Projective Geometric Algebra: A Swiss army knife for doing Cayley-Klein geometry Charles Gunn Sept. 18, 2019 at ICERM, Providence Full-featured slides available at: https://slides.com/skydog23/icerm2019. Check for updates incorporating new ideas inspired by giving the talk. This first slide will indicate whether update has occurred. 1 . 1
What is Cayley-Klein What is Cayley-Klein geometry? geometry? Example : Given a conic section in . R P 2 Q For two points and "inside" , define Q x y d ( a , b ) = log ( CR ( f , f ; x , y )) + − where are the intersections of the line , f f + − with and CR is the cross ratio. Q xy CR is invariant under projectivities d is a distance function and the white ⇒ region is a model for hyperbolic plane . H 2 1 . 2
What is Cayley-Klein What is Cayley-Klein geometry? geometry? SIGNATURE of Quadratic Form Example : (+ + −0) = (2, 1, 1) e ⋅ = ⋅ = +1, e ⋅ = −1, e ⋅ = 0, e ⋅ = 0 for i = 0 e e 1 e 2 e 3 e i e j 0 1 2 3 j 1 . 3
What is Cayley-Klein What is Cayley-Klein geometry? geometry? Signature of Q Space Symbol κ n n +1 elliptic ( n + 1, 0, 0) Ell , S H n -1 hyperbolic ( n , 1, 0) E n 0 euclidean "( n , 0, 1)" 1 . 4
3D Examples 3D Examples The Sudanese Moebius band in discovered by Sue Goodman and Dan S 3 Asimov, visualized in UNC-CH Graphics Lab, 1979. 1 . 5
3D Examples 3D Examples Tessellation of with regular right-angled dodecahedra H 3 (from "Not Knot", Geometry Center, 1993). 1 . 6
3D Examples 3D Examples The 120-cell, a tessellation of the 3-sphere S 3 (PORTAL VR, TU-Berlin, 18.09.09) 1 . 7
Cayley-Klein geometries for Cayley-Klein geometries for n = 2 = 2 Name elliptic euclidean hyperbolic signature (3,0,0) "(2,0,1)" (2,1,0) =0 =0 null points 2 2 z 2 2 2 z 2 x + y + x + y − 1 . 8
Cayley-Klein geometries for Cayley-Klein geometries for n = 2 = 2 Name elliptic euclidean hyperbolic signature (3,0,0) "(2,0,1)" (2,1,0) =0 =0 =0 null points 2 2 z 2 z 2 2 2 z 2 x + y + x + y − 1 . 8
Example Example Cayley-Klein geometries for Cayley-Klein geometries for n = 2 = 2 Name elliptic euclidean hyperbolic signature (3,0,0) "(2,0,1)" (2,1,0) =0 null points 2 2 z 2 =0 =0 x + y + z 2 2 2 z 2 x + y − =0 =0 =0 null lines* 2 2 c 2 2 b 2 2 2 c 2 a + b + a + a + b − *The line has line coordinates . ax + by + cz = 0 ( a , b , c ) 1 . 9
Question Question What is the best way What is the best way to do Cayley-Klein geometry to do Cayley-Klein geometry on the computer? on the computer? 2 . 1
Question Question What is the best way What is the best way to do Cayley-Klein geometry to do Cayley-Klein geometry on the computer? on the computer? 1993 2 . 1
Question Question What is the best way What is the best way to do Cayley-Klein geometry to do Cayley-Klein geometry on the computer? on the computer? 1993 2019 2 . 1
Vector + linear algebra Vector + linear algebra 2 . 2
Vector + linear algebra Vector + linear algebra Projective points Projective matrices 2 . 2
Vector + linear algebra Vector + linear algebra Projective points Projective matrices 2 . 3
Vector + linear algebra Vector + linear algebra Projective points Projective matrices But it's 2019 now. Can we do better? 2 . 3
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list 2 . 4
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list Coordinate-free 2 . 4
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list Uniform rep'n for points, lines, and planes Coordinate-free 2 . 4
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list Uniform rep'n for points, lines, and planes Parallel-safe meet and join operators Coordinate-free 2 . 4
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list Uniform rep'n for points, lines, and planes Parallel-safe meet and join operators Single, uniform rep'n for isometries Coordinate-free 2 . 4
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list Uniform rep'n for points, lines, and planes Parallel-safe meet and join operators Single, uniform rep'n for isometries Compact expressions for classical geometric results Coordinate-free 2 . 4
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list Uniform rep'n for points, lines, and planes Parallel-safe meet and join operators Single, uniform rep'n for isometries Compact expressions for classical geometric results Physics-ready Coordinate-free 2 . 4
