User contributions for Eric Lengyel
Jump to navigation
Jump to search
21 April 2024
- 20:4620:46, 21 April 2024 diff hist 0 Circle →Contained Points current
- 20:4520:45, 21 April 2024 diff hist −1 Sphere →Contained Points current
- 20:4520:45, 21 April 2024 diff hist −1 Circle →Contained Points
- 20:4520:45, 21 April 2024 diff hist −1 Dipole →Contained Points current
13 April 2024
- 02:0102:01, 13 April 2024 diff hist +839 N Metrics 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..." current
- 02:0002:00, 13 April 2024 diff hist 0 N File:Metric-cga-3d.svg No edit summary current
- 01:5701:57, 13 April 2024 diff hist +14 Main Page →Various properties and unary operations current
3 April 2024
- 23:0723:07, 3 April 2024 diff hist 0 File:AntiwedgeProduct.svg Eric Lengyel uploaded a new version of File:AntiwedgeProduct.svg current
- 23:0623:06, 3 April 2024 diff hist 0 File:WedgeProduct.svg Eric Lengyel uploaded a new version of File:WedgeProduct.svg current
- 23:0623:06, 3 April 2024 diff hist −152 Geometric products No edit summary current
- 23:0523:05, 3 April 2024 diff hist 0 File:GeometricAntiproduct.svg Eric Lengyel uploaded a new version of File:GeometricAntiproduct.svg current
- 23:0423:04, 3 April 2024 diff hist 0 File:GeometricProduct.svg Eric Lengyel uploaded a new version of File:GeometricProduct.svg current
- 23:0223:02, 3 April 2024 diff hist −27 Attitude No edit summary current
- 22:5922:59, 3 April 2024 diff hist 0 Containers No edit summary current
- 22:5822:58, 3 April 2024 diff hist 0 Centers No edit summary current
- 22:5722:57, 3 April 2024 diff hist 0 Carriers No edit summary current
- 22:5222:52, 3 April 2024 diff hist +70 Partners No edit summary current
31 December 2023
- 22:2922:29, 31 December 2023 diff hist +31 N Exterior product Redirected page to Exterior products current Tag: New redirect
1 December 2023
- 23:4823:48, 1 December 2023 diff hist +22 N Cocarriers Redirected page to Carriers current Tag: New redirect
- 23:4523:45, 1 December 2023 diff hist −6 Circle No edit summary
- 23:4523:45, 1 December 2023 diff hist −6 Dipole No edit summary
- 23:4323:43, 1 December 2023 diff hist 0 Sphere →Center and Container
- 23:4323:43, 1 December 2023 diff hist 0 Round point →Center and Container current
- 23:4223:42, 1 December 2023 diff hist +22 N Cocarrier Redirected page to Carriers current Tag: New redirect
- 23:4223:42, 1 December 2023 diff hist −2 Centers No edit summary
- 23:4023:40, 1 December 2023 diff hist −12 Carriers No edit summary
- 23:2323:23, 1 December 2023 diff hist 0 File:Sphere.svg Eric Lengyel uploaded a new version of File:Sphere.svg current
- 23:2323:23, 1 December 2023 diff hist 0 File:Circle.svg Eric Lengyel uploaded a new version of File:Circle.svg current
- 23:2323:23, 1 December 2023 diff hist 0 File:Dipole.svg Eric Lengyel uploaded a new version of File:Dipole.svg current
- 23:2223:22, 1 December 2023 diff hist 0 File:Round.svg Eric Lengyel uploaded a new version of File:Round.svg current
17 November 2023
- 07:2407:24, 17 November 2023 diff hist +33 Circle →Contained Points
- 07:2307:23, 17 November 2023 diff hist +30 Circle →Center and Container
- 07:2207:22, 17 November 2023 diff hist 0 Dipole →Carrier and Anticarrier
- 07:2007:20, 17 November 2023 diff hist +34 Circle →Carrier and Anticarrier
- 07:1607:16, 17 November 2023 diff hist +29 Dipole →Center and Container
- 07:1307:13, 17 November 2023 diff hist +32 Dipole →Carrier and Anticarrier
- 07:1007:10, 17 November 2023 diff hist +33 Sphere →Contained Points
- 07:0607:06, 17 November 2023 diff hist +29 Sphere →Center and Container
