We can homogeneously multiply this by $$(x^2 + y^2 + z^2 + 1)/2$$ without changing the meaning of $$\mathbf p$$ to obtain

:$$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \dfrac{1}{2}(x^2 + y^2 + z^2 + 1) \mathbf e_- + \dfrac{1}{2}(x^2 + y^2 + z^2 - 1) \mathbf e_+$$ .

Regrouping the coefficients of the $$\mathbf e_-$$ and $$\mathbf e_+$$ terms lets us write

:$$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \dfrac{1}{2}(\mathbf e_- - \mathbf e_+) + \dfrac{1}{2}(x^2 + y^2 + z^2)(\mathbf e_- + \mathbf e_+)$$ .

It is convenient to define two new vectors $$\mathbf e_4$$ and $$\mathbf e_5$$ as

:$$\mathbf e_4 = \dfrac{1}{2}(\mathbf e_- - \mathbf e_+) \qquad \mathrm{and} \qquad \mathbf e_5 = \mathbf e_- + \mathbf e_+$$ .

Using these two vectors, the point $$\mathbf p$$ is finally expressed as

:$$\mathbf p = x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3 + \mathbf e_4 + \dfrac{1}{2}(x^2 + y^2 + z^2) \mathbf e_5$$ .

Numerous alternate definitions for $$\mathbf e_4$$ and $$\mathbf e_5$$ in which $$\mathbf e_4 = a(\mathbf e_- - \mathbf e_+)$$ and $$\mathbf e_5 = b(\mathbf e_- + \mathbf e_+)$$ are possible, but the definitions above where $$a = 1/2$$ and $$b = 1$$ tend to produce the cleanest formulation of the entire algebra. In particular, any definitions in which $$ab = 1/2$$ make the bivectors $$\mathbf e_- \wedge \mathbf e_+$$ and $$\mathbf e_4 \wedge \mathbf e_5$$ equal to each other.

When we consider the case that $$(x, y, z) = (0, 0, 0)$$, we can clearly identify the vector $$\mathbf e_4$$ as the origin. If we homogeneously rescale $$\mathbf p$$ as

:$$\mathbf p = \dfrac{2x}{x^2 + y^2 + z^2} \mathbf e_1 + \dfrac{2y}{x^2 + y^2 + z^2} \mathbf e_2 + \dfrac{2z}{x^2 + y^2 + z^2} \mathbf e_3 + \dfrac{2}{x^2 + y^2 + z^2} \mathbf e_4 + \mathbf e_5$$ ,

and allow the magnitude of $$(x, y, z)$$ to become arbitrarily large, then it's apparent that $$\mathbf e_5$$ represents the point at infinity.

The following diagram illustrates the image of the ''x'' axis under the homogeneous stereographic projection that we defined $$\mathbf p$$ with. Euclidean space becomes a parabolic surface called the ''horosphere''. The ''x''-''y'' plane is mapped to a paraboloid, and the full three-dimensional Euclidean space is mapped to a higher-dimensional parabolic volume.

[[Image:horosphere.svg|512px]]

== Pages ==

=== The seven types of geometric objects ===

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Circle]]
* [[Sphere]]

=== Various properties and unary operations ===

* [[Metrics]]
* [[Duals]]
* [[Carriers]]
* [[Attitude]]
* [[Centers]]
* [[Containers]]
* [[Partners]]

=== Products and other binary operations ===

* [[Geometric products]]
* [[Exterior products]]
* [[Join and meet]]
* [[Expansion]]
* [[Projections]]

=== Conformal Transformations ===

* [[Translation]]
* [[Rotation]]
* [[Dilation]]
* [[Reflection]]
* [[Inversion]]
* [[Transversion]]

File:AntiwedgeProduct.svg

2024-04-03T23:07:09Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:AntiwedgeProduct.svg

File:WedgeProduct.svg

2024-04-03T23:06:47Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:WedgeProduct.svg

Geometric products

2024-04-03T23:06:24Z

Eric Lengyel:

The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct.

== Geometric Product ==

The following Cayley table shows the geometric products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. The identity of the geometric product is the [[scalar]] basis element $$\mathbf 1$$.

Cells highlighted green correspond to nonzero contributions from the [[wedge product]].

[[Image:GeometricProduct.svg|1920px]]

== Geometric Antiproduct ==

The following Cayley table shows the geometric antiproducts between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. The identity of the geometric antiproduct is the [[antiscalar]] basis element $${\large\unicode{x1D7D9}}$$.

Cells highlighted green correspond to nonzero contributions from the [[antiwedge product]].

[[Image:GeometricAntiproduct.svg|1920px]]

== See Also ==

* [[Wedge products]]
* [[Dot products]]
* [[Duals]]

File:GeometricAntiproduct.svg

2024-04-03T23:05:02Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:GeometricAntiproduct.svg

File:GeometricProduct.svg

2024-04-03T23:04:38Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:GeometricProduct.svg

Attitude

2024-04-03T23:02:38Z

Eric Lengyel:

The ''attitude'' function, denoted by $$\operatorname{att}$$, extracts the attitude of a geometry and returns a purely directional object. The attitude function is defined as

:$$\operatorname{att}(\mathbf u) = \mathbf u \vee \overline{\mathbf e_4}$$ .

