Rotations in 3D using Geometric Algebra
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Rotations in 3D using Geometric Algebra Kusal de Alwis Mentor: Laura Iosip Directed Reading Program, Spring 2019
Transcript of Rotations in 3D using Geometric Algebra
Rotations in 3D using Geometric Algebra
Kusal de Alwis
Mentor: Laura Iosip
Directed Reading Program, Spring 2019
The Problem
Overview
οΏ½ Prior Methods for Rotations
οΏ½ What is the Geometric Algebra?
οΏ½ Rotating with Geometric Algebra
οΏ½ Further Applications
Prior Methods
Rotation Matrix - 2D
Rotation Matrix - 3D
Rotation Matrix - 3D
Quaternions β 3D
Quaternions β 3D
Quaternions β 3D
Octonions β 4D
The Geometric Algebra
Extension of Rn
οΏ½ Vectors and Scalars
οΏ½ Same old algebraοΏ½ Scalar Multiplication, Addition
οΏ½ Only one augmentationβ¦
The Geometric Product
οΏ½ Inner ProductοΏ½ Standard Dot Product
The Geometric Product
οΏ½ Inner ProductοΏ½ Standard Dot Product
οΏ½ Exterior ProductοΏ½ Another multiplication scheme
οΏ½ Represents planes
The Geometric Product
οΏ½ Inner ProductοΏ½ Standard Dot Product
οΏ½ Exterior ProductοΏ½ Another multiplication scheme
οΏ½ Represents planes
οΏ½ Geometric Product
*Only for vectors
Implications of the Geometric Product
π"π# = π" % π# + π" β§ π# = 0 + π" β§ π# = π" β§ π#
Implications of the Geometric Product
π"π# = π" % π# + π" β§ π# = 0 + π" β§ π# = π" β§ π#
Implications of the Geometric Product
π"π# = π" % π# + π" β§ π# = 0 + π" β§ π# = π" β§ π#
π" π" + π# = π"π" + π"π# = (π" % π" + π" β§ π") + (π" % π# + π" β§ π#) = 1 + π" β§ π#
Multivectors in G3
Multivectors in G3
Multivectors in G3
Multivectors in G3
Multivectors in G3
Basis of Gn
Rn
οΏ½ π"οΏ½ π#οΏ½ π,οΏ½ π-οΏ½ β¦
οΏ½ π.
οΏ½ n-dimensional space
Basis of Gn
Rn
οΏ½ π"οΏ½ π#οΏ½ π,οΏ½ π-οΏ½ β¦
οΏ½ π.
οΏ½ n-dimensional space
Gn
οΏ½ 1
οΏ½ π", π#, π,, β¦ , π.οΏ½ π" β§ π#, π" β§ π,, β¦ π" β§ π., π# β§ π,, π# β§ π-, β¦ π# β§ π. β¦π.1" β§ π.οΏ½ π" β§ π# β§ π,, π" β§ π# β§ π-, β¦
οΏ½ β¦
οΏ½ π" β§ π# β§ π, β§ π- β§ β―β§ π.1" β§ π.
οΏ½ 2n-dimensional space
Rotations in Gn
Projections & Rejections
οΏ½ Unit a normal to a plane, Arbitrary v
Projections & Rejections
οΏ½ Unit a normal to a plane, Arbitrary v
οΏ½ ProjectionοΏ½ π£4 = π % π£ π
οΏ½ RejectionοΏ½ π£β₯ = π£ β π£4 = π£ β π % π£ π
Reflection
οΏ½ Reflect v on plane perpendicular to a
οΏ½ π£ = π£β₯ + π£4
Reflection
οΏ½ Reflect v on plane perpendicular to a
οΏ½ π£ = π£β₯ + π£4οΏ½ π 9 π£ = π£β₯ β π£4
Reflection
οΏ½ Reflect v on plane perpendicular to a
οΏ½ π£ = π£β₯ + π£4οΏ½ π 9 π£ = π£β₯ β π£4 π % π£ =
12 (ππ£ + π£π)
Reflection
οΏ½ Reflect v on plane perpendicular to a
οΏ½ π£ = π£β₯ + π£4οΏ½ π 9 π£ = π£β₯ β π£4
= π£ β π % π£ π β π % π£ π
= π£ β 2 π % π£ π
= π£ β 212 ππ£ + π£π π = π£ β ππ£π β π£π# = π£ β ππ£π β π£
= βππ£π
π % π£ =12 (ππ£ + π£π)
Rotation via Double Reflection
