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 š Prior Methods for Rotations š What is the Geometric Algebra? š Rotating with Geometric Algebra š Further Applications

Prior Methods

Rotation Matrix - 2D

Rotation Matrix - 3D

Rotation Matrix - 3D

Quaternions – 3D

Quaternions – 3D

Quaternions – 3D

Octonions – 4D

The Geometric Algebra

Extension of R n š Vectors and Scalars š Same old algebra š Scalar Multiplication, Addition š Only one augmentation…

The Geometric Product š Inner Product š Standard Dot Product

The Geometric Product š Inner Product š Standard Dot Product š Exterior Product š Another multiplication scheme š Represents planes

The Geometric Product š Inner Product š Standard Dot Product š Exterior Product š Another multiplication scheme š Represents planes š Geometric Product *Only for vectors

Implications of the Geometric Product 𝑓 " 𝑓 # = 𝑓 " % 𝑓 # + 𝑓 " ∧ 𝑓 # = 0 + 𝑓 " ∧ 𝑓 # = 𝑓 " ∧ 𝑓 #

Implications of the Geometric Product 𝑓 " 𝑓 # = 𝑓 " % 𝑓 # + 𝑓 " ∧ 𝑓 # = 0 + 𝑓 " ∧ 𝑓 # = 𝑓 " ∧ 𝑓 #

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 #

Multivectors in G 3

Multivectors in G 3

Multivectors in G 3

Multivectors in G 3

Multivectors in G 3

Basis of G n R n š 𝑓 " š 𝑓 # š 𝑓 , š 𝑓 - š … š 𝑓 . š n -dimensional space

Basis of G n R n G n š 𝑓 " š 1 š 𝑓 # š 𝑓 " , 𝑓 # , 𝑓 , , … , 𝑓 . š 𝑓 , š 𝑓 " ∧ 𝑓 # , 𝑓 " ∧ 𝑓 , , … 𝑓 " ∧ 𝑓 . , 𝑓 # ∧ 𝑓 , , 𝑓 # ∧ 𝑓 - , … 𝑓 # ∧ 𝑓 . … 𝑓 .1" ∧ 𝑓 . š 𝑓 - š 𝑓 " ∧ 𝑓 # ∧ 𝑓 , , 𝑓 " ∧ 𝑓 # ∧ 𝑓 - , … š … š … š 𝑓 . š 𝑓 " ∧ 𝑓 # ∧ 𝑓 , ∧ 𝑓 - ∧ ⋯ ∧ 𝑓 .1" ∧ 𝑓 . š n -dimensional space š 2 n -dimensional space

Rotations in G n

Projections & Rejections š Unit a normal to a plane, Arbitrary v

Projections & Rejections š Unit a normal to a plane, Arbitrary v š Projection š 𝑤 4 = 𝑏 % 𝑤 𝑏 š Rejection š 𝑤 ∥ = 𝑤 − 𝑤 4 = 𝑤 − 𝑏 % 𝑤 𝑏

Reflection š Reflect v on plane perpendicular to a š 𝑤 = 𝑤 ∥ + 𝑤 4

Reflection š Reflect v on plane perpendicular to a š 𝑤 = 𝑤 ∥ + 𝑤 4 š 𝑆 9 𝑤 = 𝑤 ∥ − 𝑤 4

Reflection š Reflect v on plane perpendicular to a š 𝑤 = 𝑤 ∥ + 𝑤 4 𝑏 % 𝑤 = 1 2 (𝑏𝑤 + 𝑤𝑏) š 𝑆 9 𝑤 = 𝑤 ∥ − 𝑤 4

Reflection š Reflect v on plane perpendicular to a š 𝑤 = 𝑤 ∥ + 𝑤 4 𝑏 % 𝑤 = 1 š 𝑆 9 𝑤 = 𝑤 ∥ − 𝑤 4 2 (𝑏𝑤 + 𝑤𝑏) = 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

Rotation via Double Reflection

Rotation via Double Reflection

Rotation via Double Reflection

Rotation via Double Reflection 𝑆𝑝𝑢 #= (𝑤) = 𝑆 > 𝑆 9 𝑤 = 𝑆 > −𝑏𝑤𝑏 = −𝑐 −𝑏𝑤𝑏 𝑐 = 𝑐𝑏𝑤𝑏𝑐

Rotation via Double Reflection 𝑆𝑝𝑢 #= (𝑤) = 𝑆 > 𝑆 9 𝑤 = 𝑆 > −𝑏𝑤𝑏 = −𝑐 −𝑏𝑤𝑏 𝑐 = 𝑐𝑏𝑤𝑏𝑐 𝑏𝑐 = 𝑐 𝑏𝑏 𝑐 = 𝑐𝑐 = 1, 𝑐𝑏 = (𝑏𝑐) 1" 𝑐𝑏

Rotation via Double Reflection 𝑆𝑝𝑢 #= (𝑤) = 𝑆 > 𝑆 9 𝑤 = 𝑆 > −𝑏𝑤𝑏 = −𝑐 −𝑏𝑤𝑏 𝑐 = 𝑐𝑏𝑤𝑏𝑐 𝑏𝑐 = 𝑐 𝑏𝑏 𝑐 = 𝑐𝑐 = 1, 𝑐𝑏 = (𝑏𝑐) 1" 𝑐𝑏 𝑆𝑝𝑢 #= (𝑤) = (𝑏𝑐) 1" 𝑤𝑏𝑐

Rotors 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓 AB

Rotors 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓 AB

Rotors 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓 AB 𝑏𝑐 = 𝑏 𝑐 𝑓 AB = 𝑓 =B

