User contributions for Eric Lengyel

1 May 2024

• 01:3601:36, 1 May 2024 diff hist +156‎ Symmetries in Projective Geometric Algebra ‎No edit summary current

29 April 2024

• 01:5301:53, 29 April 2024 diff hist +476‎ Exercises ‎No edit summary current
• 01:4801:48, 29 April 2024 diff hist +918‎ N Exercises ‎ Created page with "These are exercises accompanying the book [https://www.amazon.com/dp/B0CXY8C72T/?tag=terathon-20 ''Projective Geometric Algebra Illuminated'']. == Exercises for Chapter 2 == '''1.''' Show that Equation (2.35) properly constructs a line containing two points $$\mathbf p$$ and $$\mathbf q$$ with non-unit weights by considering $$\mathbf p / p_w \wedge \mathbf q / q_w$$ and then scaling by $$p_wq_w$$. '''2.''' Let $$\mathbf u$$ be a basis element of the 4D projective alg..."

21 April 2024

• 21:2821:28, 21 April 2024 diff hist +22‎ Main Page ‎ →‎Products and other binary operations current
• 21:2821:28, 21 April 2024 diff hist 0‎ Euclidean angle ‎No edit summary current
• 21:2721:27, 21 April 2024 diff hist +3,083‎ N 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..."
• 21:2721:27, 21 April 2024 diff hist 0‎ N File:Angle line line.svg ‎No edit summary current
• 21:2721:27, 21 April 2024 diff hist 0‎ N File:Angle plane line.svg ‎No edit summary current
• 21:2721:27, 21 April 2024 diff hist 0‎ N File:Angle plane plane.svg ‎No edit summary current
• 21:1521:15, 21 April 2024 diff hist +22‎ Euclidean distance ‎No edit summary current
• 21:1521:15, 21 April 2024 diff hist +287‎ Euclidean distance ‎No edit summary

13 April 2024

• 23:5223:52, 13 April 2024 diff hist +76‎ Transformation groups ‎No edit summary current
• 23:5123:51, 13 April 2024 diff hist +92‎ Flector ‎No edit summary current
• 23:5123:51, 13 April 2024 diff hist +86‎ Motor ‎No edit summary current
• 23:5023:50, 13 April 2024 diff hist +73‎ Plane ‎No edit summary current
• 23:5023:50, 13 April 2024 diff hist +71‎ Line ‎No edit summary current
• 23:4923:49, 13 April 2024 diff hist +73‎ Point ‎No edit summary current
• 23:3323:33, 13 April 2024 diff hist 0‎ Duality ‎No edit summary current
• 23:3323:33, 13 April 2024 diff hist 0‎ Unitization ‎No edit summary current
• 23:3323:33, 13 April 2024 diff hist 0‎ Geometric constraint ‎No edit summary current
• 23:3223:32, 13 April 2024 diff hist 0‎ Geometric norm ‎No edit summary current
• 23:3223:32, 13 April 2024 diff hist 0‎ Reverses ‎No edit summary current
• 23:3223:32, 13 April 2024 diff hist 0‎ Duals ‎No edit summary current
• 23:3223:32, 13 April 2024 diff hist 0‎ Bulk and weight ‎No edit summary current
• 23:3123:31, 13 April 2024 diff hist 0‎ Metrics ‎No edit summary current
• 23:3123:31, 13 April 2024 diff hist 0‎ Complements ‎No edit summary current
• 23:3123:31, 13 April 2024 diff hist 0‎ Grade and antigrade ‎No edit summary current
• 23:3023:30, 13 April 2024 diff hist −40‎ Euclidean distance ‎No edit summary
• 23:2823:28, 13 April 2024 diff hist +68‎ Projections ‎No edit summary current
• 23:2723:27, 13 April 2024 diff hist +102‎ Interior products ‎No edit summary current
• 23:2723:27, 13 April 2024 diff hist +19‎ N Weight dual ‎ Redirected page to Duals current Tag: New redirect
• 23:2723:27, 13 April 2024 diff hist +19‎ N Bulk dual ‎ Redirected page to Duals current Tag: New redirect
• 23:2623:26, 13 April 2024 diff hist +89‎ Dot products ‎No edit summary current
• 23:2523:25, 13 April 2024 diff hist −202‎ Geometric products ‎No edit summary current
• 23:2423:24, 13 April 2024 diff hist −153‎ Exterior products ‎No edit summary current
• 05:1605:16, 13 April 2024 diff hist +70‎ Duality ‎No edit summary
• 05:1505:15, 13 April 2024 diff hist +66‎ Unitization ‎No edit summary
• 05:1405:14, 13 April 2024 diff hist +78‎ Geometric constraint ‎No edit summary
• 05:1405:14, 13 April 2024 diff hist +71‎ Geometric norm ‎No edit summary
• 05:1305:13, 13 April 2024 diff hist +66‎ Attitude ‎No edit summary current
• 05:1105:11, 13 April 2024 diff hist +79‎ Reverses ‎No edit summary
• 05:1005:10, 13 April 2024 diff hist +75‎ Grade and antigrade ‎No edit summary
• 05:0805:08, 13 April 2024 diff hist +13‎ Metrics ‎ →‎In the Book
• 05:0805:08, 13 April 2024 diff hist +71‎ Bulk and weight ‎No edit summary
• 05:0705:07, 13 April 2024 diff hist +79‎ Metrics ‎No edit summary
• 05:0605:06, 13 April 2024 diff hist +65‎ Complements ‎No edit summary
• 05:0505:05, 13 April 2024 diff hist +1,145‎ Duals ‎No edit summary
• 01:5401:54, 13 April 2024 diff hist +8‎ Duals ‎No edit summary
• 01:5401:54, 13 April 2024 diff hist +7‎ Bulk and weight ‎No edit summary
• 01:5401:54, 13 April 2024 diff hist +21‎ N Metric antiexomorphism matrix ‎ Redirected page to Metrics current Tag: New redirect
• 01:5301:53, 13 April 2024 diff hist +21‎ N Metric exomorphism matrix ‎ Redirected page to Metrics current Tag: New redirect
• 01:5301:53, 13 April 2024 diff hist +8‎ Dot products ‎No edit summary
• 01:5301:53, 13 April 2024 diff hist +486‎ Metrics ‎No edit summary
• 01:4801:48, 13 April 2024 diff hist +674‎ N 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..."
• 01:4801:48, 13 April 2024 diff hist 0‎ N File:Antimetric-rga-3d.svg ‎No edit summary current
• 01:4501:45, 13 April 2024 diff hist 0‎ N File:Metric-rga-3d.svg ‎No edit summary current
• 01:4001:40, 13 April 2024 diff hist +14‎ Main Page ‎ →‎Various properties and unary operations
• 01:3601:36, 13 April 2024 diff hist +40‎ Duals ‎No edit summary

