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Combined display of all available logs of Rigid Geometric Algebra. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 21:27, 21 April 2024 Eric Lengyel talk contribs created page Euclidean angle (Created page with "The cosine of the Euclidean angle $$\cos \phi(\mathbf a, \mathbf b)$$ between two geometric objects '''a''' and '''b''' can be measured by the homogeneous magnitude given by :$$\cos \phi(\mathbf a, \mathbf b) = \left\Vert \mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606}\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \right\Vert_\unicode{x25CB}\left\Vert\mathbf b \right\Vert_\unicode{x25CB}$$. In the case that the grades of $$\mathbf a$$ and $$\mathbf b...")
- 21:27, 21 April 2024 Eric Lengyel talk contribs created page File:Angle line line.svg
- 21:27, 21 April 2024 Eric Lengyel talk contribs uploaded File:Angle line line.svg
- 21:27, 21 April 2024 Eric Lengyel talk contribs created page File:Angle plane line.svg
- 21:27, 21 April 2024 Eric Lengyel talk contribs uploaded File:Angle plane line.svg
- 21:27, 21 April 2024 Eric Lengyel talk contribs created page File:Angle plane plane.svg
- 21:27, 21 April 2024 Eric Lengyel talk contribs uploaded File:Angle plane plane.svg
- 23:27, 13 April 2024 Eric Lengyel talk contribs created page Weight dual (Redirected page to Duals) Tag: New redirect
- 23:27, 13 April 2024 Eric Lengyel talk contribs created page Bulk dual (Redirected page to Duals) Tag: New redirect
- 01:54, 13 April 2024 Eric Lengyel talk contribs created page Metric antiexomorphism matrix (Redirected page to Metrics) Tag: New redirect
- 01:53, 13 April 2024 Eric Lengyel talk contribs created page Metric exomorphism matrix (Redirected page to Metrics) Tag: New redirect
- 01:48, 13 April 2024 Eric Lengyel talk contribs created page Metrics (Created page with "The ''metric'' used in the 4D rigid geometric algebra over 3D Euclidean space is the $$4 \times 4$$ matrix $$\mathfrak g$$ given by :$$\mathfrak g = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\\end{bmatrix}$$ . The ''metric exomorphism matrix'' $$\mathbf G$$, often just called the "metric" itself, corresponding to the metric $$\mathfrak g$$ is the $$16 \times 16$$ matrix shown below. 420px The ''metri...")
- 01:48, 13 April 2024 Eric Lengyel talk contribs created page File:Antimetric-rga-3d.svg
- 01:48, 13 April 2024 Eric Lengyel talk contribs uploaded File:Antimetric-rga-3d.svg
- 01:45, 13 April 2024 Eric Lengyel talk contribs created page File:Metric-rga-3d.svg
- 01:45, 13 April 2024 Eric Lengyel talk contribs uploaded File:Metric-rga-3d.svg
- 06:16, 12 April 2024 Eric Lengyel talk contribs created page Dual (Redirected page to Duals) Tag: New redirect
- 06:15, 12 April 2024 Eric Lengyel talk contribs created page Duals (Created page with "The ''metric dual'' or just "dual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$ and defined as :$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf{Gu}}$$ , where $$\mathbf G$$ is the extended metric tensor. The ''metric antidual'' or just "antidual" of an object $$\mathbf u$$ is denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and defined as :$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \over...")
- 06:45, 5 April 2024 Eric Lengyel talk contribs deleted page File:Groups.svg (Deleted old revision 20240405064511!Groups.svg)
- 06:45, 5 April 2024 Eric Lengyel talk contribs uploaded a new version of File:Groups.svg
- 06:43, 5 April 2024 Eric Lengyel talk contribs deleted page File:Duality.svg (Deleted old revision 20240405064301!Duality.svg)
- 06:43, 5 April 2024 Eric Lengyel talk contribs uploaded a new version of File:Duality.svg
- 06:41, 5 April 2024 Eric Lengyel talk contribs deleted page File:Improper isom.svg (Deleted old revision 20240405064147!Improper_isom.svg)
- 06:41, 5 April 2024 Eric Lengyel talk contribs uploaded a new version of File:Improper isom.svg
- 06:41, 5 April 2024 Eric Lengyel talk contribs deleted page File:Proper isom.svg (Deleted old revision 20240405064103!Proper_isom.svg)
- 06:41, 5 April 2024 Eric Lengyel talk contribs uploaded a new version of File:Proper isom.svg
- 06:03, 5 April 2024 Eric Lengyel talk contribs deleted page File:Line infinity.svg (Deleted old revision 20240405060354!Line_infinity.svg)
- 06:03, 5 April 2024 Eric Lengyel talk contribs uploaded a new version of File:Line infinity.svg
- 06:03, 5 April 2024 Eric Lengyel talk contribs deleted page File:Line.svg (Deleted old revision 20240405060328!Line.svg)
- 06:03, 5 April 2024 Eric Lengyel talk contribs uploaded a new version of File:Line.svg
- 06:02, 5 April 2024 Eric Lengyel talk contribs deleted page File:Plane.svg (Deleted old revision 20240405060254!Plane.svg)
- 06:02, 5 April 2024 Eric Lengyel talk contribs uploaded a new version of File:Plane.svg
- 06:02, 5 April 2024 Eric Lengyel talk contribs deleted page File:Point.svg (Deleted old revision 20240405060200!Point.svg)
- 06:02, 5 April 2024 Eric Lengyel talk contribs uploaded a new version of File:Point.svg
- 00:59, 9 February 2024 Eric Lengyel talk contribs moved page Geometric property to Geometric constraint without leaving a redirect
- 02:12, 23 January 2024 Eric Lengyel talk contribs deleted page File:GeometricAntiproduct201.svg (Deleted old revision 20240123021158!GeometricAntiproduct201.svg)
- 02:11, 23 January 2024 Eric Lengyel talk contribs uploaded a new version of File:GeometricAntiproduct201.svg
- 02:11, 23 January 2024 Eric Lengyel talk contribs deleted page File:GeometricProduct201.svg (Deleted old revision 20240123021134!GeometricProduct201.svg)
- 02:11, 23 January 2024 Eric Lengyel talk contribs uploaded a new version of File:GeometricProduct201.svg
- 02:10, 23 January 2024 Eric Lengyel talk contribs deleted page File:AntiwedgeProduct.svg (Deleted old revision 20240123021002!AntiwedgeProduct.svg)
- 02:10, 23 January 2024 Eric Lengyel talk contribs uploaded a new version of File:AntiwedgeProduct.svg
