Summer School: Mathematical Methods in Robotics

Transcription

1 Summer School: Mathematical Methods in Robotics Part IV: Projective Geometry Harald Löwe TU Braunschweig, Institute Computational Mathematics 2009/07/16 Löwe (TU Braunschweig) Math. robotics 2009/07/16 1 / 27

2 Projective Spaces and Homogeneous Coordinates Projective Spaces Definition The projective n space P n is the set of all 1 dimensional subspaces of R n+1. P n = Grass(1, n + 1) is a Grassmannian and, thus, is a homogenous manifold P n = SO(n + 1)/ O(n). Löwe (TU Braunschweig) Math. robotics 2009/07/16 2 / 27

3 Projective Spaces and Homogeneous Coordinates Geometric Objects Elements of P n are called points. A 2 dimensional subspace of R n+1 is called a line of P n. A point p P is on a line L iff p is a subspace of L. A plane of P n is a 3 dimensional subspace. An (n 1) dimensional subspace of P n is called a hyperplane. The join of two geometric objects X and Y of P n is defined as XY := X + Y. The intersection of two geometric objects X and Y of P n is defined as X Y. Join and intersection of geometric objects are geometric objects again. Löwe (TU Braunschweig) Math. robotics 2009/07/16 3 / 27

4 Projective Spaces and Homogeneous Coordinates Homogeneous Coordinates Every point p P n is a 1 dimensional subspace of R n+1. Choose a basis of R n+1 and write x 0 x 0 x 1 x 1 p =. x n := R. x n Notice: The x i s depend on the basis. homogeneous coordinates Löwe (TU Braunschweig) Math. robotics 2009/07/16 4 / 27

5 Projective Spaces and Homogeneous Coordinates Coordinates for Lines Let L be a line of P n, i.e. a 2 dimensional subspace of R n+1. Choose a basis {x, y} of L. W.r.t. a basis of R n+1, we may write L = [x y] = x 0 y 0 x 1 y 1. x n. y n. Löwe (TU Braunschweig) Math. robotics 2009/07/16 5 / 27

6 Projective Spaces and Homogeneous Coordinates Notice that [x y] = [x + λy y] for all λ. Using Gauß algorithm, we obtain e.g. x 0 y x 1 y x 2 y 2 = x 2 y 2 ; elements.... ξ η ξx 2 + ηy 2.. x n y n x n y n ξx n + ηy n This subspace is the graph of the linear function ( ) x 2 y 2 ( f L : R 2 R n 1 ξ ξ ; η.. η x n y n ) Löwe (TU Braunschweig) Math. robotics 2009/07/16 6 / 27

7 Projective Spaces and Homogeneous Coordinates Thus, locally at the line E 01 = span{e 0, e 1 }, we have a chart x 2 y 2 x 2 y 2.. R (n 1) 2... x n y n x n These are the Grassmann coordinates. The chart region equals the set of lines with trivial intersection with span{e 2,..., e n }. y n Löwe (TU Braunschweig) Math. robotics 2009/07/16 7 / 27

8 Embedding of the Euclidian Space Euclidian Spaces Let E n = R n be the n dimensional Euclidian space with its usual geometric objects (points, lines,... ). Embed E n into P n as usual: (We are writing row vectors now.) E P; x [1, x]. This embedding depends on the basis of R n+1! The image of a euclidian line u + Rv is contained in the projective line [ ] 1 u. 0 v Löwe (TU Braunschweig) Math. robotics 2009/07/16 8 / 27

9 Embedding of the Euclidian Space Löwe (TU Braunschweig) Math. robotics 2009/07/16 9 / 27

10 Embedding of the Euclidian Space Circles We start with the circle C : ξ1 2 + ξ2 2 = 1 in E2 (which is a quadric). Embedding E 2 into P 2 yields C : [1 ξ 1 ξ 2 ] with ξ1 2 + ξ2 2 = 1. In homogeneous coordinates, this equation reads x1 2 + x 2 2 = x 0 2. Choose e 0 = e 0 + e 2, e 1 = e 1, e 2 = e 0 e 2 as new basis. From (x 0 x 2 )(x 0 + x 2 ) + x1 2 = 0 obtain the equation x 2 1 4x 0 x 2 = 0. Go back to E 2 (equation x 0 = 1): C : ξ 2 = ξ1 2 /4. This is a parabola. Thus, a parabola is a projective circle with one point removed. Löwe (TU Braunschweig) Math. robotics 2009/07/16 10 / 27