The following table lists the attitude for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Attitude
|-
| style="padding: 12px;" | [[Flat point]]
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$
| style="padding: 12px;" | $$\operatorname{att}(\mathbf p) = p_w \mathbf e_5$$
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\operatorname{att}(\boldsymbol l) = l_{vx} \mathbf e_{15} + l_{vy} \mathbf e_{25} + l_{vz} \mathbf e_{35}$$
|-
| style="padding: 12px;" | [[Plane]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\operatorname{att}(\mathbf g) = g_x \mathbf e_{235} + g_y \mathbf e_{315} + g_z \mathbf e_{125}$$
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\operatorname{att}(\mathbf a) = a_w \mathbf 1$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\operatorname{att}(\mathbf d) = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\operatorname{att}(\mathbf c) = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\operatorname{att}(\mathbf s) = s_u \mathbf e_{321} + s_x \mathbf e_{235} + s_y \mathbf e_{315} + s_z \mathbf e_{125}$$
|}

The [[round points]] contained by a round object ([[dipole]], [[circle]], or [[sphere]]) differ from the object's center by a multiple of the object's attitude. In general, any point $$\mathbf p$$ contained in a grade $$k$$ object $$\mathbf u$$ is given by

:$$\mathbf p = \operatorname{cen}(\mathbf u) + \operatorname{att}(\mathbf u) \vee \alpha^\unicode["segoe ui symbol"]{x2605}$$ ,

where $$\alpha$$ is a Euclidean $$(k - 2)$$-vector. If $$\mathbf u$$ is a dipole, then $$\alpha$$ is a scalar, if $$\mathbf u$$ is a circle, then $$\alpha$$ is a vector $$x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3$$, and if $$\mathbf u$$ is a sphere, then $$\alpha$$ is a bivector $$x \mathbf e_{23} + y \mathbf e_{31} + z \mathbf e_{12}$$.

== See Also ==

* [[Duals]]
* [[Carriers]]
* [[Centers]]
* [[Containers]]

Containers

2024-04-03T22:59:40Z

Eric Lengyel:

The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf u$$ is denoted by $$\operatorname{con}(\mathbf u)$$, and it is given by the [[expansion]] of $$\mathbf u$$ into its own [[carrier]]:

:$$\operatorname{con}(\mathbf u) = \mathbf u \wedge \operatorname{car}(\mathbf u)^\unicode["segoe ui symbol"]{x2606}$$ .

The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.

The following table lists the containers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Container
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf a) =
&-a_w^2 \mathbf e_{1234} \\
&+ a_xa_w \mathbf e_{4235} \\
&+ a_ya_w \mathbf e_{4315} \\
&+ a_za_w \mathbf e_{4125} \\
&+ (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}\end{split}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf d)
&= (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_{1234} \\
&+ (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\,\mathbf e_{4235} \\
&+ (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\,\mathbf e_{4315} \\
&+ (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\,\mathbf e_{4125} \\
&+ (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\,\mathbf e_{3215}\end{split}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf c) =
-\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\,\mathbf e_{1234} \\
+\,&(c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\,\mathbf e_{4235} \\
+\,&(c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\,\mathbf e_{4315} \\
+\,&(c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\,\mathbf e_{4125} \\
+\,&(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\,\mathbf e_{3215}\end{split}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf s)
&= s_u^2 \mathbf e_{1234} \\
&+ s_xs_u \mathbf e_{4235} \\
&+ s_ys_u \mathbf e_{4315} \\
&+ s_zs_u \mathbf e_{4125} \\
&+ s_ws_u \mathbf e_{3215}\end{split}$$
|}

== See Also ==

* [[Centers]]
* [[Carriers]]
* [[Partners]]
* [[Attitude]]

Centers

2024-04-03T22:58:57Z

Eric Lengyel:

The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf u$$ is denoted by $$\operatorname{cen}(\mathbf u)$$, and it is given by the [[meet]] of $$\mathbf u$$ and its own [[cocarrier]]:

:$$\operatorname{cen}(\mathbf u) = \operatorname{ccr}(\mathbf u) \vee \mathbf u$$ .

The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

The following table lists the centers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Center
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf a) =
{\phantom +}\,&a_xa_w \mathbf e_1 \\
+\,&a_ya_w \mathbf e_2 \\
+\,&a_za_w \mathbf e_3 \\
+\,&a_w^2 \mathbf e_4 \\
+\,&a_wa_u \mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf d) =
{\phantom +}\,&(d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\,\mathbf e_1 \\
+\,&(d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\,\mathbf e_2 \\
+\,&(d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\,\mathbf e_3 \\
+\,&(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_4 \\
+\,&(d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\,\mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf c) =
{\phantom +}\,&(c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\,\mathbf e_1 \\
+\,&(c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\,\mathbf e_2 \\
+\,&(c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\,\mathbf e_3 \\
+\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\,\mathbf e_4 \\
+\,&(c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\,\mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf s) =
-\,&s_xs_u \mathbf e_1 \\
-\,&s_ys_u \mathbf e_2 \\
-\,&s_zs_u \mathbf e_3 \\
+\,&s_u^2 \mathbf e_4 \\
+\,&(s_x^2 + s_y^2 + s_z^2 - s_ws_u)\,\mathbf e_5\end{split}$$
|}

== See Also ==

* [[Containers]]
* [[Carriers]]
* [[Partners]]
* [[Attitude]]

Carriers

2024-04-03T22:57:45Z

Eric Lengyel:

== Carrier ==

The ''carrier'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the lowest dimensional flat object (a [[flat point]], [[line]], or [[plane]]) that contains it. The carrier of an object $$\mathbf u$$ is denoted by $$\operatorname{car}(\mathbf u)$$, and it is calculated by simply multiplying $$\mathbf u$$ by $$\mathbf e_5$$ with the [[wedge product]] to extract the round part of $$\mathbf u$$ as a flat geometry:

:$$\operatorname{car}(\mathbf u) = \mathbf u \wedge \mathbf e_5$$ .

The following table lists the carriers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Carrier
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\operatorname{car}(\mathbf a) = a_x \mathbf e_{15} + a_y \mathbf e_{25} + a_z \mathbf e_{35} + a_w \mathbf e_{45}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\operatorname{car}(\mathbf d) = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\operatorname{car}(\mathbf c) = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\operatorname{car}(\mathbf s) = s_u {\large\unicode{x1d7d9}}$$
|}

== Cocarrier ==

The ''cocarrier'' of a round object is the carrier of its [[antidual]]. The cocarrier of an object $$\mathbf u$$ is denoted by $$\operatorname{ccr}(\mathbf u)$$, and it is calculated by

:$$\operatorname{ccr}(\mathbf u) = \mathbf u^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5$$ .

The cocarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and cocarrier can be used to calculate the center of an object $$\mathbf u$$ as a [[flat point]] with the formula $$\operatorname{ccr}(\mathbf u) \vee \operatorname{car}(\mathbf u)$$.

The following table lists the anticarriers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Cocarrier
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\operatorname{ccr}(\mathbf a) = a_w {\large\unicode{x1d7d9}}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\operatorname{ccr}(\mathbf d) = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\operatorname{ccr}(\mathbf c) = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\operatorname{ccr}(\mathbf s) = s_x \mathbf e_{15} + s_y \mathbf e_{25} + s_z \mathbf e_{35} - s_u \mathbf e_{45}$$
|}

== See Also ==

* [[Attitude]]
* [[Centers]]
* [[Containers]]
* [[Partners]]

Partners

2024-04-03T22:52:16Z

Eric Lengyel:

The ''partner'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the round object having the same center, same [[carrier]], and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf u$$ is denoted by $$\operatorname{par}(\mathbf u)$$, and it is given by the [[meet]] of the [[container]] of $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and the [[carrier]] of $$\mathbf u$$:

:$$\operatorname{par}(\mathbf u) = (-1)^{\operatorname{gr}(\mathbf u) + 1}\operatorname{con}(\mathbf u^\unicode["segoe ui symbol"]{x2606}) \vee \operatorname{car}(\mathbf u)$$ .

The [[dot product]] between a round object and its partner is always zero. They are orthogonal:

:$$\mathbf u \mathbin{\unicode{x25CF}} \operatorname{par}(\mathbf u) = 0$$ .

The following table lists the partners for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Partner
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\begin{split}\operatorname{par}(\mathbf a) &= a_xa_w^2 \mathbf e_1 + a_ya_w^2 \mathbf e_2 + a_za_w^2 \mathbf e_3 + a_w^3 \mathbf e_4 + (a_x^2 + a_y^2 + a_z^2 - a_wa_u)\,a_w \mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{par}(\mathbf d) =\,&(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)(d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{pw} \mathbf e_{45}) \\
+\,&(d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})(d_{vx} \mathbf e_{15} + d_{vy} \mathbf e_{25} + d_{vz} \mathbf e_{35}) \\
+\,&(d_{mz} d_{vy} - d_{my} d_{vz})\,d_{pw}\mathbf e_{15} + (d_{mx} d_{vz} - d_{mz} d_{vx})\,d_{pw}\mathbf e_{25} + (d_{my} d_{vx} - d_{mx} d_{vy})\,d_{pw}\mathbf e_{35}\end{split}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{par}(\mathbf c) =\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)(c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435}) \\
+\,&(c_{gw}^2 - c_{vx}^2 - c_{vy}^2 - c_{vz}^2 - c_{gx} c_{mx} - c_{gy} c_{my} - c_{gz} c_{mz})(c_{gx} \mathbf e_{235} + c_{gy} \mathbf e_{315} + c_{gz} \mathbf e_{125}) \\
+\,&(c_{vy} c_{gz} - c_{vz} c_{gy})\,c_{gw}\mathbf e_{235} + (c_{vz} c_{gx} - c_{vx} c_{gz})\,c_{gw}\mathbf e_{315} + (c_{vx} c_{gy} - c_{vy} c_{gx})\,c_{gw}\mathbf e_{125}\end{split}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{par}(\mathbf s) &= s_u^3 \mathbf e_{1234} + s_xs_u^2 \mathbf e_{4235} + s_ys_u^2 \mathbf e_{4315} + s_zs_u^2 \mathbf e_{4125} + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\,s_u \mathbf e_{3215}\end{split}$$
|}

== See Also ==

* [[Carriers]]
* [[Centers]]
* [[Containers]]
* [[Attitude]]

Exterior product

2023-12-31T22:29:00Z

Eric Lengyel: Redirected page to Exterior products

#REDIRECT [[Exterior products]]

Cocarriers

2023-12-01T23:48:38Z

Eric Lengyel: Redirected page to Carriers

#REDIRECT [[Carriers]]

Circle

2023-12-01T23:45:56Z

Eric Lengyel:

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form

:$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .

If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the [[point at infinity]], and it is thus a straight [[line]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

The various properties of a circle are summarized in the following table.

[[Image:circle.svg|800px]]

== Constraints ==

A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.

:$$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
:$$\mathbf g \cdot \mathbf v = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

== Carrier and Cocarrier ==

The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]

:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

The [[cocarrier]] of a circle $$\mathbf c$$ is the [[line]]

:$$\operatorname{ccr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

The carrier and cocarrier meet at the flat center of the circle, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf c) \vee \operatorname{ccr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

:$$\operatorname{cen}(\mathbf c) = \operatorname{ccr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

:$$\operatorname{con}(\mathbf c) = \mathbf c \wedge \operatorname{car}(\mathbf c)^\unicode["segoe ui symbol"]{x2606} = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .

== Norms ==

The radius of a circle $$\mathbf c$$ is given by

:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a circle $$\mathbf c$$ is given by

:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \underline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{x2606}$$ .

That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Sphere]]

Dipole

2023-12-01T23:45:06Z

Eric Lengyel:

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form

:$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ .

If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ components are all zero, then the dipole contains the [[point at infinity]], and it is thus a [[flat point]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as

:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .

The various properties of a dipole are summarized in the following table.

[[Image:dipole.svg|800px]]

== Constraints ==

A valid dipole $$\mathbf d$$ must satisfy the following constraints, where $$\mathbf v = (d_{vx}, d_{vy}, d_{vz})$$, $$\mathbf m = (d_{mx}, d_{my}, d_{mz})$$, and $$\mathbf p = (d_{px}, d_{py}, d_{pz})$$ are treated as ordinary 3D vectors.

:$$\mathbf p \times \mathbf v - d_{pw}\mathbf m = \mathbf 0$$
:$$\mathbf p \cdot \mathbf m = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.

