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Rotations in 3D using Geometric Algebra
Kusal de Alwis Mentor: Laura Iosip Directed Reading Program, Spring 2019
Rotations in 3D using Geometric Algebra Kusal de Alwis Mentor: - - PowerPoint PPT Presentation
Rotations in 3D using Geometric Algebra Kusal de Alwis Mentor: - - PowerPoint PPT Presentation
Rotations in 3D using Geometric Algebra Kusal de Alwis Mentor: Laura Iosip Directed Reading Program, Spring 2019 The Problem Overview Prior Methods for Rotations What is the Geometric Algebra? Rotating with Geometric Algebra
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Overview
Prior Methods for Rotations What is the Geometric Algebra? Rotating with Geometric Algebra Further Applications
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Prior Methods
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Rotation Matrix - 2D
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Rotation Matrix - 3D
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Rotation Matrix - 3D
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Quaternions – 3D
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Quaternions – 3D
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Quaternions – 3D
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Octonions – 4D
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The Geometric Algebra
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Extension of Rn
Vectors and Scalars Same old algebra
Scalar Multiplication, Addition
Only one augmentation…
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The Geometric Product
Inner Product
Standard Dot Product
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The Geometric Product
Inner Product
Standard Dot Product
Exterior Product
Another multiplication scheme Represents planes
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The Geometric Product
Inner Product
Standard Dot Product
Exterior Product
Another multiplication scheme Represents planes
Geometric Product *Only for vectors
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Implications of the Geometric Product
𝑓"𝑓# = 𝑓" % 𝑓# + 𝑓" ∧ 𝑓# = 0 + 𝑓" ∧ 𝑓# = 𝑓" ∧ 𝑓#
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Implications of the Geometric Product
𝑓"𝑓# = 𝑓" % 𝑓# + 𝑓" ∧ 𝑓# = 0 + 𝑓" ∧ 𝑓# = 𝑓" ∧ 𝑓#
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Implications of the Geometric Product
𝑓"𝑓# = 𝑓" % 𝑓# + 𝑓" ∧ 𝑓# = 0 + 𝑓" ∧ 𝑓# = 𝑓" ∧ 𝑓# 𝑓" 𝑓" + 𝑓# = 𝑓"𝑓" + 𝑓"𝑓# = (𝑓" % 𝑓" + 𝑓" ∧ 𝑓") + (𝑓" % 𝑓# + 𝑓" ∧ 𝑓#) = 1 + 𝑓" ∧ 𝑓#
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Multivectors in G3
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Multivectors in G3
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Multivectors in G3
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Multivectors in G3