- 02:09, 23 January 2024 Eric Lengyel talk contribs deleted page File:WedgeProduct.svg (Deleted old revision 20240123020946!WedgeProduct.svg)
- 02:09, 23 January 2024 Eric Lengyel talk contribs uploaded a new version of File:WedgeProduct.svg
- 02:08, 23 January 2024 Eric Lengyel talk contribs deleted page File:GeometricAntiproduct.svg (Deleted old revision 20240123020801!GeometricAntiproduct.svg)
- 02:08, 23 January 2024 Eric Lengyel talk contribs uploaded a new version of File:GeometricAntiproduct.svg
- 02:07, 23 January 2024 Eric Lengyel talk contribs deleted page File:GeometricProduct.svg (Deleted old revision 20240123020734!GeometricProduct.svg)
- 02:07, 23 January 2024 Eric Lengyel talk contribs uploaded a new version of File:GeometricProduct.svg
- 08:01, 22 January 2024 Eric Lengyel talk contribs deleted page File:Complements.svg (Deleted old revision 20240122080107!Complements.svg)
- 08:01, 22 January 2024 Eric Lengyel talk contribs uploaded a new version of File:Complements.svg
- 08:00, 22 January 2024 Eric Lengyel talk contribs deleted page File:Reverses.svg (Deleted old revision 20240122080005!Reverses.svg)
- 08:00, 22 January 2024 Eric Lengyel talk contribs uploaded a new version of File:Reverses.svg
- 07:57, 22 January 2024 Eric Lengyel talk contribs created page Weight contraction (Created page with "#REDIRECTION Interior products")
- 07:57, 22 January 2024 Eric Lengyel talk contribs created page Bulk contraction (Created page with "#REDIRECTION Interior products")
- 07:57, 22 January 2024 Eric Lengyel talk contribs created page Bulk expansion (Created page with "#REDIRECTION Interior products")
- 07:57, 22 January 2024 Eric Lengyel talk contribs created page Weight expansion (Created page with "#REDIRECTION Interior products")
- 07:56, 22 January 2024 Eric Lengyel talk contribs deleted page File:InteriorAntiproduct.svg
- 07:56, 22 January 2024 Eric Lengyel talk contribs deleted page File:InteriorProduct.svg
- 07:36, 22 January 2024 Eric Lengyel talk contribs created page File:Expansions.svg
- 07:36, 22 January 2024 Eric Lengyel talk contribs uploaded File:Expansions.svg
- 07:35, 22 January 2024 Eric Lengyel talk contribs created page File:Constractions.svg
- 07:35, 22 January 2024 Eric Lengyel talk contribs uploaded File:Constractions.svg
- 21:59, 21 January 2024 Eric Lengyel talk contribs created page File:Dots.svg
- 21:59, 21 January 2024 Eric Lengyel talk contribs uploaded File:Dots.svg
- 21:57, 21 January 2024 Eric Lengyel talk contribs deleted page File:AntidotProduct.svg
- 21:57, 21 January 2024 Eric Lengyel talk contribs deleted page File:DotProduct.svg
- 21:54, 21 January 2024 Eric Lengyel talk contribs deleted page File:GeometricAntiproduct.svg (Deleted old revision 20240121215421!GeometricAntiproduct.svg)
- 21:54, 21 January 2024 Eric Lengyel talk contribs uploaded a new version of File:GeometricAntiproduct.svg
- 21:54, 21 January 2024 Eric Lengyel talk contribs deleted page File:GeometricProduct.svg (Deleted old revision 20240121215401!GeometricProduct.svg)
- 21:54, 21 January 2024 Eric Lengyel talk contribs uploaded a new version of File:GeometricProduct.svg
- 21:48, 21 January 2024 Eric Lengyel talk contribs deleted page File:AntiwedgeProduct.svg (Deleted old revision 20240121214810!AntiwedgeProduct.svg)
- 21:48, 21 January 2024 Eric Lengyel talk contribs uploaded a new version of File:AntiwedgeProduct.svg
- 21:47, 21 January 2024 Eric Lengyel talk contribs deleted page File:WedgeProduct.svg (Deleted old revision 20240121214726!WedgeProduct.svg)
- 21:47, 21 January 2024 Eric Lengyel talk contribs uploaded a new version of File:WedgeProduct.svg
- 21:43, 21 January 2024 Eric Lengyel talk contribs deleted page File:Basis.svg (Deleted old revision 20240121214319!Basis.svg)
- 21:43, 21 January 2024 Eric Lengyel talk contribs uploaded a new version of File:Basis.svg
- 07:44, 23 October 2023 Eric Lengyel talk contribs deleted page File:AntidotProduct.svg (Deleted old revision 20231023074433!AntidotProduct.svg)
- 07:44, 23 October 2023 Eric Lengyel talk contribs uploaded a new version of File:AntidotProduct.svg
- 07:44, 23 October 2023 Eric Lengyel talk contribs deleted page File:DotProduct.svg (Deleted old revision 20231023074416!DotProduct.svg)
- 07:44, 23 October 2023 Eric Lengyel talk contribs uploaded a new version of File:DotProduct.svg
- 07:32, 23 October 2023 Eric Lengyel talk contribs created page File:Plane connect line.svg
- 07:32, 23 October 2023 Eric Lengyel talk contribs uploaded File:Plane connect line.svg
- 07:32, 23 October 2023 Eric Lengyel talk contribs created page File:Line connect point.svg
- 07:32, 23 October 2023 Eric Lengyel talk contribs uploaded File:Line connect point.svg
- 07:31, 23 October 2023 Eric Lengyel talk contribs created page File:Plane connect point.svg
- 07:31, 23 October 2023 Eric Lengyel talk contribs uploaded File:Plane connect point.svg
- 07:31, 23 October 2023 Eric Lengyel talk contribs deleted page File:Plane weight join line.svg
- 07:30, 23 October 2023 Eric Lengyel talk contribs deleted page File:Line weight join point.svg
- 07:30, 23 October 2023 Eric Lengyel talk contribs deleted page File:Plane weight join point.svg
- 07:01, 23 October 2023 Eric Lengyel talk contribs deleted page File:Complements.svg (Deleted old revision 20231023070104!Complements.svg)
- 07:01, 23 October 2023 Eric Lengyel talk contribs uploaded a new version of File:Complements.svg
- 06:54, 23 October 2023 Eric Lengyel talk contribs deleted page File:Complements.svg (Deleted old revision 20231023065433!Complements.svg)
- 06:54, 23 October 2023 Eric Lengyel talk contribs uploaded a new version of File:Complements.svg
- 00:49, 27 August 2023 Eric Lengyel talk contribs created page File:Plane onto line.svg
- 00:49, 27 August 2023 Eric Lengyel talk contribs uploaded File:Plane onto line.svg
- 00:49, 27 August 2023 Eric Lengyel talk contribs created page File:Line onto point.svg
- 00:49, 27 August 2023 Eric Lengyel talk contribs uploaded File:Line onto point.svg
- 00:49, 27 August 2023 Eric Lengyel talk contribs created page File:Plane onto point.svg