== Carrier and Cocarrier ==

The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]

:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .

The [[cocarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

:$$\operatorname{ccr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .

The carrier and cocarrier meet at the flat center of the dipole, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf d) \vee \operatorname{ccr}(\mathbf d) = (d_{vz} d_{my} - d_{vx} d_{pw} - d_{vy} d_{mz})\mathbf e_{15} + (d_{vx} d_{mz} - d_{vy} d_{pw} - d_{vz} d_{mx})\mathbf e_{25} + (d_{vy} d_{mx} - d_{vz} d_{pw} - d_{vx} d_{my})\mathbf e_{35} - (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{cen}(\mathbf d) = \operatorname{ccr}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{con}(\mathbf d) = \mathbf d \wedge \operatorname{car}(\mathbf d)^\unicode["segoe ui symbol"]{x2606} = (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\mathbf e_{4235} + (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\mathbf e_{4315} + (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\mathbf e_{4125} + (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .

A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf d = \operatorname{car}(\mathbf d) \vee \operatorname{con}(\mathbf d)$$ .

== Norms ==

The radius of a dipole $$\mathbf d$$ is given by

:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a dipole $$\mathbf d$$ is given by

:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \underline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .

The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\alpha) = \operatorname{cen}(\mathbf d) + \alpha \operatorname{att}(\mathbf d)$$ .

That is, $$\mathbf d \wedge \mathbf p(\alpha) = 0$$ for all real numbers $$\alpha$$. In particular, the two points on the surface of a dipole are given by the parameter $$\alpha_R = \pm \left\Vert\mathbf d\right\Vert_R$$ (the weighted radius). When $$\mathbf d$$ is a real dipole, this is precisely where the radius of $$\mathbf p(\alpha)$$ is zero. For smaller absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is real, and for larger absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is imaginary. When $$\mathbf d$$ is an imaginary dipole, the round point $$\mathbf p(\alpha)$$ is always imaginary, and it has an absolute radius at least as large as the dipole itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Circle]]
* [[Sphere]]

Sphere

2023-12-01T23:43:38Z

Eric Lengyel: /* Center and Container */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''sphere'' $$\mathbf s$$ is a quadrivector having the general form

:$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ .

If the $$s_u$$ component is zero, then the sphere contains the [[point at infinity]], and it is thus a flat [[plane]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a sphere can be formulated as

:$$\mathbf s = -\mathbf e_{1234} + p_x \mathbf e_{4235} + p_y \mathbf e_{4315} + p_z \mathbf e_{4125} - \dfrac{p^2 - r^2}{2} \mathbf e_{3215}$$ .

The various properties of a sphere are summarized in the following table.

[[Image:sphere.svg|512px]]

== Center and Container ==

The round [[center]] of a sphere $$\mathbf s$$ is the [[round point]] having the same center and radius as $$\mathbf s$$, and it is given by

:$$\operatorname{cen}(\mathbf s) = \operatorname{ccr}(\mathbf s) \vee \mathbf s = -s_xs_u \mathbf e_1 - s_ys_u \mathbf e_2 - s_zs_u \mathbf e_3 + s_u^2 \mathbf e_4 + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\mathbf e_5$$ .

The [[container]] of a sphere $$\mathbf s$$ is the sphere itself with a different weight:

:$$\operatorname{con}(\mathbf s) = \mathbf s \wedge \operatorname{car}(\mathbf s)^\unicode["segoe ui symbol"]{x2606} = s_u^2 \mathbf e_{1234} + s_xs_u \mathbf e_{4235} + s_ys_u \mathbf e_{4315} + s_zs_u \mathbf e_{4125} + s_ws_u \mathbf e_{3215}$$ .

== Norms ==

The radius of a sphere $$\mathbf s$$ is given by

:$$\operatorname{rad}(\mathbf s) = \dfrac{\left\Vert\mathbf s\right\Vert_R}{\left\Vert\mathbf s\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 - 2s_ws_u}{s_u^2}}$$ .

== Contained Points ==

The [[attitude]] of a sphere $$\mathbf s$$ is given by

:$$\operatorname{att}(\mathbf s) = \mathbf s \vee \underline{\mathbf e_4} = s_u \mathbf e_{321} + s_x \mathbf e_{235} + s_t \mathbf e_{315} + s_z \mathbf e_{125}$$ .

The set of round points contained by a sphere $$\mathbf s$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + \operatorname{att}(\mathbf s) \vee (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^\unicode["segoe ui symbol"]{x2605}$$ .

That is, $$\mathbf s \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all bivectors $$\boldsymbol \alpha$$. In particular, points on the surface of a sphere are given when $$\boldsymbol \alpha$$ has magnitude $$\left\Vert\mathbf s\right\Vert_R$$ (the weighted radius), and this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Circle]]

Round point

2023-12-01T23:43:19Z

Eric Lengyel: /* Center and Container */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''round point'' $$\mathbf a$$ is a vector having the general form

:$$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ .

Given a position $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a round point can be formulated as

:$$\mathbf a = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + \mathbf e_4 + \dfrac{p^2 + r^2}{2} \mathbf e_5$$ .

The various properties of a sphere are summarized in the following table.

[[Image:round.svg|512px]]

== Center and Container ==

The round [[center]] of a round point $$\mathbf a$$ is the round point itself with a different weight:

:$$\operatorname{cen}(\mathbf a) = \operatorname{ccr}(\mathbf a) \vee \mathbf a = a_xa_w \mathbf e_1 + a_ya_w \mathbf e_2 + a_za_w \mathbf e_3 + a_w^2 \mathbf e_4 + a_wa_u \mathbf e_5$$ .