12 April 2024

• 20:3920:39, 12 April 2024 diff hist +247‎ Complements ‎No edit summary
• 06:2706:27, 12 April 2024 diff hist +12‎ Bulk and weight ‎ →‎See Also
• 06:2606:26, 12 April 2024 diff hist +12‎ Complements ‎ →‎See Also
• 06:2606:26, 12 April 2024 diff hist 0‎ Duals ‎ →‎See Also
• 06:2606:26, 12 April 2024 diff hist +669‎ Duals ‎No edit summary
• 06:1806:18, 12 April 2024 diff hist +486‎ Duals ‎No edit summary
• 06:1606:16, 12 April 2024 diff hist +19‎ N Dual ‎ Redirected page to Duals current Tag: New redirect
• 06:1506:15, 12 April 2024 diff hist +560‎ N 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..."
• 06:0906:09, 12 April 2024 diff hist +12‎ Main Page ‎ →‎Various properties and unary operations

8 April 2024

• 05:1605:16, 8 April 2024 diff hist −1,993‎ Rotation ‎ →‎Calculation current
• 05:1205:12, 8 April 2024 diff hist −2,051‎ Flector ‎ →‎Flector Transformations
• 05:0105:01, 8 April 2024 diff hist −2,112‎ Motor ‎ →‎Motor Transformations

5 April 2024

• 06:4506:45, 5 April 2024 diff hist 0‎ File:Groups.svg ‎ Eric Lengyel uploaded a new version of File:Groups.svg current
• 06:4306:43, 5 April 2024 diff hist 0‎ File:Duality.svg ‎ Eric Lengyel uploaded a new version of File:Duality.svg current
• 06:4106:41, 5 April 2024 diff hist 0‎ File:Improper isom.svg ‎ Eric Lengyel uploaded a new version of File:Improper isom.svg current
• 06:4106:41, 5 April 2024 diff hist 0‎ File:Proper isom.svg ‎ Eric Lengyel uploaded a new version of File:Proper isom.svg current
• 06:0306:03, 5 April 2024 diff hist 0‎ File:Line infinity.svg ‎ Eric Lengyel uploaded a new version of File:Line infinity.svg current
• 06:0306:03, 5 April 2024 diff hist 0‎ File:Line.svg ‎ Eric Lengyel uploaded a new version of File:Line.svg current
• 06:0306:03, 5 April 2024 diff hist +2‎ Plane ‎No edit summary
• 06:0206:02, 5 April 2024 diff hist 0‎ File:Plane.svg ‎ Eric Lengyel uploaded a new version of File:Plane.svg current
• 06:0206:02, 5 April 2024 diff hist +2‎ Point ‎No edit summary
• 06:0206:02, 5 April 2024 diff hist 0‎ File:Point.svg ‎ Eric Lengyel uploaded a new version of File:Point.svg current

28 February 2024

• 03:0003:00, 28 February 2024 diff hist 0‎ Reciprocal translation ‎ Undo revision 299 by Eric Lengyel (talk) current Tag: Undo
• 02:5702:57, 28 February 2024 diff hist 0‎ Reciprocal translation ‎ →‎Example Tag: Reverted

9 February 2024

• 01:0301:03, 9 February 2024 diff hist +10‎ Transflection ‎No edit summary current
• 01:0201:02, 9 February 2024 diff hist +2‎ Flector ‎No edit summary
• 01:0101:01, 9 February 2024 diff hist +2‎ Motor ‎No edit summary
• 01:0101:01, 9 February 2024 diff hist +2‎ Line ‎No edit summary
• 01:0001:00, 9 February 2024 diff hist +2‎ Geometric norm ‎ →‎See Also
• 01:0001:00, 9 February 2024 diff hist −13‎ Geometric constraint ‎No edit summary
• 00:5900:59, 9 February 2024 diff hist +2‎ Main Page ‎ →‎Various properties and unary operations
• 00:5900:59, 9 February 2024 diff hist 0‎ m Geometric constraint ‎ Eric Lengyel moved page Geometric property to Geometric constraint without leaving a redirect

8 February 2024

• 10:1810:18, 8 February 2024 diff hist +77‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary current
• 10:1210:12, 8 February 2024 diff hist 0‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 00:5200:52, 8 February 2024 diff hist +77‎ Translation ‎No edit summary current