Rotors 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓 AB 𝑏𝑐 = 𝑏 𝑐 𝑓 AB = 𝑓 =B 𝑆𝑝𝑢 #= 𝑤 = 𝑏𝑐 1" 𝑤𝑏𝑐 = 𝑓 1=B 𝑤𝑓 =B

Rotors 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑣 𝑤 𝑓 AB 𝑏𝑐 = 𝑏 𝑐 𝑓 AB = 𝑓 =B 𝑆𝑝𝑢 #= 𝑤 = 𝑏𝑐 1" 𝑤𝑏𝑐 = 𝑓 1=B 𝑤𝑓 =B 𝑆𝑝𝑢 A,B 𝑤 = 𝑓 1A A #B 𝑤𝑓 #B

Relating back 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 = 𝑥 + 𝑦𝑓 " ∧ 𝑓 # + 𝑧𝑓 # ∧ 𝑓 , + 𝑨𝑓 " ∧ 𝑓 ,

Relating back 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 𝑟 = 𝑥 + 𝑦𝑗 + 𝑧𝑘 + 𝑨𝑙 = 𝑥 + 𝑦𝑓 " ∧ 𝑓 # + 𝑧𝑓 # ∧ 𝑓 , + 𝑨𝑓 " ∧ 𝑓 ,

Relating back 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 𝑟 = 𝑥 + 𝑦𝑗 + 𝑧𝑘 + 𝑨𝑙 = 𝑥 + 𝑦𝑓 " ∧ 𝑓 # + 𝑧𝑓 # ∧ 𝑓 , + 𝑨𝑓 " ∧ 𝑓 , 𝑓 " ∧ 𝑓 # = 𝑗 𝑓 # ∧ 𝑓 , = 𝑘 𝑓 " ∧ 𝑓 , = 𝑙

Relating back 𝑆 = 𝑣𝑤 = 𝑣 % 𝑤 + 𝑣 ∧ 𝑤 𝑟 = 𝑥 + 𝑦𝑗 + 𝑧𝑘 + 𝑨𝑙 = 𝑥 + 𝑦𝑓 " ∧ 𝑓 # + 𝑧𝑓 # ∧ 𝑓 , + 𝑨𝑓 " ∧ 𝑓 , 𝑓 " ∧ 𝑓 # = 𝑗 𝑓 # ∧ 𝑓 , = 𝑘 𝑓 " ∧ 𝑓 , = 𝑙 𝑅𝑣𝑏𝑢𝑓𝑠𝑜𝑗𝑝𝑜𝑡 ⊂ 𝐻𝑓𝑝𝑛𝑓𝑢𝑠𝑗𝑑 𝐵𝑚𝑕𝑓𝑐𝑠𝑏

Further Applications

Linear Algebra – System of Equations 𝑦 " 𝑧 " 𝛽 𝛾 = 𝑐 " 𝑦 # 𝑧 # 𝑐 #

Linear Algebra – System of Equations 𝑦 " 𝑧 " 𝛽 𝛾 = 𝑐 " 𝛽𝑦 + 𝛾𝑧 = 𝑐 𝑦 # 𝑧 # 𝑐 #

Linear Algebra – System of Equations 𝛽𝑦 + 𝛾𝑧 = 𝑐

Linear Algebra – System of Equations 𝛽𝑦 + 𝛾𝑧 = 𝑐 𝛽𝑦 ∧ 𝑧 + 𝛾𝑧 ∧ 𝑧 = 𝑐 ∧ 𝑧

Linear Algebra – System of Equations 𝛽𝑦 + 𝛾𝑧 = 𝑐 𝛽𝑦 ∧ 𝑧 + 𝛾𝑧 ∧ 𝑧 = 𝑐 ∧ 𝑧 𝑧 ∧ 𝑧 = 0 𝛽𝑦 ∧ 𝑧 = 𝑐 ∧ 𝑧

Linear Algebra – System of Equations 𝛽𝑦 + 𝛾𝑧 = 𝑐 𝛽𝑦 ∧ 𝑧 + 𝛾𝑧 ∧ 𝑧 = 𝑐 ∧ 𝑧 𝑧 ∧ 𝑧 = 0 𝛽𝑦 ∧ 𝑧 = 𝑐 ∧ 𝑧 >∧X 𝛽 = Y∧X

Linear Algebra – System of Equations 𝛽𝑦 + 𝛾𝑧 = 𝑐 𝛽𝑦 ∧ 𝑧 + 𝛾𝑧 ∧ 𝑧 = 𝑐 ∧ 𝑧 𝑧 ∧ 𝑧 = 0 𝛽𝑦 ∧ 𝑧 = 𝑐 ∧ 𝑧 >∧X >∧Y Y∧> 𝛽 = 𝛾 = X∧Y = Y∧X Y∧X

Linear Algebra – System of Equations 𝑦 " 𝑧 " 𝑦 ∧ 𝑧 = 𝑦 # 𝑧 #

Linear Algebra – System of Equations 𝑦 " 𝑧 " 𝑦 ∧ 𝑧 = 𝑦 # 𝑧 # 𝑐 " 𝑧 " 𝛽 = 𝑐 ∧ 𝑧 𝑐 # 𝑧 # 𝑦 ∧ 𝑧 = 𝑦 " 𝑧 " 𝑦 # 𝑧 #

Other Use Cases š Geometric Calculus š Calculus in G n

Other Use Cases š Geometric Calculus š Calculus in G n š Homogeneous Geometric Algebra š Represent objects not centered at the origin š Projective Geometry

Other Use Cases š Geometric Calculus š Calculus in G n š Homogeneous Geometric Algebra š Represent objects not centered at the origin š Projective Geometry š Conformal Geometric Algebra š Better representations of points, lines, spheres, etc. š Generalized operations for transformations