The [[container]] of a round point $$\mathbf a$$ is the sphere having the same center and radius as $$\mathbf a$$, and it is given by

:$$\operatorname{con}(\mathbf a) = \mathbf a \wedge \operatorname{car}(\mathbf a)^\unicode["segoe ui symbol"]{x2606} = -a_w^2 \mathbf e_{1234} + a_xa_w \mathbf e_{4235} + a_ya_w \mathbf e_{4315} + a_za_w \mathbf e_{4125} + (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}$$ .

== Norms ==

The radius of a round point $$\mathbf a$$ is given by

:$$\operatorname{rad}(\mathbf a) = \dfrac{\left\Vert\mathbf a\right\Vert_R}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{2a_wa_u - a_x^2 - a_y^2 - a_z^2}{a_w^2}}$$ .

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Dipole]]
* [[Circle]]
* [[Sphere]]

Cocarrier

2023-12-01T23:42:33Z

Eric Lengyel: Redirected page to Carriers

#REDIRECT [[Carriers]]

Centers

2023-12-01T23:42:23Z

Eric Lengyel:

The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[cocarrier]]:

:$$\operatorname{cen}(\mathbf x) = \operatorname{ccr}(\mathbf x) \vee \mathbf x$$ .

The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

The following table lists the centers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Center
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf a) =
{\phantom +}\,&a_xa_w \mathbf e_1 \\
+\,&a_ya_w \mathbf e_2 \\
+\,&a_za_w \mathbf e_3 \\
+\,&a_w^2 \mathbf e_4 \\
+\,&a_wa_u \mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf d) =
{\phantom +}\,&(d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\,\mathbf e_1 \\
+\,&(d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\,\mathbf e_2 \\
+\,&(d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\,\mathbf e_3 \\
+\,&(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_4 \\
+\,&(d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\,\mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf c) =
{\phantom +}\,&(c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\,\mathbf e_1 \\
+\,&(c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\,\mathbf e_2 \\
+\,&(c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\,\mathbf e_3 \\
+\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\,\mathbf e_4 \\
+\,&(c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\,\mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf s) =
-\,&s_xs_u \mathbf e_1 \\
-\,&s_ys_u \mathbf e_2 \\
-\,&s_zs_u \mathbf e_3 \\
+\,&s_u^2 \mathbf e_4 \\
+\,&(s_x^2 + s_y^2 + s_z^2 - s_ws_u)\,\mathbf e_5\end{split}$$
|}

== See Also ==

* [[Containers]]
* [[Carriers]]
* [[Partners]]
* [[Attitude]]

Carriers

2023-12-01T23:40:44Z

Eric Lengyel:

== Carrier ==

The ''carrier'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the lowest dimensional flat object (a [[flat point]], [[line]], or [[plane]]) that contains it. The carrier of an object $$\mathbf x$$ is denoted by $$\operatorname{car}(\mathbf x)$$, and it is calculated by simply multiplying $$\mathbf x$$ by $$\mathbf e_5$$ with the [[wedge product]] to extract the round part of $$\mathbf x$$ as a flat geometry:

:$$\operatorname{car}(\mathbf x) = \mathbf x \wedge \mathbf e_5$$ .

The following table lists the carriers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Carrier
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\operatorname{car}(\mathbf a) = a_x \mathbf e_{15} + a_y \mathbf e_{25} + a_z \mathbf e_{35} + a_w \mathbf e_{45}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\operatorname{car}(\mathbf d) = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\operatorname{car}(\mathbf c) = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\operatorname{car}(\mathbf s) = s_u {\large\unicode{x1d7d9}}$$
|}

== Cocarrier ==

The ''cocarrier'' of a round object is the carrier of its [[antidual]]. The cocarrier of an object $$\mathbf x$$ is denoted by $$\operatorname{ccr}(\mathbf x)$$, and it is calculated by

:$$\operatorname{ccr}(\mathbf x) = \mathbf x^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5$$ .

The cocarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and cocarrier can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{car}(\mathbf x) \vee \operatorname{ccr}(\mathbf x)$$.

The following table lists the anticarriers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Cocarrier
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\operatorname{ccr}(\mathbf a) = a_w {\large\unicode{x1d7d9}}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\operatorname{ccr}(\mathbf d) = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\operatorname{ccr}(\mathbf c) = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\operatorname{ccr}(\mathbf s) = s_x \mathbf e_{15} + s_y \mathbf e_{25} + s_z \mathbf e_{35} - s_u \mathbf e_{45}$$
|}

== See Also ==

* [[Attitude]]
* [[Centers]]
* [[Containers]]
* [[Partners]]

File:Sphere.svg

2023-12-01T23:23:45Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:Sphere.svg

File:Circle.svg

2023-12-01T23:23:32Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:Circle.svg

File:Dipole.svg

2023-12-01T23:23:17Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:Dipole.svg

File:Round.svg

2023-12-01T23:22:58Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:Round.svg

Circle

2023-11-17T07:24:46Z

Eric Lengyel: /* Contained Points */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form

:$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .

If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the [[point at infinity]], and it is thus a straight [[line]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

The various properties of a circle are summarized in the following table.

[[Image:circle.svg|800px]]

== Constraints ==

A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.

:$$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
:$$\mathbf g \cdot \mathbf v = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

== Carrier and Anticarrier ==

The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]

:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

The [[anticarrier]] of a circle $$\mathbf c$$ is the [[line]]

:$$\operatorname{acr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

The carrier and anticarrier meet at the flat center of the circle, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf c) \vee \operatorname{acr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

:$$\operatorname{cen}(\mathbf c) = \operatorname{acr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

:$$\operatorname{con}(\mathbf c) = \mathbf c \wedge \operatorname{car}(\mathbf c)^\unicode["segoe ui symbol"]{x2606} = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .

== Norms ==

The radius of a circle $$\mathbf c$$ is given by

:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a circle $$\mathbf c$$ is given by

:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \underline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{x2606}$$ .

That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Sphere]]

Circle

2023-11-17T07:23:52Z

Eric Lengyel: /* Center and Container */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form

:$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .

If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the [[point at infinity]], and it is thus a straight [[line]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

The various properties of a circle are summarized in the following table.