11 Projective Transformations Projective Transformations Definition Let A GL(n + 1). Then the bijective map P n P n ; [x] [A x] is called projective transformation of P n. This yields an action of GL(n + 1) on P n. Two matrices A and B belong to the same projective transformation iff B = t A, t 0. Therefore, we may use homogeneous coordinates for matrices and put PGL(n) := {[A] A GL(n + 1)}. Löwe (TU Braunschweig) Math. robotics 2009/07/16 11 / 27

12 Projective Transformations Translations Look at the projective transformation x 0 a 1 0 : x 1 b 0 1 x 2 x 0 x 1 + x 0 a x 2 + x 0 b This corresponds to translations in E 2 (defined by x 0 = 1). Löwe (TU Braunschweig) Math. robotics 2009/07/16 12 / 27

13 Projective Transformations The Main Theorem of Projective Geometry Theorem Let ϕ : P n P n (n 2) be a bijective map which maps lines onto lines. Then there exists a matrix A GL(n + 1) such that ϕ([x]) = [A] [x] for all [x] P n. Löwe (TU Braunschweig) Math. robotics 2009/07/16 13 / 27

14 Projective Transformations Action of PGL(n) on the lines Let L be a line of P n, say L = [x y] for some x, y R n+1. If [A] PGL(n), then [A] maps the points [x] and [y] to [Ax] and [Ay], respectively. We derive that [A](L) = [Ax Ay]. Generalize this to higher dimensional geometric objects like planes or hyperplanes of P n. Löwe (TU Braunschweig) Math. robotics 2009/07/16 14 / 27

15 Plücker Coordinates Lines of E 3 Ususally, we describe a line L in space by a position vector p and a vector v in direction of L: L = p + R v. We may replace v by λv (i.e. v is a homogeneous vector ). We may replace p by p + µv but not by a multiple of p. Thus, p and v are different objects. Löwe (TU Braunschweig) Math. robotics 2009/07/16 15 / 27

16 Plücker Coordinates Lines of P 3 Try to obtain a better representation of lines in the projective 3 space. Consider a line [ x0 x L := 1 x 2 x 3 y 0 y 1 y 2 y 3 Since x and y are linearly independent, the rank of the matrix equals 2. Therefore, two of the column vectors are linearly independent, i.e. not all determinants ( ) ( ) ( ) x0 x l 01 = det 1 x0 x, l y 0 y 02 = det 2 x0 x ; l 1 y 0 y 03 = det 3 2 y 0 y 3 ( ) ( ) ( ) x2 x l 23 = det 3 x3 x, l y 2 y 31 = det 1 x1 x ; l 3 y 3 y 12 = det 2 1 y 1 y 2 vanish. ]. Löwe (TU Braunschweig) Math. robotics 2009/07/16 16 / 27

17 Plücker Coordinates Not all determinants ( ) xi x l ij = det j, (i, j) = (0, 1), (0, 2), (0, 3), (2, 3), (3, 1), (1, 2) vanish. y i It follows that y j γ(l) := [l 01, l 02, l 03, l 23, l 31, l 12 ] is an element of the projective 5 space P 5. Check that γ(l) doesn t depend on the particular choice of a basis {x, y} of L. Therefore, we obtain a map γ : L P 5, where L denotes the set of lines of P 3. This map is called Plücker map or Klein map. Löwe (TU Braunschweig) Math. robotics 2009/07/16 17 / 27

18 Plücker Coordinates Affine Lines Let L : p + R v be a line of E 3. Then L corresponds to the projective line [ ] 1 p1 p L = 2 p 3 0 v 1 v 2 v 3 We derive that its Plücker image equals γ(l) = [v 1, v 2, v 3, p 2 v 3 p 3 v 2, p 3 v 1 p 1 v 3, p 1 v 2 p 2 v 2 ]. Thus, γ(l) = [v, p v]. Löwe (TU Braunschweig) Math. robotics 2009/07/16 18 / 27

19 Plücker Coordinates Affine Lines Let L be an affine line and let x, y be two points on L. Then L corresponds to the projective line [ ] [ 1 x1 x L = 2 x 3 1 x1 x = 2 x 3 1 y 1 y 2 y 3 0 y 1 x 1 y 2 x 2 y 3 x 3 ] We derive that its Plücker image equals γ(l) = [y x, x (y x)] = [y x, x y]. Löwe (TU Braunschweig) Math. robotics 2009/07/16 19 / 27