7 February 2024

• 23:3723:37, 7 February 2024 diff hist −8‎ Rotation ‎No edit summary

1 February 2024

• 06:5106:51, 1 February 2024 diff hist 0‎ Reverses ‎No edit summary

23 January 2024

• 02:1202:12, 23 January 2024 diff hist +2‎ Rigid Geometric Algebra for 2D Space ‎ →‎Geometric Products current
• 02:1102:11, 23 January 2024 diff hist 0‎ File:GeometricAntiproduct201.svg ‎ Eric Lengyel uploaded a new version of File:GeometricAntiproduct201.svg current
• 02:1102:11, 23 January 2024 diff hist 0‎ File:GeometricProduct201.svg ‎ Eric Lengyel uploaded a new version of File:GeometricProduct201.svg current
• 02:1002:10, 23 January 2024 diff hist −168‎ Exterior products ‎No edit summary Tag: Manual revert
• 02:1002:10, 23 January 2024 diff hist 0‎ File:AntiwedgeProduct.svg ‎ Eric Lengyel uploaded a new version of File:AntiwedgeProduct.svg current
• 02:0902:09, 23 January 2024 diff hist 0‎ File:WedgeProduct.svg ‎ Eric Lengyel uploaded a new version of File:WedgeProduct.svg current
• 02:0902:09, 23 January 2024 diff hist 0‎ Main Page ‎ →‎Products and other binary operations
• 02:0802:08, 23 January 2024 diff hist −216‎ Geometric products ‎No edit summary
• 02:0802:08, 23 January 2024 diff hist 0‎ File:GeometricAntiproduct.svg ‎ Eric Lengyel uploaded a new version of File:GeometricAntiproduct.svg current
• 02:0702:07, 23 January 2024 diff hist 0‎ File:GeometricProduct.svg ‎ Eric Lengyel uploaded a new version of File:GeometricProduct.svg current

22 January 2024

• 08:0108:01, 22 January 2024 diff hist 0‎ Reverses ‎No edit summary
• 08:0108:01, 22 January 2024 diff hist 0‎ Complements ‎No edit summary
• 08:0108:01, 22 January 2024 diff hist 0‎ File:Complements.svg ‎ Eric Lengyel uploaded a new version of File:Complements.svg current
• 08:0008:00, 22 January 2024 diff hist 0‎ File:Reverses.svg ‎ Eric Lengyel uploaded a new version of File:Reverses.svg current
• 07:5807:58, 22 January 2024 diff hist +24‎ Main Page ‎ →‎Products and other binary operations
• 07:5707:57, 22 January 2024 diff hist +34‎ N Weight contraction ‎ Created page with "#REDIRECTION Interior products" current
• 07:5707:57, 22 January 2024 diff hist +34‎ N Bulk contraction ‎ Created page with "#REDIRECTION Interior products" current
• 07:5707:57, 22 January 2024 diff hist +34‎ N Bulk expansion ‎ Created page with "#REDIRECTION Interior products" current
• 07:5707:57, 22 January 2024 diff hist +34‎ N Weight expansion ‎ Created page with "#REDIRECTION Interior products" current
• 07:5607:56, 22 January 2024 diff hist +106‎ Interior products ‎No edit summary
• 07:3607:36, 22 January 2024 diff hist 0‎ N File:Expansions.svg ‎No edit summary current
• 07:3507:35, 22 January 2024 diff hist 0‎ N File:Constractions.svg ‎No edit summary current

21 January 2024

• 22:0222:02, 21 January 2024 diff hist −97‎ Dot products ‎No edit summary
• 21:5921:59, 21 January 2024 diff hist 0‎ N File:Dots.svg ‎No edit summary current
• 21:5521:55, 21 January 2024 diff hist +12‎ Geometric products ‎No edit summary
• 21:5421:54, 21 January 2024 diff hist +216‎ Geometric products ‎No edit summary
• 21:5421:54, 21 January 2024 diff hist 0‎ File:GeometricAntiproduct.svg ‎ Eric Lengyel uploaded a new version of File:GeometricAntiproduct.svg
• 21:5421:54, 21 January 2024 diff hist 0‎ File:GeometricProduct.svg ‎ Eric Lengyel uploaded a new version of File:GeometricProduct.svg
• 21:5121:51, 21 January 2024 diff hist +168‎ Exterior products ‎No edit summary Tag: Reverted
• 21:4821:48, 21 January 2024 diff hist 0‎ File:AntiwedgeProduct.svg ‎ Eric Lengyel uploaded a new version of File:AntiwedgeProduct.svg
• 21:4721:47, 21 January 2024 diff hist 0‎ File:WedgeProduct.svg ‎ Eric Lengyel uploaded a new version of File:WedgeProduct.svg
• 21:4321:43, 21 January 2024 diff hist 0‎ File:Basis.svg ‎ Eric Lengyel uploaded a new version of File:Basis.svg current

10 January 2024

• 08:1808:18, 10 January 2024 diff hist 0‎ Rigid Geometric Algebra for 2D Space ‎ →‎Flectors
• 00:4900:49, 10 January 2024 diff hist +71‎ Reverses ‎No edit summary

1 January 2024

• 05:1005:10, 1 January 2024 diff hist +212‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary

2 December 2023

• 08:3008:30, 2 December 2023 diff hist −18‎ Main Page ‎ →‎Products and other binary operations

25 November 2023

• 08:1208:12, 25 November 2023 diff hist +3‎ Magnitude ‎No edit summary current

26 October 2023

• 20:3420:34, 26 October 2023 diff hist +60‎ Join and meet ‎No edit summary current

23 October 2023

• 07:3507:35, 23 October 2023 diff hist +7‎ Dot products ‎No edit summary
• 07:3407:34, 23 October 2023 diff hist 0‎ Main Page ‎No edit summary
• 07:3207:32, 23 October 2023 diff hist 0‎ N File:Plane connect line.svg ‎No edit summary current
• 07:3207:32, 23 October 2023 diff hist 0‎ N File:Line connect point.svg ‎No edit summary current
• 07:3107:31, 23 October 2023 diff hist 0‎ N File:Plane connect point.svg ‎No edit summary current
• 07:3107:31, 23 October 2023 diff hist −12‎ Join and meet ‎No edit summary
• 07:2807:28, 23 October 2023 diff hist +16‎ Join and meet ‎No edit summary
• 07:2407:24, 23 October 2023 diff hist 0‎ Geometric constraint ‎No edit summary
• 07:2307:23, 23 October 2023 diff hist 0‎ Geometric norm ‎No edit summary
• 07:2007:20, 23 October 2023 diff hist 0‎ Attitude ‎No edit summary
• 07:1907:19, 23 October 2023 diff hist 0‎ Unitization ‎No edit summary
• 07:1507:15, 23 October 2023 diff hist −4‎ Bulk and weight ‎No edit summary
• 07:1507:15, 23 October 2023 diff hist +125‎ Bulk and weight ‎No edit summary
• 07:0207:02, 23 October 2023 diff hist 0‎ Complements ‎No edit summary
• 07:0107:01, 23 October 2023 diff hist 0‎ File:Complements.svg ‎ Eric Lengyel uploaded a new version of File:Complements.svg
• 06:5406:54, 23 October 2023 diff hist 0‎ File:Complements.svg ‎ Eric Lengyel uploaded a new version of File:Complements.svg
• 06:0706:07, 23 October 2023 diff hist −3,459‎ Complements ‎No edit summary
• 02:2502:25, 23 October 2023 diff hist +1‎ Projective Geometric Algebra Done Right ‎No edit summary current