[[Image:circle.svg|800px]]

== Constraints ==

A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.

:$$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
:$$\mathbf g \cdot \mathbf v = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

== Carrier and Anticarrier ==

The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]

:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

The [[anticarrier]] of a circle $$\mathbf c$$ is the [[line]]

:$$\operatorname{acr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

The carrier and anticarrier meet at the flat center of the circle, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf c) \vee \operatorname{acr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

:$$\operatorname{cen}(\mathbf c) = \operatorname{acr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

:$$\operatorname{con}(\mathbf c) = \mathbf c \wedge \operatorname{car}(\mathbf c)^\unicode["segoe ui symbol"]{x2606} = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .

== Norms ==

The radius of a circle $$\mathbf c$$ is given by

:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a circle $$\mathbf c$$ is given by

:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \underline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^* \vee \operatorname{att}(\mathbf c)$$ .

That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Sphere]]

Dipole

2023-11-17T07:22:14Z

Eric Lengyel: /* Carrier and Anticarrier */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form

:$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ .

If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ components are all zero, then the dipole contains the [[point at infinity]], and it is thus a [[flat point]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as

:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .

The various properties of a dipole are summarized in the following table.

[[Image:dipole.svg|800px]]

== Constraints ==

A valid dipole $$\mathbf d$$ must satisfy the following constraints, where $$\mathbf v = (d_{vx}, d_{vy}, d_{vz})$$, $$\mathbf m = (d_{mx}, d_{my}, d_{mz})$$, and $$\mathbf p = (d_{px}, d_{py}, d_{pz})$$ are treated as ordinary 3D vectors.

:$$\mathbf p \times \mathbf v - d_{pw}\mathbf m = \mathbf 0$$
:$$\mathbf p \cdot \mathbf m = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.

== Carrier and Anticarrier ==

The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]

:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .

The [[anticarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

:$$\operatorname{acr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .

The carrier and anticarrier meet at the flat center of the dipole, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf d) \vee \operatorname{acr}(\mathbf d) = (d_{vz} d_{my} - d_{vx} d_{pw} - d_{vy} d_{mz})\mathbf e_{15} + (d_{vx} d_{mz} - d_{vy} d_{pw} - d_{vz} d_{mx})\mathbf e_{25} + (d_{vy} d_{mx} - d_{vz} d_{pw} - d_{vx} d_{my})\mathbf e_{35} - (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{cen}(\mathbf d) = \operatorname{acr}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{con}(\mathbf d) = \mathbf d \wedge \operatorname{car}(\mathbf d)^\unicode["segoe ui symbol"]{x2606} = (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\mathbf e_{4235} + (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\mathbf e_{4315} + (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\mathbf e_{4125} + (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .

A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf d = \operatorname{car}(\mathbf d) \vee \operatorname{con}(\mathbf d)$$ .

== Norms ==

The radius of a dipole $$\mathbf d$$ is given by

:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a dipole $$\mathbf d$$ is given by

:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \underline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .

The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\alpha) = \operatorname{cen}(\mathbf d) + \alpha \operatorname{att}(\mathbf d)$$ .

That is, $$\mathbf d \wedge \mathbf p(\alpha) = 0$$ for all real numbers $$\alpha$$. In particular, the two points on the surface of a dipole are given by the parameter $$\alpha_R = \pm \left\Vert\mathbf d\right\Vert_R$$ (the weighted radius). When $$\mathbf d$$ is a real dipole, this is precisely where the radius of $$\mathbf p(\alpha)$$ is zero. For smaller absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is real, and for larger absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is imaginary. When $$\mathbf d$$ is an imaginary dipole, the round point $$\mathbf p(\alpha)$$ is always imaginary, and it has an absolute radius at least as large as the dipole itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Circle]]
* [[Sphere]]

Circle

2023-11-17T07:20:27Z

Eric Lengyel: /* Carrier and Anticarrier */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form

:$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .

If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the [[point at infinity]], and it is thus a straight [[line]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

The various properties of a circle are summarized in the following table.

[[Image:circle.svg|800px]]

== Constraints ==

A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.

:$$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
:$$\mathbf g \cdot \mathbf v = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

== Carrier and Anticarrier ==

The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]

:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

The [[anticarrier]] of a circle $$\mathbf c$$ is the [[line]]

:$$\operatorname{acr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

The carrier and anticarrier meet at the flat center of the circle, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf c) \vee \operatorname{acr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

:$$\operatorname{cen}(\mathbf c) = -\operatorname{car}(\mathbf c^*) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

:$$\operatorname{con}(\mathbf c) = \operatorname{car}(\mathbf c)^* \wedge \mathbf c = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .

== Norms ==

The radius of a circle $$\mathbf c$$ is given by

:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a circle $$\mathbf c$$ is given by

:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \underline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^* \vee \operatorname{att}(\mathbf c)$$ .

That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Sphere]]

Dipole

2023-11-17T07:16:25Z

Eric Lengyel: /* Center and Container */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form

:$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ .

If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ components are all zero, then the dipole contains the [[point at infinity]], and it is thus a [[flat point]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as

:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .

The various properties of a dipole are summarized in the following table.

[[Image:dipole.svg|800px]]

== Constraints ==

A valid dipole $$\mathbf d$$ must satisfy the following constraints, where $$\mathbf v = (d_{vx}, d_{vy}, d_{vz})$$, $$\mathbf m = (d_{mx}, d_{my}, d_{mz})$$, and $$\mathbf p = (d_{px}, d_{py}, d_{pz})$$ are treated as ordinary 3D vectors.

:$$\mathbf p \times \mathbf v - d_{pw}\mathbf m = \mathbf 0$$
:$$\mathbf p \cdot \mathbf m = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.

== Carrier and Anticarrier ==

The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]

:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .

The [[anticarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

:$$\operatorname{acr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .

The carrier and anticarrier meet at the flat center of the dipole, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf d) \vee \operatorname{acr}(\mathbf d) = (d_{vx} d_{pw} - d_{vz} d_{my} + d_{vy} d_{mz})\mathbf e_{15} + (d_{vy} d_{pw} - d_{vx} d_{mz} + d_{vz} d_{mx})\mathbf e_{25} + (d_{vz} d_{pw} - d_{vy} d_{mx} + d_{vx} d_{my})\mathbf e_{35} + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{cen}(\mathbf d) = \operatorname{acr}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{con}(\mathbf d) = \mathbf d \wedge \operatorname{car}(\mathbf d)^\unicode["segoe ui symbol"]{x2606} = (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\mathbf e_{4235} + (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\mathbf e_{4315} + (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\mathbf e_{4125} + (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .

A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf d = \operatorname{car}(\mathbf d) \vee \operatorname{con}(\mathbf d)$$ .

== Norms ==

The radius of a dipole $$\mathbf d$$ is given by

:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a dipole $$\mathbf d$$ is given by

:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \underline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .

The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\alpha) = \operatorname{cen}(\mathbf d) + \alpha \operatorname{att}(\mathbf d)$$ .

That is, $$\mathbf d \wedge \mathbf p(\alpha) = 0$$ for all real numbers $$\alpha$$. In particular, the two points on the surface of a dipole are given by the parameter $$\alpha_R = \pm \left\Vert\mathbf d\right\Vert_R$$ (the weighted radius). When $$\mathbf d$$ is a real dipole, this is precisely where the radius of $$\mathbf p(\alpha)$$ is zero. For smaller absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is real, and for larger absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is imaginary. When $$\mathbf d$$ is an imaginary dipole, the round point $$\mathbf p(\alpha)$$ is always imaginary, and it has an absolute radius at least as large as the dipole itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Circle]]
* [[Sphere]]

Dipole

2023-11-17T07:13:43Z

Eric Lengyel: /* Carrier and Anticarrier */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form

:$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ .

If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ components are all zero, then the dipole contains the [[point at infinity]], and it is thus a [[flat point]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as

:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .

The various properties of a dipole are summarized in the following table.

[[Image:dipole.svg|800px]]

== Constraints ==

A valid dipole $$\mathbf d$$ must satisfy the following constraints, where $$\mathbf v = (d_{vx}, d_{vy}, d_{vz})$$, $$\mathbf m = (d_{mx}, d_{my}, d_{mz})$$, and $$\mathbf p = (d_{px}, d_{py}, d_{pz})$$ are treated as ordinary 3D vectors.

:$$\mathbf p \times \mathbf v - d_{pw}\mathbf m = \mathbf 0$$
:$$\mathbf p \cdot \mathbf m = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.

== Carrier and Anticarrier ==

The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]

:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .

The [[anticarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

:$$\operatorname{acr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .

The carrier and anticarrier meet at the flat center of the dipole, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf d) \vee \operatorname{acr}(\mathbf d) = (d_{vx} d_{pw} - d_{vz} d_{my} + d_{vy} d_{mz})\mathbf e_{15} + (d_{vy} d_{pw} - d_{vx} d_{mz} + d_{vz} d_{mx})\mathbf e_{25} + (d_{vz} d_{pw} - d_{vy} d_{mx} + d_{vx} d_{my})\mathbf e_{35} + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{cen}(\mathbf d) = -\operatorname{car}(\mathbf d^*) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{con}(\mathbf d) = \operatorname{car}(\mathbf d)^* \wedge \mathbf d = -(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_{4235} + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_{4315} + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_{4125} - (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .

A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf d = \operatorname{car}(\mathbf d) \vee \operatorname{con}(\mathbf d)$$ .

== Norms ==

The radius of a dipole $$\mathbf d$$ is given by

:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a dipole $$\mathbf d$$ is given by

:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \underline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .

The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\alpha) = \operatorname{cen}(\mathbf d) + \alpha \operatorname{att}(\mathbf d)$$ .

That is, $$\mathbf d \wedge \mathbf p(\alpha) = 0$$ for all real numbers $$\alpha$$. In particular, the two points on the surface of a dipole are given by the parameter $$\alpha_R = \pm \left\Vert\mathbf d\right\Vert_R$$ (the weighted radius). When $$\mathbf d$$ is a real dipole, this is precisely where the radius of $$\mathbf p(\alpha)$$ is zero. For smaller absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is real, and for larger absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is imaginary. When $$\mathbf d$$ is an imaginary dipole, the round point $$\mathbf p(\alpha)$$ is always imaginary, and it has an absolute radius at least as large as the dipole itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Circle]]
* [[Sphere]]

Sphere

2023-11-17T07:10:00Z

Eric Lengyel: /* Contained Points */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''sphere'' $$\mathbf s$$ is a quadrivector having the general form

:$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ .

If the $$s_u$$ component is zero, then the sphere contains the [[point at infinity]], and it is thus a flat [[plane]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a sphere can be formulated as

:$$\mathbf s = -\mathbf e_{1234} + p_x \mathbf e_{4235} + p_y \mathbf e_{4315} + p_z \mathbf e_{4125} - \dfrac{p^2 - r^2}{2} \mathbf e_{3215}$$ .

The various properties of a sphere are summarized in the following table.

[[Image:sphere.svg|512px]]

== Center and Container ==

The round [[center]] of a sphere $$\mathbf s$$ is the [[round point]] having the same center and radius as $$\mathbf s$$, and it is given by

:$$\operatorname{cen}(\mathbf s) = \operatorname{acr}(\mathbf s) \vee \mathbf s = -s_xs_u \mathbf e_1 - s_ys_u \mathbf e_2 - s_zs_u \mathbf e_3 + s_u^2 \mathbf e_4 + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\mathbf e_5$$ .