20 Plücker Coordinates Lines At Infinity Let L be a line at infinity, i.e. a projective line which is contained in x 0 = 0. Then [ ] 0 x1 x L = 2 x 3 0 y 1 y 2 y 3 We determine the Plücker coordinates of L: γ(l) = [0, x y]. Löwe (TU Braunschweig) Math. robotics 2009/07/16 20 / 27

21 Plücker Coordinates An affine line L = p + Rv has Plücker coordinates γ(l) = [v, p v] = [x, y]. We compute the scalar product x y = v p v = 0. If L is a line at infinity, then its Plücker coordinates γ(l) = [x, y] satisfy x = 0 and, thus, x y = 0. Theorem If L is a line of P 3, then its Plücker coordinates γ(l) = [x, y] satisfy x y = x T y = 0. Löwe (TU Braunschweig) Math. robotics 2009/07/16 21 / 27

22 Plücker Coordinates The Klein Quadrik Definition The Klein quadric M 4 2 is the set of all points [x, y] P5 which fulfill x T y = x 1 y 1 + x 2 y 2 + x 3 y 3 = 0. M 4 2 is a projective quadric, i.e. a set of projective points satisfying some homogeneous quadratic equation. The Plücker map γ assigns to every line L L a point γ(l) M 4 2 on the Klein quadric. Löwe (TU Braunschweig) Math. robotics 2009/07/16 22 / 27

23 Plücker Coordinates Theorem γ : L M 4 2 is bijective, i.e. 1 Different lines have different Plücker coordinates. 2 Every point on the Klein quadric is a Plücker coordinate of some line. The points [0, y] correspond to a line at infinity [ ] 0 a L = with a b = y. 0 b We derive that L = γ 1 ([0, y]) = y. Computing γ 1 ([x, y]), x 0, x T y = 0, means to solve v = x and p v = y. This implies γ 1 ([x, y]) = x y x 2 + R x. Löwe (TU Braunschweig) Math. robotics 2009/07/16 23 / 27

24 Plücker Coordinates The Action of PGL(3) If L L contains x and y, then its Plücker coordinates are ( ) xi x l ij = det j = x i y j x j y i. Consider an element [a kl ] PGL(3). The new coordinates are x i = k a ikx k and y j = l a jly l. Consequently, the new Plücker coordinates are l y i y j ij = x i y j x j y i ( ) ( ) ( ) ( ) = a ik x k a jl y l a jl x l a ik y k = k,l k a ik a jl (x k y l x l y k ) = k,l l l a ik a jl l kl. k Löwe (TU Braunschweig) Math. robotics 2009/07/16 24 / 27

25 Plücker Coordinates The transformation [a kl ] of P 3 induces a linear transformation of Plücker coordinates: l ij = ( ) xk x a ik a jl l kl where l kl = det l. y k y l k,l Note: l lk = l kl, l kk = 0 l ij = (a i0 a j1 a i1 a j0 )l 01 + (a i0 a j2 a i2 a j0 )l 02 + (a i0 a j3 a i3 a j0 )l (a i2 a j3 a i3 a j2 )l 23 + (a i3 a j1 a i1 a j3 )l 31 + (a i1 a j2 a i2 a j1 )l 12 ( ) ai0 a = det i1 l a j0 a j1 We can read the (6 6) matrix C directly from the equations. Note that C is regular (all possible Plücker coordinats are linear combinations of the columns and there are 6 linearly independent Plücker coordinates). Löwe (TU Braunschweig) Math. robotics 2009/07/16 25 / 27

26 Plücker Coordinates Some Results Consider two elements L 1 = [x, y], L 2 = [u, v] M 4 2. L 1, L 2 generate a 1 dimensional projective subspace G of P 5. We have that L 1 L 2 G M 4 2. Moreover, if L 1, L 2 are skew lines, then G M 4 2 = {L 1, L 2 }. In particular, a flat line pencil is represented by a 1 dimensional projective subspace which is completely contained in M 4 2. Let E P 5 be a 2 dimensional projective subspace. Then E M 4 2 holds if and only if either E is a line pencil in P 3 or is the set of all lines in a plane of P 3. If E M 4 2 doesn t contain any 1 dimensional subspaces of P5, then E is a regulus. Löwe (TU Braunschweig) Math. robotics 2009/07/16 26 / 27

27 Plücker Coordinates We remark that a point [p 0 p] is an element of the line with Plücker coordinates [x, y] if and only if p T y = 0 and p o y + p x = 0. Moreover, a line with Plücker coordinates [x, y] is contained in the plane [u 0, u] if and only if u T x = 0 and u 0 x + u y = 0. Löwe (TU Braunschweig) Math. robotics 2009/07/16 27 / 27