19 October 2023

• 06:0906:09, 19 October 2023 diff hist +205‎ Projections ‎No edit summary
• 01:2801:28, 19 October 2023 diff hist −24‎ Main Page ‎ →‎Pages
• 01:2701:27, 19 October 2023 diff hist −1,291‎ Complements ‎No edit summary

2 October 2023

• 19:2719:27, 2 October 2023 diff hist +2‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary

1 October 2023

• 00:3100:31, 1 October 2023 diff hist 0‎ Reciprocal translation ‎ →‎Reciprocal Translation to Horizon

10 September 2023

• 03:4103:41, 10 September 2023 diff hist −1‎ Symmetries in Projective Geometric Algebra ‎ →‎Norms

5 September 2023

• 20:1920:19, 5 September 2023 diff hist +6‎ Translation ‎ →‎See Also
• 20:1920:19, 5 September 2023 diff hist +6‎ Rotation ‎ →‎See Also

27 August 2023

• 00:5600:56, 27 August 2023 diff hist −6‎ Projections ‎No edit summary
• 00:5000:50, 27 August 2023 diff hist −23‎ Projections ‎ →‎See Also
• 00:4900:49, 27 August 2023 diff hist 0‎ N File:Plane onto line.svg ‎No edit summary current
• 00:4900:49, 27 August 2023 diff hist 0‎ N File:Line onto point.svg ‎No edit summary current
• 00:4900:49, 27 August 2023 diff hist 0‎ N File:Plane onto point.svg ‎No edit summary current
• 00:4900:49, 27 August 2023 diff hist 0‎ N File:Line onto plane.svg ‎No edit summary current
• 00:4800:48, 27 August 2023 diff hist 0‎ N File:Point onto line.svg ‎No edit summary current
• 00:4800:48, 27 August 2023 diff hist 0‎ N File:Point onto plane.svg ‎No edit summary current
• 00:4800:48, 27 August 2023 diff hist +30‎ Projections ‎No edit summary

26 August 2023

• 21:5921:59, 26 August 2023 diff hist −750‎ Projections ‎No edit summary
• 20:1920:19, 26 August 2023 diff hist +99‎ Join and meet ‎No edit summary
• 20:1820:18, 26 August 2023 diff hist −41‎ Unitization ‎No edit summary
• 08:4408:44, 26 August 2023 diff hist +33‎ Rigid Geometric Algebra for 2D Space ‎ →‎Join and Meet
• 08:4408:44, 26 August 2023 diff hist +84‎ Rigid Geometric Algebra for 2D Space ‎ →‎Projections
• 08:3308:33, 26 August 2023 diff hist +20‎ Rigid Geometric Algebra for 2D Space ‎ →‎Projections
• 01:1301:13, 26 August 2023 diff hist +24‎ Rigid Geometric Algebra for 2D Space ‎ →‎Projections
• 01:0601:06, 26 August 2023 diff hist −112‎ Rigid Geometric Algebra for 2D Space ‎No edit summary
• 01:0001:00, 26 August 2023 diff hist −132‎ Rigid Geometric Algebra for 2D Space ‎ →‎Projections
• 00:4800:48, 26 August 2023 diff hist +28‎ N Weight norms ‎ Redirected page to Geometric norm current Tag: New redirect
• 00:4700:47, 26 August 2023 diff hist +28‎ N Bulk norms ‎ Redirected page to Geometric norm current Tag: New redirect
• 00:4600:46, 26 August 2023 diff hist −296‎ Rigid Geometric Algebra for 2D Space ‎ →‎Join and Meet
• 00:4300:43, 26 August 2023 diff hist +28‎ N Geometric norms ‎ Redirected page to Geometric norm current Tag: New redirect
• 00:4300:43, 26 August 2023 diff hist 0‎ File:GeometricAntiproduct201.svg ‎ Eric Lengyel uploaded a new version of File:GeometricAntiproduct201.svg
• 00:4300:43, 26 August 2023 diff hist 0‎ File:GeometricProduct201.svg ‎ Eric Lengyel uploaded a new version of File:GeometricProduct201.svg
• 00:4100:41, 26 August 2023 diff hist −168‎ Geometric products ‎No edit summary
• 00:4000:40, 26 August 2023 diff hist 0‎ File:GeometricAntiproduct.svg ‎ Eric Lengyel uploaded a new version of File:GeometricAntiproduct.svg
• 00:4000:40, 26 August 2023 diff hist 0‎ File:GeometricProduct.svg ‎ Eric Lengyel uploaded a new version of File:GeometricProduct.svg
• 00:3400:34, 26 August 2023 diff hist +21‎ Transformation groups ‎No edit summary
• 00:3100:31, 26 August 2023 diff hist 0‎ File:Groups.svg ‎ Eric Lengyel uploaded a new version of File:Groups.svg
• 00:2800:28, 26 August 2023 diff hist +56‎ Reciprocal rotation ‎No edit summary current
• 00:2700:27, 26 August 2023 diff hist +33‎ N Reciprocal rotations ‎ Redirected page to Reciprocal rotation current Tag: New redirect
• 00:2700:27, 26 August 2023 diff hist +36‎ N Reciprocal translations ‎ Redirected page to Reciprocal translation current Tag: New redirect
• 00:2700:27, 26 August 2023 diff hist +60‎ Reciprocal translation ‎No edit summary
• 00:2600:26, 26 August 2023 diff hist 0‎ m Reciprocal translation ‎ Eric Lengyel moved page Dual translation to Reciprocal translation without leaving a redirect
• 00:2500:25, 26 August 2023 diff hist 0‎ m Reciprocal rotation ‎ Eric Lengyel moved page Dual rotation to Reciprocal rotation without leaving a redirect
• 00:2500:25, 26 August 2023 diff hist +54‎ Transformation groups ‎No edit summary
• 00:2000:20, 26 August 2023 diff hist −18‎ Join and meet ‎ →‎See Also
• 00:2000:20, 26 August 2023 diff hist −693‎ Join and meet ‎No edit summary