- 00:49, 27 August 2023 Eric Lengyel talk contribs uploaded File:Plane onto point.svg
- 00:49, 27 August 2023 Eric Lengyel talk contribs created page File:Line onto plane.svg
- 00:49, 27 August 2023 Eric Lengyel talk contribs uploaded File:Line onto plane.svg
- 00:48, 27 August 2023 Eric Lengyel talk contribs created page File:Point onto line.svg
- 00:48, 27 August 2023 Eric Lengyel talk contribs uploaded File:Point onto line.svg
- 00:48, 27 August 2023 Eric Lengyel talk contribs created page File:Point onto plane.svg
- 00:48, 27 August 2023 Eric Lengyel talk contribs uploaded File:Point onto plane.svg
- 00:47, 27 August 2023 Eric Lengyel talk contribs deleted page File:Plane line.svg
- 00:47, 27 August 2023 Eric Lengyel talk contribs deleted page File:Line point.svg
- 00:47, 27 August 2023 Eric Lengyel talk contribs deleted page File:Plane point.svg
- 00:47, 27 August 2023 Eric Lengyel talk contribs deleted page File:Line plane.svg
- 00:47, 27 August 2023 Eric Lengyel talk contribs deleted page File:Point line.svg
- 00:46, 27 August 2023 Eric Lengyel talk contribs deleted page File:Point plane.svg
- 00:48, 26 August 2023 Eric Lengyel talk contribs created page Weight norms (Redirected page to Geometric norm) Tag: New redirect
- 00:47, 26 August 2023 Eric Lengyel talk contribs created page Bulk norms (Redirected page to Geometric norm) Tag: New redirect
- 00:43, 26 August 2023 Eric Lengyel talk contribs created page Geometric norms (Redirected page to Geometric norm) Tag: New redirect
- 00:43, 26 August 2023 Eric Lengyel talk contribs deleted page File:GeometricAntiproduct201.svg (Deleted old revision 20230826004312!GeometricAntiproduct201.svg)
- 00:43, 26 August 2023 Eric Lengyel talk contribs uploaded a new version of File:GeometricAntiproduct201.svg
- 00:43, 26 August 2023 Eric Lengyel talk contribs deleted page File:GeometricProduct201.svg (Deleted old revision 20230826004302!GeometricProduct201.svg)
- 00:43, 26 August 2023 Eric Lengyel talk contribs uploaded a new version of File:GeometricProduct201.svg
- 00:40, 26 August 2023 Eric Lengyel talk contribs deleted page File:GeometricAntiproduct.svg (Deleted old revision 20230826004029!GeometricAntiproduct.svg)
- 00:40, 26 August 2023 Eric Lengyel talk contribs uploaded a new version of File:GeometricAntiproduct.svg
- 00:40, 26 August 2023 Eric Lengyel talk contribs deleted page File:GeometricProduct.svg (Deleted old revision 20230826004011!GeometricProduct.svg)
- 00:40, 26 August 2023 Eric Lengyel talk contribs uploaded a new version of File:GeometricProduct.svg
- 00:31, 26 August 2023 Eric Lengyel talk contribs deleted page File:Groups.svg (Deleted old revision 20230826003116!Groups.svg)
- 00:31, 26 August 2023 Eric Lengyel talk contribs uploaded a new version of File:Groups.svg
- 00:27, 26 August 2023 Eric Lengyel talk contribs created page Reciprocal rotations (Redirected page to Reciprocal rotation) Tag: New redirect
- 00:27, 26 August 2023 Eric Lengyel talk contribs created page Reciprocal translations (Redirected page to Reciprocal translation) Tag: New redirect
- 00:26, 26 August 2023 Eric Lengyel talk contribs moved page Dual translation to Reciprocal translation without leaving a redirect
- 00:25, 26 August 2023 Eric Lengyel talk contribs moved page Dual rotation to Reciprocal rotation without leaving a redirect
- 00:02, 26 August 2023 Eric Lengyel talk contribs deleted page File:Plane weight join line.svg (Deleted old revision 20230826000237!Plane_weight_join_line.svg)
- 00:02, 26 August 2023 Eric Lengyel talk contribs uploaded a new version of File:Plane weight join line.svg
- 00:02, 26 August 2023 Eric Lengyel talk contribs deleted page File:Line weight join point.svg (Deleted old revision 20230826000216!Line_weight_join_point.svg)
- 00:02, 26 August 2023 Eric Lengyel talk contribs uploaded a new version of File:Line weight join point.svg
- 00:01, 26 August 2023 Eric Lengyel talk contribs deleted page File:Plane weight join point.svg (Deleted old revision 20230826000155!Plane_weight_join_point.svg)
- 00:01, 26 August 2023 Eric Lengyel talk contribs uploaded a new version of File:Plane weight join point.svg
- 18:15, 25 August 2023 Eric Lengyel talk contribs deleted page File:AntidotProduct.svg (Deleted old revision 20230825181529!AntidotProduct.svg)
- 18:15, 25 August 2023 Eric Lengyel talk contribs uploaded a new version of File:AntidotProduct.svg
- 18:15, 25 August 2023 Eric Lengyel talk contribs deleted page File:DotProduct.svg (Deleted old revision 20230825181510!DotProduct.svg)
- 18:15, 25 August 2023 Eric Lengyel talk contribs uploaded a new version of File:DotProduct.svg
- 05:12, 5 August 2023 Eric Lengyel talk contribs created page File:Antispace5.svg
- 05:12, 5 August 2023 Eric Lengyel talk contribs uploaded File:Antispace5.svg
- 03:56, 5 August 2023 Eric Lengyel talk contribs created page File:Antispace4.svg
- 03:56, 5 August 2023 Eric Lengyel talk contribs uploaded File:Antispace4.svg
- 03:56, 5 August 2023 Eric Lengyel talk contribs created page File:Antispace3.svg
- 03:56, 5 August 2023 Eric Lengyel talk contribs uploaded File:Antispace3.svg
- 03:55, 5 August 2023 Eric Lengyel talk contribs created page File:Antispace2.svg
- 03:55, 5 August 2023 Eric Lengyel talk contribs uploaded File:Antispace2.svg
- 03:55, 5 August 2023 Eric Lengyel talk contribs created page File:Antispace1.svg
- 03:55, 5 August 2023 Eric Lengyel talk contribs uploaded File:Antispace1.svg
- 02:19, 5 August 2023 Eric Lengyel talk contribs created page Space-Antispace Transform Correspondence in Projective Geometric Algebra (Created page with "'''By Eric Lengyel'''<br /> May 20, 2022 The concept of duality can be understood geometrically in an ''n''-dimensional projective setting by considering both the subspace that an object occupies and the complementary subspace that the object concurrently does not occupy. The dimensionalities of these two components always sum to ''n'', and they represent the ''space'' and ''antispace'' associated with the object. (Antispace is also known as negative space or counterspa...")