- 07:0507:05, 17 November 2023 diff hist +30 Round point →Center and Container
- 06:5906:59, 17 November 2023 diff hist 0 File:Plane.svg Eric Lengyel uploaded a new version of File:Plane.svg current
- 06:5906:59, 17 November 2023 diff hist 0 File:Line.svg Eric Lengyel uploaded a new version of File:Line.svg current
- 06:5906:59, 17 November 2023 diff hist 0 File:Point.svg Eric Lengyel uploaded a new version of File:Point.svg current
- 06:5606:56, 17 November 2023 diff hist −35 Centers No edit summary
- 06:5406:54, 17 November 2023 diff hist −155 Containers No edit summary
- 06:4906:49, 17 November 2023 diff hist +4 Carriers →Anticarrier
- 06:4706:47, 17 November 2023 diff hist 0 File:Sphere.svg Eric Lengyel uploaded a new version of File:Sphere.svg
- 06:4606:46, 17 November 2023 diff hist 0 File:Circle.svg Eric Lengyel uploaded a new version of File:Circle.svg
- 06:4606:46, 17 November 2023 diff hist 0 File:Dipole.svg Eric Lengyel uploaded a new version of File:Dipole.svg
- 06:4606:46, 17 November 2023 diff hist 0 File:Round.svg Eric Lengyel uploaded a new version of File:Round.svg
24 October 2023
- 06:5006:50, 24 October 2023 diff hist 0 Expansion No edit summary current
- 06:4606:46, 24 October 2023 diff hist 0 Expansion No edit summary
- 06:4006:40, 24 October 2023 diff hist 0 Expansion No edit summary
- 06:1906:19, 24 October 2023 diff hist 0 Expansion No edit summary
23 October 2023
- 07:1607:16, 23 October 2023 diff hist −4 Duals No edit summary current
- 03:2803:28, 23 October 2023 diff hist +464 Duals No edit summary
- 03:2303:23, 23 October 2023 diff hist +19 N Antidual Redirected page to Duals current Tag: New redirect
- 03:2103:21, 23 October 2023 diff hist −23 Duals No edit summary
- 03:1803:18, 23 October 2023 diff hist +2 Main Page →Products and other binary operations
- 03:1803:18, 23 October 2023 diff hist +27 N Join Redirected page to Join and meet current Tag: New redirect
- 03:1803:18, 23 October 2023 diff hist 0 m Expansion Eric Lengyel moved page Connect to Expansion without leaving a redirect
- 03:1703:17, 23 October 2023 diff hist +2 Join and meet →See Also current
- 03:1703:17, 23 October 2023 diff hist +27 N Meet Redirected page to Join and meet current Tag: New redirect
- 03:1703:17, 23 October 2023 diff hist +37 Projections No edit summary current
- 03:1403:14, 23 October 2023 diff hist +11 Expansion No edit summary
- 03:1203:12, 23 October 2023 diff hist −4 Expansion No edit summary
- 03:0803:08, 23 October 2023 diff hist 0 Expansion No edit summary
- 02:4602:46, 23 October 2023 diff hist −483 Expansion No edit summary
- 02:4102:41, 23 October 2023 diff hist 0 Expansion No edit summary
- 02:3802:38, 23 October 2023 diff hist 0 File:Point connect round.svg Eric Lengyel uploaded a new version of File:Point connect round.svg current
- 02:3702:37, 23 October 2023 diff hist 0 File:Dipole connect round.svg Eric Lengyel uploaded a new version of File:Dipole connect round.svg current
- 02:3702:37, 23 October 2023 diff hist 0 File:Line connect dipole.svg Eric Lengyel uploaded a new version of File:Line connect dipole.svg current
- 02:3702:37, 23 October 2023 diff hist 0 File:Circle connect dipole.svg Eric Lengyel uploaded a new version of File:Circle connect dipole.svg current
- 02:3602:36, 23 October 2023 diff hist 0 File:Line connect point.svg Eric Lengyel uploaded a new version of File:Line connect point.svg current
- 02:3602:36, 23 October 2023 diff hist 0 File:Circle connect point.svg Eric Lengyel uploaded a new version of File:Circle connect point.svg current
- 02:3502:35, 23 October 2023 diff hist 0 File:Line connect round.svg Eric Lengyel uploaded a new version of File:Line connect round.svg current