The [[container]] of a sphere $$\mathbf s$$ is the sphere itself with a different weight:

:$$\operatorname{con}(\mathbf s) = \mathbf s \wedge \operatorname{car}(\mathbf s)^\unicode["segoe ui symbol"]{x2606} = s_u^2 \mathbf e_{1234} + s_xs_u \mathbf e_{4235} + s_ys_u \mathbf e_{4315} + s_zs_u \mathbf e_{4125} + s_ws_u \mathbf e_{3215}$$ .

== Norms ==

The radius of a sphere $$\mathbf s$$ is given by

:$$\operatorname{rad}(\mathbf s) = \dfrac{\left\Vert\mathbf s\right\Vert_R}{\left\Vert\mathbf s\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 - 2s_ws_u}{s_u^2}}$$ .

== Contained Points ==

The [[attitude]] of a sphere $$\mathbf s$$ is given by

:$$\operatorname{att}(\mathbf s) = \mathbf s \vee \underline{\mathbf e_4} = s_u \mathbf e_{321} + s_x \mathbf e_{235} + s_t \mathbf e_{315} + s_z \mathbf e_{125}$$ .

The set of round points contained by a sphere $$\mathbf s$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + \operatorname{att}(\mathbf s) \vee (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^\unicode["segoe ui symbol"]{x2605}$$ .

That is, $$\mathbf s \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all bivectors $$\boldsymbol \alpha$$. In particular, points on the surface of a sphere are given when $$\boldsymbol \alpha$$ has magnitude $$\left\Vert\mathbf s\right\Vert_R$$ (the weighted radius), and this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Circle]]

Sphere

2023-11-17T07:06:54Z

Eric Lengyel: /* Center and Container */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''sphere'' $$\mathbf s$$ is a quadrivector having the general form

:$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ .

If the $$s_u$$ component is zero, then the sphere contains the [[point at infinity]], and it is thus a flat [[plane]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a sphere can be formulated as

:$$\mathbf s = -\mathbf e_{1234} + p_x \mathbf e_{4235} + p_y \mathbf e_{4315} + p_z \mathbf e_{4125} - \dfrac{p^2 - r^2}{2} \mathbf e_{3215}$$ .

The various properties of a sphere are summarized in the following table.

[[Image:sphere.svg|512px]]

== Center and Container ==

The round [[center]] of a sphere $$\mathbf s$$ is the [[round point]] having the same center and radius as $$\mathbf s$$, and it is given by

:$$\operatorname{cen}(\mathbf s) = \operatorname{acr}(\mathbf s) \vee \mathbf s = -s_xs_u \mathbf e_1 - s_ys_u \mathbf e_2 - s_zs_u \mathbf e_3 + s_u^2 \mathbf e_4 + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\mathbf e_5$$ .

The [[container]] of a sphere $$\mathbf s$$ is the sphere itself with a different weight:

:$$\operatorname{con}(\mathbf s) = \mathbf s \wedge \operatorname{car}(\mathbf s)^\unicode["segoe ui symbol"]{x2606} = s_u^2 \mathbf e_{1234} + s_xs_u \mathbf e_{4235} + s_ys_u \mathbf e_{4315} + s_zs_u \mathbf e_{4125} + s_ws_u \mathbf e_{3215}$$ .

== Norms ==

The radius of a sphere $$\mathbf s$$ is given by

:$$\operatorname{rad}(\mathbf s) = \dfrac{\left\Vert\mathbf s\right\Vert_R}{\left\Vert\mathbf s\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 - 2s_ws_u}{s_u^2}}$$ .

== Contained Points ==

The [[attitude]] of a sphere $$\mathbf s$$ is given by

:$$\operatorname{att}(\mathbf s) = \mathbf s \vee \underline{\mathbf e_4} = s_x \mathbf e_{235} + s_t \mathbf e_{315} + s_z \mathbf e_{125} + s_u \mathbf e_{321}$$ .

The set of round points contained by a sphere $$\mathbf s$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^* \vee \operatorname{att}(\mathbf s)$$ .

That is, $$\mathbf s \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all bivectors $$\boldsymbol \alpha$$. In particular, points on the surface of a sphere are given when $$\boldsymbol \alpha$$ has magnitude $$\left\Vert\mathbf s\right\Vert_R$$ (the weighted radius), and this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Circle]]

Round point

2023-11-17T07:05:02Z

Eric Lengyel: /* Center and Container */

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''round point'' $$\mathbf a$$ is a vector having the general form

:$$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ .

Given a position $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a round point can be formulated as

:$$\mathbf a = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + \mathbf e_4 + \dfrac{p^2 + r^2}{2} \mathbf e_5$$ .

The various properties of a sphere are summarized in the following table.

[[Image:round.svg|512px]]

== Center and Container ==

The round [[center]] of a round point $$\mathbf a$$ is the round point itself with a different weight:

:$$\operatorname{cen}(\mathbf a) = \operatorname{acr}(\mathbf a) \vee \mathbf a = a_xa_w \mathbf e_1 + a_ya_w \mathbf e_2 + a_za_w \mathbf e_3 + a_w^2 \mathbf e_4 + a_wa_u \mathbf e_5$$ .

The [[container]] of a round point $$\mathbf a$$ is the sphere having the same center and radius as $$\mathbf a$$, and it is given by

:$$\operatorname{con}(\mathbf a) = \mathbf a \wedge \operatorname{car}(\mathbf a)^\unicode["segoe ui symbol"]{x2606} = -a_w^2 \mathbf e_{1234} + a_xa_w \mathbf e_{4235} + a_ya_w \mathbf e_{4315} + a_za_w \mathbf e_{4125} + (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}$$ .

== Norms ==

The radius of a round point $$\mathbf a$$ is given by

:$$\operatorname{rad}(\mathbf a) = \dfrac{\left\Vert\mathbf a\right\Vert_R}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{2a_wa_u - a_x^2 - a_y^2 - a_z^2}{a_w^2}}$$ .

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Dipole]]
* [[Circle]]
* [[Sphere]]

File:Plane.svg

2023-11-17T06:59:36Z

Eric Lengyel: Eric Lengyel uploaded a new version of File:Plane.svg