- 01:58, 5 August 2023 Eric Lengyel talk contribs created page Antiprojection (Redirected page to Projections) Tag: New redirect
- 01:13, 5 August 2023 Eric Lengyel talk contribs created page Symmetries in Projective Geometric Algebra (Created page with "__NOTOC__ '''By Eric Lengyel'''<br /> October 19, 2020 This is a continuation of the post Projective Geometric Algebra Done Right from earlier this year. That post introduces the geometric antiproduct and the notation $$\unicode{x27D1}$$ and $$\unicode{x27C7}$$ used in this post. As before, everything in this post assumes familiarity with Chapter 4 of FGED1. I have been thinking about the geometric product and geometric antiproduct on a low-priority thread for most...")
- 23:37, 4 August 2023 Eric Lengyel talk contribs created page Projective Geometric Algebra Done Right (Created page with "January 23, 2020 This is part one of a two-part series. The second part can be found here: Symmetries in Projective Geometric Algebra. In the last chapter of FGED1, I provided a practical introduction to projective Grassmann algebra. Most people working in computer graphics are familiar with four-dimensional homogeneous coordinates, and projective Grassmann algebra extends this concept to a larger algebraic structure under which the wedge and antiwedge products per...")
- 05:06, 1 August 2023 Eric Lengyel talk contribs created page Euclidean distances (Redirected page to Euclidean distance) Tag: New redirect
- 05:05, 1 August 2023 Eric Lengyel talk contribs created page Magnitudes (Redirected page to Magnitude) Tag: New redirect
- 05:04, 1 August 2023 Eric Lengyel talk contribs created page Attitudes (Redirected page to Attitude) Tag: New redirect
- 08:48, 15 July 2023 Eric Lengyel talk contribs created page Meet (Redirected page to Join and meet) Tag: New redirect
- 08:48, 15 July 2023 Eric Lengyel talk contribs created page Join (Redirected page to Join and meet) Tag: New redirect
- 06:59, 15 July 2023 Eric Lengyel talk contribs created page Transflections (Redirected page to Transflection) Tag: New redirect
- 06:58, 15 July 2023 Eric Lengyel talk contribs created page Inversions (Redirected page to Inversion) Tag: New redirect
- 06:57, 15 July 2023 Eric Lengyel talk contribs created page Antireverse (Redirected page to Reverses) Tag: New redirect
- 06:56, 15 July 2023 Eric Lengyel talk contribs created page Motors (Redirected page to Motor) Tag: New redirect
- 06:56, 15 July 2023 Eric Lengyel talk contribs created page Flectors (Redirected page to Flector) Tag: New redirect
- 06:52, 15 July 2023 Eric Lengyel talk contribs created page Interior antiproduct (Redirected page to Interior products) Tag: New redirect
- 06:51, 15 July 2023 Eric Lengyel talk contribs created page Interior product (Redirected page to Interior products) Tag: New redirect
- 06:51, 15 July 2023 Eric Lengyel talk contribs created page Antidot product (Redirected page to Dot products) Tag: New redirect
- 06:51, 15 July 2023 Eric Lengyel talk contribs created page Dot product (Redirected page to Dot products) Tag: New redirect
- 06:51, 15 July 2023 Eric Lengyel talk contribs created page Commutator (Redirected page to Commutators) Tag: New redirect
- 06:50, 15 July 2023 Eric Lengyel talk contribs created page Projection (Redirected page to Projections) Tag: New redirect
- 06:50, 15 July 2023 Eric Lengyel talk contribs created page Antiwedge products (Redirected page to Exterior products) Tag: New redirect
- 06:50, 15 July 2023 Eric Lengyel talk contribs created page Antiwedge product (Redirected page to Exterior products) Tag: New redirect
- 06:50, 15 July 2023 Eric Lengyel talk contribs created page Wedge product (Redirected page to Exterior products) Tag: New redirect
- 06:49, 15 July 2023 Eric Lengyel talk contribs created page Exterior antiproduct (Redirected page to Exterior products) Tag: New redirect
- 06:49, 15 July 2023 Eric Lengyel talk contribs created page Exterior product (Redirected page to Exterior products) Tag: New redirect
- 06:49, 15 July 2023 Eric Lengyel talk contribs created page Geometric product (Redirected page to Geometric products) Tag: New redirect
- 06:48, 15 July 2023 Eric Lengyel talk contribs created page Geometric antiproduct (Redirected page to Geometric product) Tag: New redirect
- 06:48, 15 July 2023 Eric Lengyel talk contribs created page Scalar (Redirected page to Scalars and antiscalars) Tag: New redirect
- 06:48, 15 July 2023 Eric Lengyel talk contribs created page Bulk norm (Redirected page to Geometric norm) Tag: New redirect
- 06:47, 15 July 2023 Eric Lengyel talk contribs created page Weight norm (Redirected page to Geometric norm) Tag: New redirect
- 06:47, 15 July 2023 Eric Lengyel talk contribs created page Antiscalar (Redirected page to Scalars and antiscalars) Tag: New redirect
- 06:47, 15 July 2023 Eric Lengyel talk contribs created page Unitized (Redirected page to Unitization) Tag: New redirect
- 06:46, 15 July 2023 Eric Lengyel talk contribs created page Weight (Redirected page to Bulk and weight) Tag: New redirect
- 06:46, 15 July 2023 Eric Lengyel talk contribs created page Bulk (Redirected page to Bulk and weight) Tag: New redirect
- 06:46, 15 July 2023 Eric Lengyel talk contribs created page Reverse (Redirected page to Reverses) Tag: New redirect
- 06:45, 15 July 2023 Eric Lengyel talk contribs created page Complement (Redirected page to Complements) Tag: New redirect
- 06:45, 15 July 2023 Eric Lengyel talk contribs created page Grades (Redirected page to Grade and antigrade) Tag: New redirect
- 06:45, 15 July 2023 Eric Lengyel talk contribs created page Trivectors (Redirected page to Trivector) Tag: New redirect
- 06:44, 15 July 2023 Eric Lengyel talk contribs created page Bivectors (Redirected page to Bivector) Tag: New redirect
- 06:44, 15 July 2023 Eric Lengyel talk contribs created page Translations (Redirected page to Translation) Tag: New redirect