- 02:3502:35, 23 October 2023 diff hist 0 File:Circle connect round.svg Eric Lengyel uploaded a new version of File:Circle connect round.svg current
- 02:3502:35, 23 October 2023 diff hist 0 File:Plane connect line.svg Eric Lengyel uploaded a new version of File:Plane connect line.svg current
- 02:3502:35, 23 October 2023 diff hist 0 File:Sphere connect line.svg Eric Lengyel uploaded a new version of File:Sphere connect line.svg current
- 02:3402:34, 23 October 2023 diff hist 0 File:Plane connect circle.svg Eric Lengyel uploaded a new version of File:Plane connect circle.svg current
- 02:3402:34, 23 October 2023 diff hist 0 File:Sphere connect circle.svg Eric Lengyel uploaded a new version of File:Sphere connect circle.svg current
- 02:3402:34, 23 October 2023 diff hist 0 File:Plane connect point.svg Eric Lengyel uploaded a new version of File:Plane connect point.svg current
- 02:3302:33, 23 October 2023 diff hist 0 File:Sphere connect point.svg Eric Lengyel uploaded a new version of File:Sphere connect point.svg current
- 02:3302:33, 23 October 2023 diff hist 0 File:Plane connect dipole.svg Eric Lengyel uploaded a new version of File:Plane connect dipole.svg current
- 02:3302:33, 23 October 2023 diff hist 0 File:Sphere connect dipole.svg Eric Lengyel uploaded a new version of File:Sphere connect dipole.svg current
- 02:3302:33, 23 October 2023 diff hist 0 File:Plane connect round.svg Eric Lengyel uploaded a new version of File:Plane connect round.svg current
- 02:3102:31, 23 October 2023 diff hist 0 File:Sphere connect round.svg Eric Lengyel uploaded a new version of File:Sphere connect round.svg current
27 August 2023
- 01:0101:01, 27 August 2023 diff hist +33 Partners No edit summary
- 01:0001:00, 27 August 2023 diff hist +33 Containers No edit summary
- 01:0001:00, 27 August 2023 diff hist +33 Centers No edit summary
- 01:0001:00, 27 August 2023 diff hist +33 Carriers →Anticarrier
- 00:5900:59, 27 August 2023 diff hist +297 Duals No edit summary
- 00:5800:58, 27 August 2023 diff hist +594 Expansion No edit summary
25 August 2023
- 22:5422:54, 25 August 2023 diff hist 0 N File:AntiwedgeProduct.svg No edit summary
- 22:5422:54, 25 August 2023 diff hist 0 N File:WedgeProduct.svg No edit summary
- 22:5422:54, 25 August 2023 diff hist +38 Exterior products No edit summary current
- 22:5022:50, 25 August 2023 diff hist −2 Expansion No edit summary
- 22:4522:45, 25 August 2023 diff hist −4 Expansion No edit summary
- 22:4222:42, 25 August 2023 diff hist +2 Expansion No edit summary
- 22:3822:38, 25 August 2023 diff hist +2 Expansion No edit summary
- 22:3422:34, 25 August 2023 diff hist 0 Expansion No edit summary
- 22:3022:30, 25 August 2023 diff hist −2 Expansion No edit summary
- 22:2222:22, 25 August 2023 diff hist +5 Expansion No edit summary
- 22:1122:11, 25 August 2023 diff hist +4 Expansion No edit summary
- 21:5521:55, 25 August 2023 diff hist −25 Partners No edit summary
- 21:4721:47, 25 August 2023 diff hist +155 Containers No edit summary
- 21:3521:35, 25 August 2023 diff hist −131 Centers No edit summary
- 21:3121:31, 25 August 2023 diff hist −2 Carriers →Anticarrier
- 21:0521:05, 25 August 2023 diff hist +85 Duals No edit summary
6 August 2023
- 07:1207:12, 6 August 2023 diff hist +4 Carriers →Anticarrier
- 03:5803:58, 6 August 2023 diff hist +20 N Spheres Redirected page to Sphere current Tag: New redirect
- 03:5803:58, 6 August 2023 diff hist +20 N Circles Redirected page to Circle current Tag: New redirect
- 03:5803:58, 6 August 2023 diff hist +20 N Dipoles Redirected page to Dipole current Tag: New redirect