25 August 2023

• 19:2219:22, 25 August 2023 diff hist −49‎ Geometric constraint ‎No edit summary
• 19:2019:20, 25 August 2023 diff hist −259‎ Geometric norm ‎No edit summary
• 18:2918:29, 25 August 2023 diff hist +49‎ Dot products ‎No edit summary
• 18:2818:28, 25 August 2023 diff hist +195‎ Dot products ‎No edit summary

6 August 2023

• 05:3605:36, 6 August 2023 diff hist 0‎ Motor ‎No edit summary
• 05:3605:36, 6 August 2023 diff hist +4‎ Motor ‎No edit summary
• 05:2605:26, 6 August 2023 diff hist −7‎ Motor ‎No edit summary

5 August 2023

• 05:1705:17, 5 August 2023 diff hist +202‎ Main Page ‎No edit summary
• 05:1505:15, 5 August 2023 diff hist +1‎ Projective Geometric Algebra Done Right ‎No edit summary
• 05:1505:15, 5 August 2023 diff hist +1‎ Symmetries in Projective Geometric Algebra ‎No edit summary
• 05:1505:15, 5 August 2023 diff hist +1‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 05:1405:14, 5 August 2023 diff hist +174‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 05:1205:12, 5 August 2023 diff hist 0‎ N File:Antispace5.svg ‎No edit summary current
• 05:1105:11, 5 August 2023 diff hist +5,183‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 04:5104:51, 5 August 2023 diff hist +3,929‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 04:3804:38, 5 August 2023 diff hist +2,165‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 04:1804:18, 5 August 2023 diff hist +3,260‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 03:5703:57, 5 August 2023 diff hist +15‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 03:5603:56, 5 August 2023 diff hist 0‎ N File:Antispace4.svg ‎No edit summary current
• 03:5603:56, 5 August 2023 diff hist 0‎ N File:Antispace3.svg ‎No edit summary current
• 03:5503:55, 5 August 2023 diff hist 0‎ N File:Antispace2.svg ‎No edit summary current
• 03:5503:55, 5 August 2023 diff hist 0‎ N File:Antispace1.svg ‎No edit summary current
• 03:5303:53, 5 August 2023 diff hist +6,930‎ Space-Antispace Transform Correspondence in Projective Geometric Algebra ‎No edit summary
• 02:1902:19, 5 August 2023 diff hist +1,294‎ N 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..."
• 01:5801:58, 5 August 2023 diff hist +25‎ N Antiprojection ‎ Redirected page to Projections current Tag: New redirect
• 01:5701:57, 5 August 2023 diff hist +67‎ Symmetries in Projective Geometric Algebra ‎No edit summary
• 01:5001:50, 5 August 2023 diff hist 0‎ Projective Geometric Algebra Done Right ‎No edit summary
• 01:4601:46, 5 August 2023 diff hist +22‎ Projective Geometric Algebra Done Right ‎No edit summary
• 01:1701:17, 5 August 2023 diff hist +594‎ Symmetries in Projective Geometric Algebra ‎ →‎Conclusion
• 01:1301:13, 5 August 2023 diff hist +27,118‎ N 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..."

4 August 2023

• 23:3823:38, 4 August 2023 diff hist −51‎ Projective Geometric Algebra Done Right ‎No edit summary
• 23:3723:37, 4 August 2023 diff hist +16,568‎ N 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..."
• 07:4607:46, 4 August 2023 diff hist +236‎ Translation ‎No edit summary

3 August 2023

• 01:3801:38, 3 August 2023 diff hist +2‎ Euclidean distance ‎No edit summary
• 01:3801:38, 3 August 2023 diff hist −17‎ Euclidean distance ‎No edit summary

1 August 2023

• 05:2305:23, 1 August 2023 diff hist −57‎ Euclidean distance ‎No edit summary
• 05:0705:07, 1 August 2023 diff hist +28‎ Magnitude ‎ →‎Examples
• 05:0605:06, 1 August 2023 diff hist +32‎ N Euclidean distances ‎ Redirected page to Euclidean distance current Tag: New redirect
• 05:0605:06, 1 August 2023 diff hist +13‎ Euclidean distance ‎ →‎See Also
• 05:0505:05, 1 August 2023 diff hist +23‎ N Magnitudes ‎ Redirected page to Magnitude current Tag: New redirect
• 05:0405:04, 1 August 2023 diff hist +22‎ N Attitudes ‎ Redirected page to Attitude current Tag: New redirect
• 05:0405:04, 1 August 2023 diff hist −305‎ Euclidean distance ‎No edit summary