- 06:44, 15 July 2023 Eric Lengyel talk contribs created page Reflections (Redirected page to Reflection) Tag: New redirect
- 06:44, 15 July 2023 Eric Lengyel talk contribs created page Rotations (Redirected page to Rotation) Tag: New redirect
- 06:44, 15 July 2023 Eric Lengyel talk contribs created page Planes (Redirected page to Plane) Tag: New redirect
- 06:44, 15 July 2023 Eric Lengyel talk contribs created page Lines (Redirected page to Line) Tag: New redirect
- 06:43, 15 July 2023 Eric Lengyel talk contribs created page Points (Redirected page to Point) Tag: New redirect
- 06:43, 15 July 2023 Eric Lengyel talk contribs created page Trivector (Created page with "A ''trivector'' in a geometric algebra is an element composed entirely of components having grade 3. == See Also == * Vector * Bivector * Antivector")
- 06:42, 15 July 2023 Eric Lengyel talk contribs created page Bivector (Created page with "A ''bivector'' in a geometric algebra is an element composed entirely of components having grade 2. == See Also == * Vector * Trivector * Antivector")
- 06:42, 15 July 2023 Eric Lengyel talk contribs created page Antivector (Created page with "An ''antivector'' in a geometric algebra is an element composed entirely of components having antigrade 1. In an ''n''-dimensional geometric algebra, these are the elements having grade $$n - 1$$. == See Also == * Vector * Bivector * Trivector")
- 06:41, 15 July 2023 Eric Lengyel talk contribs created page Antivectors (Redirected page to Antivector) Tag: New redirect
- 06:40, 15 July 2023 Eric Lengyel talk contribs created page Vector (Created page with "A ''vector'' in a geometric algebra is an element composed entirely of components having grade 1. == See Also == * Bivector * Trivector * Antivector")
- 06:40, 15 July 2023 Eric Lengyel talk contribs created page Vectors (Redirected page to Vector) Tag: New redirect
- 06:39, 15 July 2023 Eric Lengyel talk contribs created page Antiscalars (Redirected page to Scalars and antiscalars) Tag: New redirect
- 06:39, 15 July 2023 Eric Lengyel talk contribs created page Scalars (Redirected page to Scalars and antiscalars) Tag: New redirect
- 06:37, 15 July 2023 Eric Lengyel talk contribs created page Antigrade (Redirected page to Grade and antigrade) Tag: New redirect
- 06:36, 15 July 2023 Eric Lengyel talk contribs created page Grade (Redirected page to Grade and antigrade) Tag: New redirect
- 06:36, 15 July 2023 Eric Lengyel talk contribs created page Commutators (Created page with "In geometric algebra, there are four ''commutator'' products defined as follows. :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b - \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b + \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode...")
- 06:35, 15 July 2023 Eric Lengyel talk contribs created page Point (Created page with "400px|thumb|right|'''Figure 1.''' A point is the intersection of a 4D vector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''point'' $$\mathbf p$$ is a vector having the general form :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ . All points possess the geometric property. The bulk of a point is given by its $$x$$, $$y$$, and $$z$$ coordinates, and...")
- 06:35, 15 July 2023 Eric Lengyel talk contribs created page File:Point.svg
- 06:35, 15 July 2023 Eric Lengyel talk contribs uploaded File:Point.svg
- 06:33, 15 July 2023 Eric Lengyel talk contribs created page Duality (Created page with "480px|thumb|right|'''Figure 1.''' The coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as the one-dimensional span of a single vector representing a homogeneous point or as the $$(n - 1)$$-dimensional span of all orthogonal vectors representing a homogeneous plane. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other. The concept of duality can be understood geometric...")
- 06:33, 15 July 2023 Eric Lengyel talk contribs created page File:Duality.svg
- 06:33, 15 July 2023 Eric Lengyel talk contribs uploaded File:Duality.svg
- 06:30, 15 July 2023 Eric Lengyel talk contribs created page Wedge products (Redirected page to Exterior products) Tag: New redirect
- 06:29, 15 July 2023 Eric Lengyel talk contribs created page Geometric products (Created page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct...")
- 06:29, 15 July 2023 Eric Lengyel talk contribs created page File:GeometricAntiproduct.svg
- 06:29, 15 July 2023 Eric Lengyel talk contribs uploaded File:GeometricAntiproduct.svg
- 06:29, 15 July 2023 Eric Lengyel talk contribs created page File:GeometricProduct.svg
- 06:29, 15 July 2023 Eric Lengyel talk contribs uploaded File:GeometricProduct.svg
- 06:28, 15 July 2023 Eric Lengyel talk contribs created page Exterior products (Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. The exterior product between two elements $$\mathbf a$$ and $$\mathbf b$$ generally combines their spatial extents, and the magnitude of the result indicates how close they are to being orthogonal. If the spatial extents of $$\m...")
- 06:28, 15 July 2023 Eric Lengyel talk contribs created page File:AntiwedgeProduct.svg
- 06:28, 15 July 2023 Eric Lengyel talk contribs uploaded File:AntiwedgeProduct.svg
- 06:28, 15 July 2023 Eric Lengyel talk contribs created page File:WedgeProduct.svg
- 06:28, 15 July 2023 Eric Lengyel talk contribs uploaded File:WedgeProduct.svg
- 06:27, 15 July 2023 Eric Lengyel talk contribs created page Interior products (Created page with "The left and right ''interior products'' are special products in geometric algebra that are useful for performing projections. These products cancel common factors in their operands and thus reduce grade. Depending on the choice of dualization function, there are several possible interior products. We define the interior products in terms of the left and right complements. Interior products are also known as contraction products. == Left and Right Interior Prod...")