Rotation via Double Reflection
Rotation via Double Reflection
Rotation via Double Reflection
π ππ‘#=(π£) = π > π 9 π£ = π > βππ£π = βπ βππ£π π = πππ£ππ
Rotation via Double Reflection
π ππ‘#=(π£) = π > π 9 π£ = π > βππ£π = βπ βππ£π π = πππ£ππ
ππ ππ = π ππ π = ππ = 1, ππ = (ππ)1"
Rotation via Double Reflection
π ππ‘#=(π£) = π > π 9 π£ = π > βππ£π = βπ βππ£π π = πππ£ππ
ππ ππ = π ππ π = ππ = 1, ππ = (ππ)1"
π ππ‘#=(π£) = (ππ)1"π£ππ
Rotors
π = π’π£ = π’ % π£ + π’ β§ π£ = π’ π£ πAB
Rotors
π = π’π£ = π’ % π£ + π’ β§ π£ = π’ π£ πAB
Rotors
π = π’π£ = π’ % π£ + π’ β§ π£ = π’ π£ πABππ = π π πAB = π=B
Rotors
π = π’π£ = π’ % π£ + π’ β§ π£ = π’ π£ πAB
ππ = π π πAB = π=B
π ππ‘#= π£ = ππ 1"π£ππ = π1=Bπ£π=B
Rotors
π = π’π£ = π’ % π£ + π’ β§ π£ = π’ π£ πAB
ππ = π π πAB = π=B
π ππ‘#= π£ = ππ 1"π£ππ = π1=Bπ£π=B
π ππ‘A,B π£ = π1A#Bπ£π
A#B
Relating back
π = π’π£ = π’ % π£ + π’ β§ π£= π€ + π₯π" β§ π# + π¦π# β§ π, + π§π" β§ π,
Relating back
π = π’π£ = π’ % π£ + π’ β§ π£ π = π€ + π₯π + π¦π + π§π= π€ + π₯π" β§ π# + π¦π# β§ π, + π§π" β§ π,
Relating back
π = π’π£ = π’ % π£ + π’ β§ π£ π = π€ + π₯π + π¦π + π§π
π" β§ π# = π π# β§ π, = π π" β§ π, = π
= π€ + π₯π" β§ π# + π¦π# β§ π, + π§π" β§ π,
Relating back
π = π’π£ = π’ % π£ + π’ β§ π£ π = π€ + π₯π + π¦π + π§π
π" β§ π# = π π# β§ π, = π π" β§ π, = π
ππ’ππ‘πππππππ β πΊπππππ‘πππ π΄ππππππ
= π€ + π₯π" β§ π# + π¦π# β§ π, + π§π" β§ π,
Further Applications
Linear Algebra β System of Equations
π₯" π¦"π₯# π¦#
πΌπ½ = π"
π#
Linear Algebra β System of Equations
πΌπ₯ + π½π¦ = ππ₯" π¦"π₯# π¦#
πΌπ½ = π"
π#
Linear Algebra β System of Equations
πΌπ₯ + π½π¦ = π
Linear Algebra β System of Equations
πΌπ₯ + π½π¦ = π
πΌπ₯ β§ π¦ + π½π¦ β§ π¦ = π β§ π¦
Linear Algebra β System of Equations
πΌπ₯ + π½π¦ = π
πΌπ₯ β§ π¦ + π½π¦ β§ π¦ = π β§ π¦
π¦ β§ π¦ = 0πΌπ₯ β§ π¦ = π β§ π¦
Linear Algebra β System of Equations
πΌπ₯ + π½π¦ = π
πΌπ₯ β§ π¦ + π½π¦ β§ π¦ = π β§ π¦
π¦ β§ π¦ = 0πΌπ₯ β§ π¦ = π β§ π¦
πΌ = >β§XYβ§X
Linear Algebra β System of Equations
πΌπ₯ + π½π¦ = π
πΌπ₯ β§ π¦ + π½π¦ β§ π¦ = π β§ π¦
π¦ β§ π¦ = 0πΌπ₯ β§ π¦ = π β§ π¦
πΌ = >β§XYβ§X
π½ = >β§YXβ§Y
= Yβ§>Yβ§X
Linear Algebra β System of Equations
π₯ β§ π¦ =π₯" π¦"π₯# π¦#
Linear Algebra β System of Equations
π₯ β§ π¦ =π₯" π¦"π₯# π¦#
πΌ =π β§ π¦π₯ β§ π¦
=
π" π¦"π# π¦#π₯" π¦"π₯# π¦#
Other Use Cases
οΏ½ Geometric CalculusοΏ½ Calculus in Gn
Other Use Cases
οΏ½ Geometric CalculusοΏ½ Calculus in Gn
οΏ½ Homogeneous Geometric AlgebraοΏ½ Represent objects not centered at the originοΏ½ Projective Geometry
Other Use Cases
οΏ½ Geometric CalculusοΏ½ Calculus in Gn
οΏ½ Homogeneous Geometric AlgebraοΏ½ Represent objects not centered at the originοΏ½ Projective Geometry
οΏ½ Conformal Geometric AlgebraοΏ½ Better representations of points, lines, spheres, etc.οΏ½ Generalized operations for transformations
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