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list Uniform rep'n for points, lines, and planes Parallel-safe meet and join operators Single, uniform rep'n for isometries Compact expressions for classical geometric results Physics-ready Coordinate-free Metric-neutral 2 . 4
Cayley-Klein programmer's wish list Cayley-Klein programmer's wish list Uniform rep'n for points, lines, and planes Parallel-safe meet and join operators Single, uniform rep'n for isometries Compact expressions for classical geometric results Physics-ready Backwards compatible Coordinate-free Metric-neutral 2 . 4
Partial solutions: Quaternions (1843) Partial solutions: Quaternions (1843) A 4D algebra generated by units satisfying: {1, i , j , k } 2 2 2 2 1, i = j = k = 1 = −1 ij = − ji , ... 3 . 1
Quaternions Quaternions Quaternions H s + x i + y j + z k Im. quaternions IH v := x i + y j + z k ⇔ ( x , y , z ) ∈ R 3 Unit quaternions U { g ∈ H ∣ g g = 1} 3 . 2
Quaternions Quaternions Quaternions H s + x i + y j + z k Im. quaternions IH v := x i + y j + z k ⇔ ( x , y , z ) ∈ R 3 Unit quaternions U { g ∈ H ∣ g g = 1} I. Geometric product : v v − v 1 v v v = ⋅ + × 1 2 2 1 2 3 . 3
Quaternions Quaternions Quaternions H s + x i + y j + z k Im. quaternions IH v := x i + y j + z k ⇔ ( x , y , z ) ∈ R 3 Unit quaternions U { g ∈ H ∣ g g = 1} I. Geometric product : v v − v 1 v v v = ⋅ + × 1 2 2 1 2 cross product inner product 3 . 3
Quaternions Quaternions Quaternions H s + x i + y j + z k Im. quaternions IH v := x i + y j + z k ⇔ ( x , y , z ) ∈ R 3 Unit quaternions U { g ∈ H ∣ g g = 1} II. Rotations via sandwiches : 1. For , there exists so that g ∈ U x ∈ IH g = cos( t ) + sin( t ) x = e t x 2. For any , the "sandwich" v ∈ IH (≅ R ) 3 gv g rotates around the axis by an angle . v x 2 t 3. Comparison to matrices. 3 . 4
Quaternions Quaternions Quaternions H s + x i + y j + z k Im. quaternions IH v := x i + y j + z k ⇔ ( x , y , z ) ∈ R 3 Unit quaternions U { g ∈ H ∣ g g = 1} Advantages Advantages I. Geometric product II. Rotations as sandwiches Disadvantages Disadvantages I. Only applies to points/vectors II. Special case R 3 3 . 5
Partial solutions: Grassmann algebra Partial solutions: Grassmann algebra Hermann Grassmann (1809-1877) Ausdehnungslehre (1844) 4 . 1
Grassmann algebra Grassmann algebra R P 2 ∗ The wedge ( ) product in and R P 2 ∧ 4 . 2
Grassmann algebra Grassmann algebra 4 . 3
Grassmann algebra Grassmann algebra Standard projective is join x ∧ y yields R P 2 ⋀ 4 . 3
Grassmann algebra Grassmann algebra Standard projective Dual projective is join is meet x ∧ y x ∧ y 2 ∗ yields yields R P R P 2 ⋀ ⋀ 4 . 3
Grassmann algebra Grassmann algebra The dual projective 2 ∗ Grassmann algebra R P ⋀ Grade Sym Generators Dim. Type ⋀ 0 0 1 1 Scalar ⋀ 1 1 { e , e , e 3 Line } 0 1 2 ⋀ 2 { E e e 2 = ∧ } 3 Point i j k ⋀ 3 3 I = e e e 1 Pseudoscalar ∧ ∧ 0 1 2 4 . 4
Grassmann algebra Grassmann algebra The dual projective 2 ∗ Grassmann algebra R P ⋀ Grade Sym Generators Dim. Type ⋀ 0 0 1 1 Scalar ⋀ 1 1 { e , e , e 3 Line } 0 1 2 ⋀ 2 { E e e 2 = ∧ } 3 Point i j k ⋀ 3 3 I = e e e 1 Pseudoscalar ∧ ∧ 0 1 2 n ∗ We will be using for the rest of the talk. R P ⋀ 4 . 4
Grassmann algebra Grassmann algebra The wedge ( ) product in R P 2 ∧ Properties of ∧ 1. Antisymmetric : For 1-vectors : x , y x ∧ y = − y ∧ x x ∧ x = 0 2. Subspace lattice : For linearly independent subspaces x ∈ ⋀ m x ∧ y ∈ ⋀ k + m , is the subspace spanned by and ⋀ k , y ∈ x y otherwise it's zero. Note: The regressive (join) product is also available. ∨ (Then it's called a Grassmann-Cayley algebra.) 4 . 5
Grassmann algebra Grassmann algebra Note: spanning subspace means different things in standard and dual setting. In 3D: Standard : a line is the Dual : a line is the subspace spanned by subspace spanned by two points. two planes. e Plane pencil g n a r t n i o P S Axis p e a r 4 . 6
Grassmann algebra Grassmann algebra Advantages Advantages 1. Points, lines, and planes are equal citizens. 2. "Parallel-safe" meet and join operators since projective. Disadvantages Disadvantages 1. Only incidence (projective), no metric. 4 . 7
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