15 July 2023

• 08:4808:48, 15 July 2023 diff hist +27‎ N Meet ‎ Redirected page to Join and meet current Tag: New redirect
• 08:4808:48, 15 July 2023 diff hist +27‎ N Join ‎ Redirected page to Join and meet current Tag: New redirect
• 06:5906:59, 15 July 2023 diff hist +27‎ N Transflections ‎ Redirected page to Transflection current Tag: New redirect
• 06:5806:58, 15 July 2023 diff hist +23‎ N Inversions ‎ Redirected page to Inversion current Tag: New redirect
• 06:5706:57, 15 July 2023 diff hist +22‎ N Antireverse ‎ Redirected page to Reverses current Tag: New redirect
• 06:5606:56, 15 July 2023 diff hist +19‎ N Motors ‎ Redirected page to Motor current Tag: New redirect
• 06:5606:56, 15 July 2023 diff hist +21‎ N Flectors ‎ Redirected page to Flector current Tag: New redirect
• 06:5306:53, 15 July 2023 diff hist −2‎ Main Page ‎No edit summary
• 06:5206:52, 15 July 2023 diff hist +31‎ N Interior antiproduct ‎ Redirected page to Interior products current Tag: New redirect
• 06:5106:51, 15 July 2023 diff hist +31‎ N Interior product ‎ Redirected page to Interior products current Tag: New redirect
• 06:5106:51, 15 July 2023 diff hist +26‎ N Antidot product ‎ Redirected page to Dot products current Tag: New redirect
• 06:5106:51, 15 July 2023 diff hist +26‎ N Dot product ‎ Redirected page to Dot products current Tag: New redirect
• 06:5106:51, 15 July 2023 diff hist +25‎ N Commutator ‎ Redirected page to Commutators current Tag: New redirect
• 06:5006:50, 15 July 2023 diff hist +25‎ N Projection ‎ Redirected page to Projections current Tag: New redirect
• 06:5006:50, 15 July 2023 diff hist +31‎ N Antiwedge products ‎ Redirected page to Exterior products current Tag: New redirect
• 06:5006:50, 15 July 2023 diff hist +31‎ N Antiwedge product ‎ Redirected page to Exterior products current Tag: New redirect
• 06:5006:50, 15 July 2023 diff hist +31‎ N Wedge product ‎ Redirected page to Exterior products current Tag: New redirect
• 06:4906:49, 15 July 2023 diff hist +31‎ N Exterior antiproduct ‎ Redirected page to Exterior products current Tag: New redirect
• 06:4906:49, 15 July 2023 diff hist +31‎ N Exterior product ‎ Redirected page to Exterior products current Tag: New redirect
• 06:4906:49, 15 July 2023 diff hist +32‎ N Geometric product ‎ Redirected page to Geometric products current Tag: New redirect
• 06:4906:49, 15 July 2023 diff hist +1‎ Geometric antiproduct ‎ Changed redirect target from Geometric product to Geometric products current Tag: Redirect target changed
• 06:4806:48, 15 July 2023 diff hist +31‎ N Geometric antiproduct ‎ Redirected page to Geometric product Tag: New redirect
• 06:4806:48, 15 July 2023 diff hist +37‎ N Scalar ‎ Redirected page to Scalars and antiscalars current Tag: New redirect
• 06:4806:48, 15 July 2023 diff hist +28‎ N Bulk norm ‎ Redirected page to Geometric norm current Tag: New redirect
• 06:4706:47, 15 July 2023 diff hist +28‎ N Weight norm ‎ Redirected page to Geometric norm current Tag: New redirect
• 06:4706:47, 15 July 2023 diff hist +37‎ N Antiscalar ‎ Redirected page to Scalars and antiscalars current Tag: New redirect
• 06:4706:47, 15 July 2023 diff hist +25‎ N Unitized ‎ Redirected page to Unitization current Tag: New redirect
• 06:4606:46, 15 July 2023 diff hist +29‎ N Weight ‎ Redirected page to Bulk and weight current Tag: New redirect
• 06:4606:46, 15 July 2023 diff hist +29‎ N Bulk ‎ Redirected page to Bulk and weight current Tag: New redirect
• 06:4606:46, 15 July 2023 diff hist +22‎ N Reverse ‎ Redirected page to Reverses current Tag: New redirect
• 06:4506:45, 15 July 2023 diff hist +25‎ N Complement ‎ Redirected page to Complements current Tag: New redirect
• 06:4506:45, 15 July 2023 diff hist +33‎ N Grades ‎ Redirected page to Grade and antigrade current Tag: New redirect
• 06:4506:45, 15 July 2023 diff hist +23‎ N Trivectors ‎ Redirected page to Trivector current Tag: New redirect
• 06:4406:44, 15 July 2023 diff hist +22‎ N Bivectors ‎ Redirected page to Bivector current Tag: New redirect
• 06:4406:44, 15 July 2023 diff hist +25‎ N Translations ‎ Redirected page to Translation current Tag: New redirect
• 06:4406:44, 15 July 2023 diff hist +24‎ N Reflections ‎ Redirected page to Reflection current Tag: New redirect
• 06:4406:44, 15 July 2023 diff hist +22‎ N Rotations ‎ Redirected page to Rotation current Tag: New redirect
• 06:4406:44, 15 July 2023 diff hist +19‎ N Planes ‎ Redirected page to Plane current Tag: New redirect
• 06:4406:44, 15 July 2023 diff hist +18‎ N Lines ‎ Redirected page to Line current Tag: New redirect
• 06:4306:43, 15 July 2023 diff hist +19‎ N Points ‎ Redirected page to Point current Tag: New redirect
• 06:4306:43, 15 July 2023 diff hist +166‎ N 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" current
• 06:4206:42, 15 July 2023 diff hist +166‎ N 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" current
• 06:4206:42, 15 July 2023 diff hist +261‎ N 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" current
• 06:4106:41, 15 July 2023 diff hist +24‎ N Antivectors ‎ Redirected page to Antivector current Tag: New redirect
• 06:4006:40, 15 July 2023 diff hist +162‎ N 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" current
• 06:4006:40, 15 July 2023 diff hist +20‎ N Vectors ‎ Redirected page to Vector current Tag: New redirect
• 06:3906:39, 15 July 2023 diff hist +37‎ N Antiscalars ‎ Redirected page to Scalars and antiscalars current Tag: New redirect
• 06:3906:39, 15 July 2023 diff hist +37‎ N Scalars ‎ Redirected page to Scalars and antiscalars current Tag: New redirect
• 06:3706:37, 15 July 2023 diff hist +33‎ N Antigrade ‎ Redirected page to Grade and antigrade current Tag: New redirect
• 06:3606:36, 15 July 2023 diff hist +33‎ N Grade ‎ Redirected page to Grade and antigrade current Tag: New redirect