- 06:27, 15 July 2023 Eric Lengyel talk contribs created page File:InteriorAntiproduct.svg
- 06:27, 15 July 2023 Eric Lengyel talk contribs uploaded File:InteriorAntiproduct.svg
- 06:27, 15 July 2023 Eric Lengyel talk contribs created page File:InteriorProduct.svg
- 06:27, 15 July 2023 Eric Lengyel talk contribs uploaded File:InteriorProduct.svg
- 06:26, 15 July 2023 Eric Lengyel talk contribs created page Dot products (Created page with "The ''dot product'' is the inner product in geometric algebra, and it makes up the scalar part of the geometric product. There are two products with symmetric properties called the dot product and antidot product. The dot product and antidot product are important for the calculation of norms. == Dot Product == The dot product between two elements $$\mathbf a$$ and $$\mathbf b$$ is written $$\mathbf a \mathbin{\unicode{x25CF}} \mathbf b$$ and r...")
- 06:26, 15 July 2023 Eric Lengyel talk contribs created page File:AntidotProduct.svg
- 06:26, 15 July 2023 Eric Lengyel talk contribs uploaded File:AntidotProduct.svg
- 06:26, 15 July 2023 Eric Lengyel talk contribs created page File:DotProduct.svg
- 06:26, 15 July 2023 Eric Lengyel talk contribs uploaded File:DotProduct.svg
- 06:24, 15 July 2023 Eric Lengyel talk contribs created page Quaternions (Redirected page to Quaternion) Tag: New redirect
- 06:23, 15 July 2023 Eric Lengyel talk contribs created page Quaternion (Created page with "__NOTOC__ A ''quaternion'' is an operator that performs a rotation about the origin in 3D space. Conventionally, a quaternion $$\mathbf q$$ is written as :$$\mathbf q = q_w + q_x \mathbf i + q_y \mathbf j + q_z \mathbf k$$ , where the "imaginary" units $$\mathbf i$$, $$\mathbf j$$, and $$\mathbf k$$ all square to $$-1$$ and multiply according to the rules :$$\mathbf{ij} = -\mathbf{ji} = \mathbf k$$ :$$\mathbf{jk} = -\mathbf{kj} = \mathbf i$$ :$$\mathbf{ki} = -\mathbf{...")
- 06:20, 15 July 2023 Eric Lengyel talk contribs created page Scalars and antiscalars (Created page with "A ''scalar'' in a geometric algebra is an element having grade 0. Scalars are just ordinary real numbers, and they do not involve any basis vectors. The basis element representing the unit scalar is denoted by $$\mathbf 1$$, a boldface number one. The unit scalar $$\mathbf 1$$ is the multiplicative identity of the geometric product. For a general element $$\mathbf a$$, the notation $$a_{\mathbf 1}$$ means the scalar component of $$\mathbf a$$. An ''antiscalar'...")
- 06:16, 15 July 2023 Eric Lengyel talk contribs created page Plane (Created page with "400px|thumb|right|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''plane'' $$\mathbf g$$ is a trivector having the general form :$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ . All planes possess the geometric property. The bulk of a plane is given by its $$w$$ coordinate, a...")
- 06:16, 15 July 2023 Eric Lengyel talk contribs created page File:Plane.svg
- 06:16, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane.svg
- 06:15, 15 July 2023 Eric Lengyel talk contribs created page Grade and antigrade (Created page with "The ''grade'' of a basis element in a geometric algebra is equal to the number of basis vectors present in its factorization. An arbitrary element whose components all have the same grade is also said to have that grade. The ''antigrade'' of a basis element is equal to the number of basis vectors absent from its factorization. The grade of an element $$\mathbf x$$ is denoted by $$\operatorname{gr}(\mathbf x)$$, and the antigrade is denoted by $$\operatorname{ag}(\mathb...")
- 06:14, 15 July 2023 Eric Lengyel talk contribs created page Reverses (Created page with "''Reverses'' are unary operations in geometric algebra that are analogs of conjugate or transpose operations. For any element $$\mathbf x$$ that is the wedge product of $$k$$ vectors, the ''reverse'' of $$\mathbf x$$, which we denote by $$\mathbf{\tilde x}$$, is the result of multiplying those same $$k$$ vectors in reverse order. For example, the reverse of $$\mathbf e_{234}$$ is $$\mathbf e_4 \wedge \mathbf e_3 \wedge \mathbf e_2$$, which we would write as $$-\math...")
- 06:14, 15 July 2023 Eric Lengyel talk contribs created page File:Reverses.svg
- 06:14, 15 July 2023 Eric Lengyel talk contribs uploaded File:Reverses.svg
- 06:12, 15 July 2023 Eric Lengyel talk contribs created page File:Plane line.svg
- 06:12, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane line.svg
- 06:12, 15 July 2023 Eric Lengyel talk contribs created page File:Line point.svg
- 06:12, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line point.svg
- 06:12, 15 July 2023 Eric Lengyel talk contribs created page File:Plane point.svg
- 06:12, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane point.svg
- 06:11, 15 July 2023 Eric Lengyel talk contribs created page File:Line plane.svg
- 06:11, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line plane.svg
- 06:11, 15 July 2023 Eric Lengyel talk contribs created page File:Point line.svg
- 06:11, 15 July 2023 Eric Lengyel talk contribs uploaded File:Point line.svg
- 06:11, 15 July 2023 Eric Lengyel talk contribs created page File:Point plane.svg
- 06:11, 15 July 2023 Eric Lengyel talk contribs uploaded File:Point plane.svg
- 06:11, 15 July 2023 Eric Lengyel talk contribs created page Projections (Created page with "Projections and antiprojections of one geometric object onto another can be accomplished using interior products as described below. The formulas on this page are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. == Projection == The geometric projection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula $$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{...")
- 06:10, 15 July 2023 Eric Lengyel talk contribs created page Euclidean distance (Created page with "The Euclidean distance between geometric objects can be measured by using commutators to calculate homogeneous magnitudes. The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. The points, lines, and p...")