- 03:5703:57, 6 August 2023 diff hist +19 N Planes Redirected page to Plane current Tag: New redirect
- 03:5703:57, 6 August 2023 diff hist +18 N Lines Redirected page to Line current Tag: New redirect
- 03:5703:57, 6 August 2023 diff hist +24 N Flat points Redirected page to Flat point current Tag: New redirect
- 03:5703:57, 6 August 2023 diff hist +25 N Round points Redirected page to Round point current Tag: New redirect
- 03:2603:26, 6 August 2023 diff hist +31 N Exterior antiproduct Redirected page to Exterior products current Tag: New redirect
- 03:2503:25, 6 August 2023 diff hist +32 N Geometric antiproduct Redirected page to Geometric products current Tag: New redirect
- 03:2503:25, 6 August 2023 diff hist +32 N Geometric product Redirected page to Geometric products current Tag: New redirect
- 03:2503:25, 6 August 2023 diff hist +31 N Antiwedge product Redirected page to Exterior products current Tag: New redirect
- 03:2503:25, 6 August 2023 diff hist +31 N Wedge product Redirected page to Exterior products current Tag: New redirect
- 03:2503:25, 6 August 2023 diff hist +25 N Projection Redirected page to Projections current Tag: New redirect
- 03:2403:24, 6 August 2023 diff hist +19 N Dual Redirected page to Duals current Tag: New redirect
- 03:2403:24, 6 August 2023 diff hist +22 N Carrier Redirected page to Carriers current Tag: New redirect
- 03:2303:23, 6 August 2023 diff hist +22 N Partner Redirected page to Partners current Tag: New redirect
- 03:2303:23, 6 August 2023 diff hist +24 N Container Redirected page to Containers current Tag: New redirect
- 03:2003:20, 6 August 2023 diff hist +688 N Transversion Created page with "__NOTOC__ A ''transversion'' is a reciprocal transformation performed by the operator :$$\mathfrak T = {\dfrac{\tau_{x\vphantom{y}}}{2} \mathbf e_{423} + \dfrac{\tau_y}{2} \mathbf e_{431} + \dfrac{\tau_z{\vphantom{y}}}{2} \mathbf e_{412} + \large\unicode{x1d7d9}}$$ . == Matrix Form == When a transversion $$\mathfrak T$$ is applied to a round point, it is equivalent to premultiplying the point by the $$5 \times 5$$ matrix :$$\begin{bmatrix} 1 & 0 & 0 & 0 & -\tau_x..." current
- 03:2003:20, 6 August 2023 diff hist +2,238 N Dilation Created page with "__NOTOC__ A ''dilation'' is a conformal transformation of Euclidean space performed by the operator :$$\mathbf D = \dfrac{1 - \sigma}{2} (c_x \mathbf e_{235} + c_y \mathbf e_{315} + c_z \mathbf e_{125} - \mathbf e_{321}) + \dfrac{1 + \sigma}{2} {\large\unicode{x1d7d9}}$$ . This operator scales an object $$\mathbf x$$ by the factor $$\sigma$$ about the center point $$\mathbf c = (c_x, c_y, c_z)$$ when used with the sandwich antiproduct $$\mathbf D \mathbin{\unicode{x27C..." current
- 03:1903:19, 6 August 2023 diff hist +764 N Rotation Created page with "__NOTOC__ A ''rotation'' is a proper isometry of Euclidean space performed by the operator :$$\mathbf R = \boldsymbol l\sin\dfrac{\phi}{2} + {\large\unicode{x1d7d9}}\cos\dfrac{\phi}{2}$$ , where $$\boldsymbol l$$ is a unitized line corresponding to the axis of rotation. This operator is identical to the [http://rigidgeometricalgebra.org/wiki/index.php?title=Rotation rotation operator in rigid geometric algebra] but with the extra factor of $$\mathbf e_5$$. It rotat..." current
- 03:1903:19, 6 August 2023 diff hist +1,718 N Translation Created page with "__NOTOC__ A ''translation'' is a proper isometry of Euclidean space performed by the operator :$$\mathbf T = {\dfrac{\tau_{x\vphantom{y}}}{2} \mathbf e_{235} + \dfrac{\tau_y}{2} \mathbf e_{315} + \dfrac{\tau_z{\vphantom{y}}}{2} \mathbf e_{125} + \large\unicode{x1d7d9}}$$ . This operator is identical to the [http://rigidgeometricalgebra.org/wiki/index.php?title=Translation translation operator in rigid geometric algebra] but with the extra factor of $$\mathbf e_5$$. It..." current