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Multivectors in G3
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Basis of Gn
Rn
𝑓" 𝑓# 𝑓, 𝑓- … 𝑓. n-dimensional space
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Basis of Gn
Rn
𝑓" 𝑓# 𝑓, 𝑓- … 𝑓. n-dimensional space
Gn
1 𝑓", 𝑓#, 𝑓,, … , 𝑓. 𝑓" ∧ 𝑓#, 𝑓" ∧ 𝑓,, … 𝑓" ∧ 𝑓., 𝑓# ∧ 𝑓,, 𝑓# ∧ 𝑓-, … 𝑓# ∧ 𝑓. … 𝑓.1" ∧ 𝑓. 𝑓" ∧ 𝑓# ∧ 𝑓,, 𝑓" ∧ 𝑓# ∧ 𝑓-, … … 𝑓" ∧ 𝑓# ∧ 𝑓, ∧ 𝑓- ∧ ⋯ ∧ 𝑓.1" ∧ 𝑓. 2n-dimensional space
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Rotations in Gn
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Projections & Rejections
Unit a normal to a plane, Arbitrary v
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Projections & Rejections
Unit a normal to a plane, Arbitrary v Projection
𝑤4 = 𝑏 % 𝑤 𝑏
Rejection
𝑤∥ = 𝑤 − 𝑤4 = 𝑤 − 𝑏 % 𝑤 𝑏
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Reflection
Reflect v on plane perpendicular to a 𝑤 = 𝑤∥ + 𝑤4
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Reflection
Reflect v on plane perpendicular to a 𝑤 = 𝑤∥ + 𝑤4 𝑆9 𝑤 = 𝑤∥ − 𝑤4
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Reflection
Reflect v on plane perpendicular to a 𝑤 = 𝑤∥ + 𝑤4 𝑆9 𝑤 = 𝑤∥ − 𝑤4 𝑏 % 𝑤 = 1 2 (𝑏𝑤 + 𝑤𝑏)
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Reflection
Reflect v on plane perpendicular to a 𝑤 = 𝑤∥ + 𝑤4 𝑆9 𝑤 = 𝑤∥ − 𝑤4 = 𝑤 − 𝑏 % 𝑤 𝑏 − 𝑏 % 𝑤 𝑏 = 𝑤 − 2 𝑏 % 𝑤 𝑏 = 𝑤 − 2 1 2 𝑏𝑤 + 𝑤𝑏 𝑏 = 𝑤 − 𝑏𝑤𝑏 − 𝑤𝑏# = 𝑤 − 𝑏𝑤𝑏 − 𝑤 = −𝑏𝑤𝑏 𝑏 % 𝑤 = 1 2 (𝑏𝑤 + 𝑤𝑏)
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Rotation via Double Reflection
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Rotation via Double Reflection
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Rotation via Double Reflection
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Rotation via Double Reflection
𝑆𝑝𝑢#=(𝑤) = 𝑆> 𝑆9 𝑤 = 𝑆> −𝑏𝑤𝑏 = −𝑐 −𝑏𝑤𝑏 𝑐 = 𝑐𝑏𝑤𝑏𝑐
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Rotation via Double Reflection
𝑆𝑝𝑢#=(𝑤) = 𝑆> 𝑆9 𝑤 = 𝑆> −𝑏𝑤𝑏 = −𝑐 −𝑏𝑤𝑏 𝑐 = 𝑐𝑏𝑤𝑏𝑐 𝑐𝑏 𝑏𝑐 = 𝑐 𝑏𝑏 𝑐 = 𝑐𝑐 = 1, 𝑐𝑏 = (𝑏𝑐)1"
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Rotation via Double Reflection
𝑆𝑝𝑢#=(𝑤) = 𝑆> 𝑆9 𝑤 = 𝑆> −𝑏𝑤𝑏 = −𝑐 −𝑏𝑤𝑏 𝑐 = 𝑐𝑏𝑤𝑏𝑐 𝑐𝑏 𝑏𝑐 = 𝑐 𝑏𝑏 𝑐 = 𝑐𝑐 = 1, 𝑐𝑏 = (𝑏𝑐)1" 𝑆𝑝𝑢#=(𝑤) = (𝑏𝑐)1"𝑤𝑏𝑐
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Rotors
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓AB
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Rotors
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓AB
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Rotors
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓AB 𝑏𝑐 = 𝑏 𝑐 𝑓AB = 𝑓=B
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Rotors
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓AB 𝑏𝑐 = 𝑏 𝑐 𝑓AB = 𝑓=B 𝑆𝑝𝑢#= 𝑤 = 𝑏𝑐 1"𝑤𝑏𝑐 = 𝑓1=B𝑤𝑓=B
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Rotors