- 06:10, 15 July 2023 Eric Lengyel talk contribs created page File:Distance line line.svg
- 06:10, 15 July 2023 Eric Lengyel talk contribs uploaded File:Distance line line.svg
- 06:10, 15 July 2023 Eric Lengyel talk contribs created page File:Distance point plane.svg
- 06:10, 15 July 2023 Eric Lengyel talk contribs uploaded File:Distance point plane.svg
- 06:09, 15 July 2023 Eric Lengyel talk contribs created page File:Distance point line.svg
- 06:09, 15 July 2023 Eric Lengyel talk contribs uploaded File:Distance point line.svg
- 06:09, 15 July 2023 Eric Lengyel talk contribs created page File:Distance point point.svg
- 06:09, 15 July 2023 Eric Lengyel talk contribs uploaded File:Distance point point.svg
- 06:06, 15 July 2023 Eric Lengyel talk contribs created page Flector (Created page with "400px|thumb|right|'''Figure 1.''' A flector represents an improper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a reflection across a plane perpendicular to the same line. A ''flector'' is an operator that performs an improper isometry in Euclidean space. Such isometries encompass all possible combinations of an odd number of reflections, inversions, transflections, and rotorefle...")
- 06:06, 15 July 2023 Eric Lengyel talk contribs created page File:Improper isom.svg
- 06:06, 15 July 2023 Eric Lengyel talk contribs uploaded File:Improper isom.svg
- 06:04, 15 July 2023 Eric Lengyel talk contribs created page File:Complements.svg
- 06:04, 15 July 2023 Eric Lengyel talk contribs uploaded File:Complements.svg
- 06:04, 15 July 2023 Eric Lengyel talk contribs created page Complements (Created page with "''Complements'' are unary operations in geometric algebra that perform a specific type of dualization. Every basis element $$\mathbf x$$ has a ''right complement'', which we denote by $$\overline{\mathbf x}$$, that satisfies the equation :$$\mathbf x \wedge \overline{\mathbf x} = {\large\unicode{x1D7D9}}$$ . There is also a ''left complement'', which we denote by $$\underline{\mathbf x}$$, that satisfies the equation :$$\underline{\mathbf x} \wedge \mathbf x = {\larg...")
- 06:03, 15 July 2023 Eric Lengyel talk contribs created page Geometric property (Created page with "An element $$\mathbf x$$ of a geometric algebra possesses the ''geometric property'' if and only if the geometric product between $$\mathbf x$$ and its own reverse is a scalar, which is given by the dot product, and the geometric antiproduct between $$\mathbf x$$ and its own antireverse is an antiscalar, which is given by the antidot product. That is, :$$\mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde x} = \mathbf x \mathbin{\unicode{x25CF}} \mathbf{\...")
- 06:03, 15 July 2023 Eric Lengyel talk contribs created page Unitization (Created page with "''Unitization'' is the process of scaling an element of a projective geometric algebra so that its weight norm becomes the antiscalar $$\large\unicode{x1D7D9}$$. An element that has a weight norm of $$\large\unicode{x1D7D9}$$ is said to be ''unitized''. An element $$\mathbf x$$ is unitized by calculating :$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\ma...")
- 06:02, 15 July 2023 Eric Lengyel talk contribs created page Geometric norm (Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf x$$, d...")
- 05:59, 15 July 2023 Eric Lengyel talk contribs created page Dual rotation (Created page with "A ''dual rotation'' is a proper isometry of dual Euclidean space. For a bulk normalized line $$\boldsymbol l$$, the specific kind of dual motor :$$\mathbf R = \boldsymbol l\sin\phi + \mathbf 1\cos\phi$$ , performs a dual rotation of an object $$\mathbf x$$ by twice the angle $$\phi$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27D1}} \mathbf x \mathbin{\unicode{x27D1}} \mathbf{\tilde R}$$. The line $$\boldsymbol l$$ and its bulk complement...")
- 05:59, 15 July 2023 Eric Lengyel talk contribs created page File:DualRotation.svg
- 05:59, 15 July 2023 Eric Lengyel talk contribs uploaded File:DualRotation.svg
- 05:59, 15 July 2023 Eric Lengyel talk contribs created page File:Rotation.svg
- 05:59, 15 July 2023 Eric Lengyel talk contribs uploaded File:Rotation.svg
- 05:59, 15 July 2023 Eric Lengyel talk contribs created page Dual translation (Created page with "__NOTOC__ A ''dual translation'' is a proper isometry of dual Euclidean space. The specific kind of dual motor :$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$ performs a perspective projection in the direction of $$\mathbf t = (t_x, t_y, t_z)$$ with the focal length given by :$$g = \dfrac{1}{2\Vert \mathbf t \Vert}$$ . == Example == The left image below shows the flow field in the ''x''-''z'' plane for the translation $...")
- 05:58, 15 July 2023 Eric Lengyel talk contribs created page File:DualTranslation.svg
- 05:58, 15 July 2023 Eric Lengyel talk contribs uploaded File:DualTranslation.svg
- 05:58, 15 July 2023 Eric Lengyel talk contribs created page File:Translation.svg
- 05:58, 15 July 2023 Eric Lengyel talk contribs uploaded File:Translation.svg
- 05:57, 15 July 2023 Eric Lengyel talk contribs created page Translation (Created page with "__NOTOC__ A ''translation'' is a proper isometry of Euclidean space. The specific kind of motor :$$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$ performs a translation by twice the displacement vector $$\mathbf t = (t_x, t_y, t_z)$$ when used as an operator in the sandwich antiproduct. This can be interpreted as a rotation about the line at infinity perpendicular to the direction $$\mathbf t$$. === Trans...")
- 05:57, 15 July 2023 Eric Lengyel talk contribs created page Rotation (Created page with "A ''rotation'' is a proper isometry of Euclidean space. For a unitized line $$\boldsymbol l$$, the specific kind of motor :$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ , performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathb...")
- 05:56, 15 July 2023 Eric Lengyel talk contribs created page Reflection (Created page with "A ''reflection'' is an improper isometry of Euclidean space. When used as an operator in the sandwich antiproduct, a unitized plane $$\mathbf F = F_{gx}\mathbf e_{423} + F_{gy}\mathbf e_{431} + F_{gz}\mathbf e_{412} + F_{gw}\mathbf e_{321}$$ is a specific kind of flector that performs a reflection through $$\mathbf F$$. == Calculation == The exact reflection calculations for points, lines, and planes are shown in the following table. {| class="wikitable"...")