- 03:1903:19, 6 August 2023 diff hist +563 N Projections Created page with "Any geometric object $$\mathbf x$$ can be projected onto another geometric object $$\mathbf y$$ of higher grade by first calculating the connect of $$\mathbf x$$ with $$\mathbf y$$ and then using the meet operation to intersect the result with $$\mathbf y$$. That is, the projection of $$\mathbf x$$ onto $$\mathbf y$$ is given by :$$(\mathbf y^* \wedge \mathbf x) \vee \mathbf y$$ . This formula is general and works for flat points, lines, planes, r..."
- 03:1903:19, 6 August 2023 diff hist +12,873 N Expansion Created page with "The ''connect'' operation is performed by taking the wedge product between the dual of an object ''A'' and another object ''B'' with lower grade. The result is an object ''C'' that is orthogonal to ''A'' and contains ''B'', allowing a projection of ''B'' onto ''A'' through a simple intersection of ''A'' and ''C''. The flat points, lines, planes, round points, dipoles, circles, and spheres appearing in the following tables are defined..."
- 03:1803:18, 6 August 2023 diff hist +14,963 N Join and meet Created page with "The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection. The flat points, lines, planes, round points, dipoles, circles, and spheres appearing in the following tables are defined as follows: :$$\mathbf p = p_x \mathbf e_{15} +..."
- 03:1803:18, 6 August 2023 diff hist −1,292 Wedge products Redirected page to Exterior products current Tag: New redirect
- 03:1803:18, 6 August 2023 diff hist +1,323 N Exterior products Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. == Exterior Product == The following Cayley table shows the exterior products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. 1440px == Exterior Anti..."
- 03:1703:17, 6 August 2023 diff hist +1,323 N Wedge products Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. == Exterior Product == The following Cayley table shows the exterior products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. 1440px == Exterior Anti..."
- 03:1703:17, 6 August 2023 diff hist +1,223 N Geometric products Created page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The following Cayley table shows the geometric products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. The identity of the geometric product is the scalar basis element $$\mathbf 1$$. Cells colored yellow correspond..."
- 03:1603:16, 6 August 2023 diff hist +3,293 N Duals Created page with "The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^*$$, is given by :$$\mathbf x^* = \mathbf{\tilde x} \mathbin{\unicode{x27D1}} {\large\unicode{x1d7d9}}$$ . The following table lists the duals for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. {| class="wikitable" ! Type !! Definition !! Dual |- | style="padding: 12px;" | Flat point | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p..."
- 03:1603:16, 6 August 2023 diff hist +3,794 N Attitude Created page with "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 :$$\operatorname{att}(\mathbf x) = \mathbf x \vee \overline{\mathbf e_4}$$ . 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 poin..."
- 03:1603:16, 6 August 2023 diff hist +3,776 N Partners Created page with "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 $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the meet of the carrier of $$\mathbf x$$ with the container of $$\mathbf x^*$$: :$$\operatorname{par}(\mathbf x) = \operatorna..."