• 06:3606:36, 15 July 2023 diff hist +1,113‎ N Commutators ‎ Created page with "In geometric algebra, there are four ''commutator'' products defined as follows. :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_- = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b - \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode{x27D1}}_+ = \dfrac{1}{2}\left(\mathbf a \mathbin{\unicode{x27D1}} \mathbf b + \mathbf b \mathbin{\unicode{x27D1}} \mathbf a\right)$$ :$$[\mathbf a, \mathbf b]^{\Large\unicode..." current
• 06:3506:35, 15 July 2023 diff hist +1,490‎ N Point ‎ Created page with "400px|thumb|right|'''Figure 1.''' A point is the intersection of a 4D vector with the 3D subspace where $$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$$ coordinates, and..."
• 06:3506:35, 15 July 2023 diff hist 0‎ N File:Point.svg ‎No edit summary
• 06:3306:33, 15 July 2023 diff hist +3,246‎ N Duality ‎ Created page with "480px|thumb|right|'''Figure 1.''' The coordinates $$(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)$$-dimensional span of all orthogonal vectors representing a homogeneous plane. Geometrically, these two interpretations are dual to each other, and their distances to the origin are reciprocals of each other. The concept of duality can be understood geometric..."
• 06:3306:33, 15 July 2023 diff hist 0‎ N File:Duality.svg ‎No edit summary
• 06:3006:30, 15 July 2023 diff hist +31‎ N Wedge products ‎ Redirected page to Exterior products current Tag: New redirect
• 06:2906:29, 15 July 2023 diff hist +5,062‎ N Geometric products ‎ Created page with "The ''geometric product'' is the fundamental product of geometric algebra. There are two products with symmetric properties called the geometric product and geometric antiproduct. == Geometric Product == The geometric product between two elements $$\mathbf a$$ and $$\mathbf b$$ has often been written by simply juxtaposing its operands as $$\mathbf{ab}$$ without the use of any infix operator. However, this clearly becomes impractical when both a product and antiproduct..."
• 06:2906:29, 15 July 2023 diff hist 0‎ N File:GeometricAntiproduct.svg ‎No edit summary
• 06:2906:29, 15 July 2023 diff hist 0‎ N File:GeometricProduct.svg ‎No edit summary
• 06:2806:28, 15 July 2023 diff hist +6,603‎ N Exterior products ‎ Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. The exterior product between two elements $$\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..."
• 06:2806:28, 15 July 2023 diff hist 0‎ N File:AntiwedgeProduct.svg ‎No edit summary
• 06:2806:28, 15 July 2023 diff hist 0‎ N File:WedgeProduct.svg ‎No edit summary
• 06:2706:27, 15 July 2023 diff hist +3,406‎ N 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..."
• 06:2606:26, 15 July 2023 diff hist +1,779‎ N 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..."
• 06:2406:24, 15 July 2023 diff hist +24‎ N Quaternions ‎ Redirected page to Quaternion current Tag: New redirect
• 06:2306:23, 15 July 2023 diff hist +3,118‎ N 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 :$$\mathbf{ij} = -\mathbf{ji} = \mathbf k$$ :$$\mathbf{jk} = -\mathbf{kj} = \mathbf i$$ :$$\mathbf{ki} = -\mathbf{..." current
• 06:2006:20, 15 July 2023 diff hist +1,034‎ N Scalars and antiscalars ‎ Created page with "A ''scalar'' in a geometric algebra is an element having grade 0. Scalars are just ordinary real numbers, and they do not involve any basis vectors. The basis element representing the unit scalar is denoted by $$\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$$. An ''antiscalar'..." current
• 06:1606:16, 15 July 2023 diff hist +1,553‎ N Plane ‎ Created page with "400px|thumb|right|'''Figure 1.''' A plane is the intersection of a 4D trivector with the 3D subspace where $$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$$ coordinate, a..."
• 06:1606:16, 15 July 2023 diff hist 0‎ N File:Plane.svg ‎No edit summary
• 06:1506:15, 15 July 2023 diff hist +863‎ N Grade and antigrade ‎ Created page with "The ''grade'' of a basis element in a geometric algebra is equal to the number of basis vectors present in its factorization. An arbitrary element whose components all have the same grade is also said to have that grade. The ''antigrade'' of a basis element is equal to the number of basis vectors absent from its factorization. The grade of an element $$\mathbf x$$ is denoted by $$\operatorname{gr}(\mathbf x)$$, and the antigrade is denoted by $$\operatorname{ag}(\mathb..."
• 06:1406:14, 15 July 2023 diff hist +2,029‎ N 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 $$-\math..."
• 06:1406:14, 15 July 2023 diff hist 0‎ N File:Reverses.svg ‎No edit summary
• 06:1106:11, 15 July 2023 diff hist +7,517‎ N Projections ‎ Created page with "Projections and antiprojections of one geometric object onto another can be accomplished using interior products as described below. The formulas on this page are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. == Projection == The geometric projection of an object $$\mathbf x$$ onto an object $$\mathbf y$$ is given by the general formula $$(\mathbf y_\unicode{x25CB} \mathbin{\unicode{..."
• 06:1006:10, 15 July 2023 diff hist +3,533‎ N 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..."
• 06:1006:10, 15 July 2023 diff hist 0‎ N File:Distance line line.svg ‎No edit summary current
• 06:1006:10, 15 July 2023 diff hist 0‎ N File:Distance point plane.svg ‎No edit summary current
• 06:0906:09, 15 July 2023 diff hist 0‎ N File:Distance point line.svg ‎No edit summary current
• 06:0906:09, 15 July 2023 diff hist 0‎ N File:Distance point point.svg ‎No edit summary current
• 06:0606:06, 15 July 2023 diff hist +12,199‎ N 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..."
• 06:0606:06, 15 July 2023 diff hist 0‎ N File:Improper isom.svg ‎No edit summary
• 06:0406:04, 15 July 2023 diff hist 0‎ N File:Complements.svg ‎No edit summary