- 05:55, 15 July 2023 Eric Lengyel talk contribs created page Inversion (Created page with "An ''inversion'' is an improper isometry of Euclidean space. When used as an operator in the sandwich antiproduct, a unitized point $$\mathbf F = F_{px}\mathbf e_1 + F_{py}\mathbf e_2 + F_{pz}\mathbf e_3 + \mathbf e_4$$ is a specific kind of flector that performs an inversion through $$\mathbf F$$. == Calculation == The exact inversion calculations for points, lines, and planes are shown in the following table. {| class="wikitable" ! Type || Inversion |-...")
- 05:54, 15 July 2023 Eric Lengyel talk contribs created page Transflection (Created page with "A ''transflection'' is an improper isometry of Euclidean space consisting of a reflection through a plane and a translation parallel to the same plane. All combinations of a reflection and a translation, even if the original translation vector is not parallel to the original reflection plane, can be formulated as a transflection with respect to some plane. The specific kind of flector :$$\mathbf F = F_{px} \mathbf e_{1} + F_{py} \mathbf e_{2} + F_{pz} \math...")
- 05:52, 15 July 2023 Eric Lengyel talk contribs created page File:Groups.svg
- 05:52, 15 July 2023 Eric Lengyel talk contribs uploaded File:Groups.svg
- 05:52, 15 July 2023 Eric Lengyel talk contribs created page Transformation groups (Created page with "In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, every Euclidean isometry of 3D space can be represented by a motor $$\mathbf Q$$ of the form :$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ or by a flector $$\mathbf F$$ of the form :$$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_...")
- 05:50, 15 July 2023 Eric Lengyel talk contribs created page Magnitude (Created page with "A ''magnitude'' is a quantity that represents a concrete distance of some kind. In rigid geometric algebra, a magnitude $$\mathbf z$$ is composed of two components, a scalar and an antiscalar, as follows: :$$\mathbf z = x\mathbf 1 + y {\large\unicode{x1d7d9}}$$ Magnitudes are homogeneous just like everything else in a projective geometric algebra. This means it has both a bulk and a weight, and it is unitized by making the magnitude of its weight one. ===...")
- 05:46, 15 July 2023 Eric Lengyel talk contribs created page File:GeometricAntiproduct201.svg
- 05:46, 15 July 2023 Eric Lengyel talk contribs uploaded File:GeometricAntiproduct201.svg
- 05:46, 15 July 2023 Eric Lengyel talk contribs created page File:GeometricProduct201.svg
- 05:46, 15 July 2023 Eric Lengyel talk contribs uploaded File:GeometricProduct201.svg
- 05:46, 15 July 2023 Eric Lengyel talk contribs created page File:Unary201.svg
- 05:46, 15 July 2023 Eric Lengyel talk contribs uploaded File:Unary201.svg
- 05:46, 15 July 2023 Eric Lengyel talk contribs created page File:Basis201.svg
- 05:46, 15 July 2023 Eric Lengyel talk contribs uploaded File:Basis201.svg
- 05:46, 15 July 2023 Eric Lengyel talk contribs created page Rigid Geometric Algebra for 2D Space (Created page with "== Introduction == thumb|right|400px|'''Table 1.''' The 8 basis elements of the 3D rigid geometric algebra. In the three-dimensional rigid geometric algebra, there are 8 graded basis elements. These are listed in Table 1. There is a single ''scalar'' basis element $$\mathbf 1$$, and its multiples correspond to the real numbers, which are values that have no dimensions. There are three ''vector'' basis elements named $$\mathbf e_1$$, $$\mathbf e_...")
- 05:40, 15 July 2023 Eric Lengyel talk contribs created page File:Plane weight join line.svg
- 05:40, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane weight join line.svg
- 05:40, 15 July 2023 Eric Lengyel talk contribs created page File:Line weight join point.svg
- 05:40, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line weight join point.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs created page File:Plane weight join point.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane weight join point.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs created page File:Line meet plane.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line meet plane.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs created page File:Plane meet plane.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs uploaded File:Plane meet plane.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs created page File:Line join point.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line join point.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs created page File:Point join point.svg
- 05:39, 15 July 2023 Eric Lengyel talk contribs uploaded File:Point join point.svg
- 05:38, 15 July 2023 Eric Lengyel talk contribs created page Join and meet (Created page with "The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection. The points, lines, and planes appearing in the following tables are defined as follows: :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ :$$\mathbf...")
- 05:34, 15 July 2023 Eric Lengyel talk contribs created page File:Skew lines.svg
- 05:34, 15 July 2023 Eric Lengyel talk contribs uploaded File:Skew lines.svg
- 05:34, 15 July 2023 Eric Lengyel talk contribs created page File:Line infinity.svg
- 05:34, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line infinity.svg
- 05:34, 15 July 2023 Eric Lengyel talk contribs created page File:Line.svg
- 05:34, 15 July 2023 Eric Lengyel talk contribs uploaded File:Line.svg
- 05:34, 15 July 2023 Eric Lengyel talk contribs created page Line (Created page with "400px|thumb|right|'''Figure 1.''' A line is the intersection of a 4D bivector with the 3D subspace where $$w = 1$$. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a ''line'' $$\boldsymbol l$$ is a bivector having the general form :$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$ . The components $$(l_{vx}, l_{vy}, l_{vz})$$ corr...")
- 05:31, 15 July 2023 Eric Lengyel talk contribs created page Bulk and weight (Created page with "The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon. An element is unitized when the magnitude of its weight is one. The following table lists the bulk and weight for the main types in the 4D rigid geometric algebra $$\mathcal...")
- 05:29, 15 July 2023 Eric Lengyel talk contribs created page Attitude (Created page with "The attitude function, denoted by $$\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}$$ . The attitude of a line is the line's direction as a vector, and the attitude of a plane is the plane's normal as a bivector. The following table lists the attitude for the main types in the 4D rigid geometric algebra...")
- 05:27, 15 July 2023 Eric Lengyel talk contribs created page File:Proper isom.svg
- 05:27, 15 July 2023 Eric Lengyel talk contribs uploaded File:Proper isom.svg
- 05:27, 15 July 2023 Eric Lengyel talk contribs created page Motor (Created page with "400px|thumb|right|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\mathbf L$$ and a displacement along the same line. A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of rotations and translations. The name motor is a portmanteau of ''motion operator'' or ''moment vector...")
- 05:24, 15 July 2023 Eric Lengyel talk contribs created page File:Basis.svg
- 05:24, 15 July 2023 Eric Lengyel talk contribs uploaded File:Basis.svg