- 03:1603:16, 6 August 2023 diff hist +4,839 N Carriers Created page with "== Carrier == The ''carrier'' of a round object (a round point, dipole, circle, or sphere) is the lowest dimensional flat object (a flat point, line, or plane) that contains it. The carrier of an object $$\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$$ as a flat geometry: :$$\operatorn..."
- 03:1603:16, 6 August 2023 diff hist +3,359 N Centers Created page with "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 anticarrier: :$$\operatorname{cen}(\mathbf x) = -\operatorname{car}(\mathbf x^*) \vee \mathbf x$$ . (The negative sign is not strictly necessary, but is included so the fu..."
- 03:1603:16, 6 August 2023 diff hist −3,338 Center Redirected page to Centers current Tag: New redirect
- 03:1503:15, 6 August 2023 diff hist +3,279 N Containers Created page with "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 $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the connect of $$\mathbf x$$ with its own carrier: :$$\operatorname{con}(\mathbf x) = \operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ . The squared radius of an object's container has the same sign as the squared radi..."
- 03:1503:15, 6 August 2023 diff hist +3,359 N Center Created page with "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 anticarrier: :$$\operatorname{cen}(\mathbf x) = -\operatorname{car}(\mathbf x^*) \vee \mathbf x$$ . (The negative sign is not strictly necessary, but is included so the fu..."
- 03:1503:15, 6 August 2023 diff hist +2,931 N Sphere Created page with "__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..."
- 03:1503:15, 6 August 2023 diff hist +5,915 N Circle Created page with "__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 $$gw$$ componen..."
- 03:1403:14, 6 August 2023 diff hist +5,689 N Dipole Created page with "__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 $$mz$$ com..."
- 03:1403:14, 6 August 2023 diff hist +1,626 N Round point Created page with "__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 :$$\mathbf a = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + \mathbf e_4 + \dfrac{p^2 + r^2}{2} \mathbf e_5$$ . Th..."
- 03:1403:14, 6 August 2023 diff hist +837 N Plane Created page with "400px|thumb|right|'''Figure 1.''' The various properties of a plane. In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''plane'' $$\mathbf g$$ is a quadrivector having the general form :$$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$ . A plane can be viewed as an infinitely large sphere containing the point at infinity. A plane in conformal geometric algebra is the precise..." current
- 03:1403:14, 6 August 2023 diff hist +878 N Line Created page with "400px|thumb|right|'''Figure 1.''' The various properties of a line. In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''line'' $$\boldsymbol l$$ is a trivector having the general form :$$\boldsymbol l = v_x \mathbf e_{415} + v_y \mathbf e_{425} + v_z \mathbf e_{435} + m_x \mathbf e_{235} + m_y \mathbf e_{315} + m_z \mathbf e_{125}$$ . A line can be viewed as an infinitely large circle that contains the point at infinity. A line in..." current
- 03:1303:13, 6 August 2023 diff hist +836 N Flat point Created page with "400px|thumb|right|'''Figure 1.''' The various properties of a flat point. In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''flat point'' $$\mathbf p$$ is a bivector having the general form :$$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$ . A flat point can be viewed as a dipole that has one end at the point at infinity. A flat point in conformal geometric algebra is the precise..." current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Wedge product.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet sphere.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet point.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet plane.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet line.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet dipole.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere meet circle.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect line.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect dipole.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Sphere connect circle.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Round join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Point join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Point connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane meet plane.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane meet line.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane meet dipole.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane meet circle.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect line.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect dipole.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:GeometricProduct.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Plane connect circle.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Dipole.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:GeometricAntiproduct.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line connect point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Line connect dipole.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Horosphere.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Dipole join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Dipole join point.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Circle.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Dipole join dipole.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Dipole connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Circle meet line.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Circle meet circle.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Circle join round.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Antiwedge product.svg Uploaded with SimpleBatchUpload current
- 03:1203:12, 6 August 2023 diff hist 0 N File:Circle connect round.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Circle connect point.svg Uploaded with SimpleBatchUpload
- 03:1203:12, 6 August 2023 diff hist 0 N File:Circle connect dipole.svg Uploaded with SimpleBatchUpload
- 03:1003:10, 6 August 2023 diff hist +4,614 Main Page No edit summary