• 06:0406:04, 15 July 2023 diff hist +7,055‎ N 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 :$$\underline{\mathbf x} \wedge \mathbf x = {\larg..."
• 06:0306:03, 15 July 2023 diff hist +2,788‎ N Geometric constraint ‎ Created page with "An element $$\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{\..."
• 06:0306:03, 15 July 2023 diff hist +2,586‎ N 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 :$$\mathbf{\hat x} = \dfrac{\mathbf x}{\left\Vert\mathbf x\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf x}{\sqrt{\mathbf x \mathbin{\unicode{x25CB}} \smash{\ma..."
• 06:0206:02, 15 July 2023 diff hist +10,552‎ N Geometric norm ‎ Created page with "The ''geometric norm'' is a measure of the magnitude of an element. It has two components called the bulk norm and the weight norm. For points, lines, and planes, the geometric norm is equal to the shortest Euclidean distance between the geometry and the origin. For motors and flectors, the geometric norm is equal to half the distance that the origin is moved by the isometry operator. == Bulk Norm == The ''bulk norm'' of an element $$\mathbf x$$, d..."
• 05:5905:59, 15 July 2023 diff hist +5,971‎ N Reciprocal rotation ‎ Created page with "A ''dual rotation'' is a proper isometry of dual Euclidean space. For a bulk normalized line $$\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$$ and its bulk complement..."
• 05:5905:59, 15 July 2023 diff hist 0‎ N File:DualRotation.svg ‎No edit summary current
• 05:5905:59, 15 July 2023 diff hist 0‎ N File:Rotation.svg ‎No edit summary current
• 05:5905:59, 15 July 2023 diff hist +2,936‎ N Reciprocal translation ‎ Created page with "__NOTOC__ A ''dual translation'' is a proper isometry of dual Euclidean space. The specific kind of dual motor :$$\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}$$ . == Example == The left image below shows the flow field in the ''x''-''z'' plane for the translation $..."
• 05:5805:58, 15 July 2023 diff hist 0‎ N File:DualTranslation.svg ‎No edit summary current
• 05:5805:58, 15 July 2023 diff hist 0‎ N File:Translation.svg ‎No edit summary current
• 05:5705:57, 15 July 2023 diff hist +2,900‎ N Translation ‎ Created page with "__NOTOC__ A ''translation'' is a proper isometry of Euclidean space. The specific kind of motor :$$\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$$. === Trans..."
• 05:5705:57, 15 July 2023 diff hist +4,695‎ N Rotation ‎ Created page with "A ''rotation'' is a proper isometry of Euclidean space. For a unitized line $$\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..."
• 05:5605:56, 15 July 2023 diff hist +3,321‎ N 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"..." current
• 05:5505:55, 15 July 2023 diff hist +2,029‎ N 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 |-..." current
• 05:5405:54, 15 July 2023 diff hist +4,564‎ N 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 :$$\mathbf F = F_{px} \mathbf e_{1} + F_{py} \mathbf e_{2} + F_{pz} \math..."
• 05:5205:52, 15 July 2023 diff hist 0‎ N File:Groups.svg ‎No edit summary
• 05:5205:52, 15 July 2023 diff hist +6,554‎ N Transformation groups ‎ Created page with "In the 4D rigid geometric algebra $$\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_..."
• 05:5005:50, 15 July 2023 diff hist +1,064‎ N 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. ===..."
• 05:4605:46, 15 July 2023 diff hist 0‎ N File:GeometricAntiproduct201.svg ‎No edit summary
• 05:4605:46, 15 July 2023 diff hist 0‎ N File:GeometricProduct201.svg ‎No edit summary
• 05:4605:46, 15 July 2023 diff hist 0‎ N File:Unary201.svg ‎No edit summary current
• 05:4605:46, 15 July 2023 diff hist 0‎ N File:Basis201.svg ‎No edit summary current
• 05:4605:46, 15 July 2023 diff hist +17,445‎ N 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 $$\mathbf e_1$$, $$\mathbf e_..."
• 05:3905:39, 15 July 2023 diff hist 0‎ N File:Line meet plane.svg ‎No edit summary current
• 05:3905:39, 15 July 2023 diff hist 0‎ N File:Plane meet plane.svg ‎No edit summary current
• 05:3905:39, 15 July 2023 diff hist 0‎ N File:Line join point.svg ‎No edit summary current
• 05:3905:39, 15 July 2023 diff hist 0‎ N File:Point join point.svg ‎No edit summary current
• 05:3805:38, 15 July 2023 diff hist +5,522‎ N Join and meet ‎ Created page with "The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection. The points, lines, and planes appearing in the following tables are defined as follows: :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ :$$\mathbf..."
• 05:3405:34, 15 July 2023 diff hist 0‎ N File:Skew lines.svg ‎No edit summary current
• 05:3405:34, 15 July 2023 diff hist 0‎ N File:Line infinity.svg ‎No edit summary
• 05:3405:34, 15 July 2023 diff hist 0‎ N File:Line.svg ‎No edit summary
• 05:3405:34, 15 July 2023 diff hist +4,278‎ N 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..."
• 05:3305:33, 15 July 2023 diff hist +695‎ Bulk and weight ‎No edit summary
• 05:3105:31, 15 July 2023 diff hist +3,283‎ N 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 $$\mathcal..."
• 05:2905:29, 15 July 2023 diff hist +1,645‎ N Attitude ‎ Created page with "The attitude function, denoted by $$\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}$$ . The attitude of a line is the line's direction as a vector, and the attitude of a plane is the plane's normal as a bivector. The following table lists the attitude for the main types in the 4D rigid geometric algebra..."
• 05:2705:27, 15 July 2023 diff hist 0‎ N File:Proper isom.svg ‎No edit summary
• 05:2705:27, 15 July 2023 diff hist +21,325‎ N Motor ‎ Created page with "400px|thumb|right|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\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..."
• 05:2405:24, 15 July 2023 diff hist 0‎ N File:Basis.svg ‎No edit summary
• 05:2305:23, 15 July 2023 diff hist +7,232‎ Main Page ‎No edit summary