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓AB 𝑏𝑐 = 𝑏 𝑐 𝑓AB = 𝑓=B 𝑆𝑝𝑢#= 𝑤 = 𝑏𝑐 1"𝑤𝑏𝑐 = 𝑓1=B𝑤𝑓=B 𝑆𝑝𝑢A,B 𝑤 = 𝑓1A
#B𝑤𝑓 A #B
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Relating back
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑥 + 𝑦𝑓" ∧ 𝑓# + 𝑧𝑓# ∧ 𝑓, + 𝑨𝑓" ∧ 𝑓,
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Relating back
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 𝑟 = 𝑥 + 𝑦𝑗 + 𝑧𝑘 + 𝑨𝑙 = 𝑥 + 𝑦𝑓" ∧ 𝑓# + 𝑧𝑓# ∧ 𝑓, + 𝑨𝑓" ∧ 𝑓,
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Relating back
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 𝑟 = 𝑥 + 𝑦𝑗 + 𝑧𝑘 + 𝑨𝑙 𝑓" ∧ 𝑓# = 𝑗 𝑓# ∧ 𝑓, = 𝑘 𝑓" ∧ 𝑓, = 𝑙 = 𝑥 + 𝑦𝑓" ∧ 𝑓# + 𝑧𝑓# ∧ 𝑓, + 𝑨𝑓" ∧ 𝑓,
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Relating back
𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 𝑟 = 𝑥 + 𝑦𝑗 + 𝑧𝑘 + 𝑨𝑙 𝑓" ∧ 𝑓# = 𝑗 𝑓# ∧ 𝑓, = 𝑘 𝑓" ∧ 𝑓, = 𝑙
𝑅𝑣𝑏𝑢𝑓𝑠𝑜𝑗𝑝𝑜𝑡 ⊂ 𝐻𝑓𝑝𝑛𝑓𝑢𝑠𝑗𝑑 𝐵𝑚𝑓𝑐𝑠𝑏
= 𝑥 + 𝑦𝑓" ∧ 𝑓# + 𝑧𝑓# ∧ 𝑓, + 𝑨𝑓" ∧ 𝑓,
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Further Applications
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Linear Algebra – System of Equations
𝑦" 𝑧" 𝑦# 𝑧# 𝛽 𝛾 = 𝑐" 𝑐#
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Linear Algebra – System of Equations
𝛽𝑦 + 𝛾𝑧 = 𝑐 𝑦" 𝑧" 𝑦# 𝑧# 𝛽 𝛾 = 𝑐" 𝑐#
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Linear Algebra – System of Equations
𝛽𝑦 + 𝛾𝑧 = 𝑐
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Linear Algebra – System of Equations
𝛽𝑦 + 𝛾𝑧 = 𝑐 𝛽𝑦 ∧ 𝑧 + 𝛾𝑧 ∧ 𝑧 = 𝑐 ∧ 𝑧
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Linear Algebra – System of Equations
𝛽𝑦 + 𝛾𝑧 = 𝑐 𝛽𝑦 ∧ 𝑧 + 𝛾𝑧 ∧ 𝑧 = 𝑐 ∧ 𝑧 𝑧 ∧ 𝑧 = 0 𝛽𝑦 ∧ 𝑧 = 𝑐 ∧ 𝑧
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Linear Algebra – System of Equations
𝛽𝑦 + 𝛾𝑧 = 𝑐 𝛽𝑦 ∧ 𝑧 + 𝛾𝑧 ∧ 𝑧 = 𝑐 ∧ 𝑧 𝑧 ∧ 𝑧 = 0 𝛽𝑦 ∧ 𝑧 = 𝑐 ∧ 𝑧 𝛽 =
>∧X Y∧X
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Linear Algebra – System of Equations
𝛽𝑦 + 𝛾𝑧 = 𝑐 𝛽𝑦 ∧ 𝑧 + 𝛾𝑧 ∧ 𝑧 = 𝑐 ∧ 𝑧 𝑧 ∧ 𝑧 = 0 𝛽𝑦 ∧ 𝑧 = 𝑐 ∧ 𝑧 𝛽 =
>∧X Y∧X
𝛾 =
>∧Y X∧Y = Y∧> Y∧X
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Linear Algebra – System of Equations
𝑦 ∧ 𝑧 = 𝑦" 𝑧" 𝑦# 𝑧#
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Linear Algebra – System of Equations
𝑦 ∧ 𝑧 = 𝑦" 𝑧" 𝑦# 𝑧# 𝛽 = 𝑐 ∧ 𝑧 𝑦 ∧ 𝑧 = 𝑐" 𝑧" 𝑐# 𝑧# 𝑦" 𝑧" 𝑦# 𝑧#
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Other Use Cases
Geometric Calculus
Calculus in Gn
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Other Use Cases
Geometric Calculus
Calculus in Gn
Homogeneous Geometric Algebra
Represent objects not centered at the origin Projective Geometry
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Other Use Cases
Geometric Calculus
Calculus in Gn
Homogeneous Geometric Algebra
Represent objects not centered at the origin Projective Geometry
Conformal Geometric Algebra
Better representations of points, lines, spheres, etc. Generalized operations for transformations
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