The 2D Projective Plane

1. The 2D Projective Plane Points and Lines

2. The 2D projective plane A point in a plane is represented by (x, y) in R2 (x, y) is also a vector Homogeneous representation point ( x1, x2 , x3 ) x= x1/ x3 y = x2/ x3 One representation of a point is ( x, y, 1)

3. Lines and points Line ax + by +c = 1 • ( x, y, 1) ( a, b, c)T =0 • xTl = 0 • l = (a, b, c)T l’ = (a’, b’, c’)T Define the vector x = l x l’ Intersection of 2 lines Where x is a cross product

4. Cross products • From the identity l. (l x l’) = l’. (l x l’) = 0 lTx = l’Tx = 0 • Thus x is a point lying in both l and l’ which is the intersection of two lines • Line joining 2 points l = x x x’

5. Ideal points and line at infinity 1 • Intersection of parallel lines l = (a, b, c) and l’ =(a, b, c’) are parallel l x l’ = (c’ – c)( b, -a, 0)T • Ignoring scale, it is a point ( b, -a, 0)T with inhomogeneous coordinates (b/0, -a/0)T It is a point at infinity where the lines meet. It is not a finite point in R2

6. Ideal points and line at infinity 2 • The points with last coordinate x3 = 0 are known as ideal points, or points at infinity. Note that this set lies on a single line , the line at infinity linf =( 0, 0, 1)T as ( 0,0,1) (x1 , x2 , 0) = 0 A line l =( a, b, c)T intersect the line at infinity in the ideal point ( b, -a, 0)T (b, -a) T represent the line’s direction

7. Ideal points and line at infinity 3 • The line at infinity can be thought of as the set of directions of all lines in the plane. • One may argument R2 by adding points with the last coordinate x3 = 0, • The resulting space is p2 is named the projective space • The geometry of p2 is known as projective geometry

8. A model of projective plane • A fruitful way of thinking : p2 is a set of rays in R3. Each vector k( x1, x2, x3)T as k varies forms a ray pass through the origin. A ray can be thought of a point in p2 In this model , the lines in p2 are planes passing through the origin

9. Points and lines of p2 are represented as rays and planes through the origin

10. A model of the projective plane • Points and lines of p2 are represented by rays and planes respectively, through the origin in R3. • x1x2 - plane represents the line at infinity linfLines in x1x2 plane represent ideal points

11. Projective plane model 2 • Two non-identical rays lie on exactly one plane, and any two planes intersect at one ray. • This is analogue of two distinct points define a line and two lines meet always intersect at a point. • Points and lines may be obtained by intersecting rays and planes by the plane x3 = 1 • The rays representing ideal points and the plane representing linfinity are parallel to the plane x3 =1

12. Duality Principle • To any theorem of 2 –dimensional projective geometry, there corresponds a dual theorem, which may be derived by interchanging the roles of points and lines in the original theorem

13. Conics and Dual conics • A conic is a curve described by a second-degree equation in the plane. • In Euclidean geometry, conics are of 3 main types: hyperbola, ellipse and parabola • In 2D projective geometry all non-degenerate conics are equivalent under projective transformations

14. Equation of conics • In inhomogeneous coordinates, a conic is given by • ax2 + bxy + cy2 + dx + ey + f = 0 • Let x = x1/x3 , y = x1/x3 , Then • ax12 + bx1x2 + cx12 + dx1x3 +ex2x3 +fx32 =0

15. Conics • In matrix form • xTC x = 0 where • C =

16. Five point define a conic • C is symmetric and only the ratios of the matrix elements are important. C has 5 degrees of freedom Put x3 =1. If C passes through xi, yi, , then ( xi2 xi yi yi2 xi yi 1) c =0 where c = ( a, b, c, d, e, f)T is the conic represented as a 6-vector

17. Constraints from 5 points c = 0

18. Tangent lines to conics • The line l tangent to a conic C at a point x on C is given by l = Cx

19. Dual Conics • There exist a dual conic C* which defines an equation on lines. • A line l tangent to the conic C satisfies • lT C*l = 0 • C* is the adjoint matrix of C • For a non-singular symmetric matrix C* = C-1

20. Degenerated conics • If C is not of full rank, then the conic is termed degnerate. Degenerate point conics include 2 lines ( rank 2) and a repeated line (rank1) • C = l mT + m lT • Is composed of two lines l and m

21. Projective transformations • A projectivity is an invertible mapping h from p2 to itself such that three points x1, x2 and x3 lie on the same line if and only if h (x1), h(x2) and h( x3) do. Projectivities form a group. Projectivity is also called A projective transformation or a homography

22. Projective transformations 2 • A mapping h :p2  p2 is a projectivity if and only if there exists a non-singular 3 x 3 matrix H such that for any point in p2represented by a vector x is true that h(x) = H x. Any point in p2 is represented as a homogenous 3-vector x, and Hx is a linear mapping of the homogeneous coordinates

23. Central projection maps points to points and lines to lines

24. Definition • A planar projective transform is represented by a non-singular 3x3 matrix H • where x’ = H x

25. Removing the projective distortion from the perspective image of a plane • Compute projective transform from point correspondence: x’=Hx

26. Four point correspondences lead to 8 such linear equations on H • Four points gives 8 constraints are used to computing H. • The points must be in general position: no 3 points are collinear. • x= H-1x’ Remove projective distortion

27. Three remarks on computing H • Knowledge of camera parameters or the pose of plane are not required • 4 points are not always necessary. Alternative approach requires less information. • Superior methods of computing projective transform are described later

28. Removing the projective distortion from a perspective image of a plane • H between image and plane can be computed without knowing the camera parameters

29. The projective transformation between two images induced by a world plane

30. The projective transformation between two images with the same camera centre

31. The projective transformation the image of a plane and the image of its shadow onto another plane

32. Transformation of lines • Under a point transformation x’ = H x, • Given xi lies on l , x’ will lie on a transformed line l’ =H-T l • In this way, l’T xi’ = lT H-1Hxi =0

33. Transformation of conics x = H-1 x’ xT =x’T H-T xTCx = x’T H-TC H-1 x’ = x’T C’x’ A conic C transform to C’= H-T C H-1 A dual conic transform to C*’ = HC*HT

34. Distortion arising under central projection Images of a tiled floor: Similarity, Affine, Projective

35. A hierarchy of transformations • Class 1: Isometries • An isometry is represented as where e = 1 or -1

36. Euclidean transformation • If e = 1, the isometry is orientation- preserving and is an Euclidean transformation If e = -1, the isometry reverse orientation HE is a planar Euclidean transformation with R a 2x2 rotation matrix

37. Groups and orientation • An isometry is orientation-preserving if R has determinant 1. The isometries form a group. • This also applies to the cases of • Similarity and affine transformation

38. Class ll: Similarity transformations

39. Class lll:Affine transformation

40. Decomposing an affine matrix A • A = R (q) R(-f) D R(f) • Where R (q) and R(f) are rotation by q and f respectively. D is a diagonal matrix. • This follows from the SVD where U and V are orthogonal matrices • A = (UVT)(VDVT)= R (q) (R(-f) D R(f))

41. Invariants • (i) parallel lines • (ii) Ratio of lengths of parallel line segments • (iii) Ratio of areas

42. Distortion arising from a planar affine transformation

43. Class lV: Projective transformation • Vector v is a 2-vector and v is a scaler. The matrix has 9 elements with only their ratio significant.

44. Comparing affine and projective transformationsIdeal points are mapped to finite points in projective transformations

45. Decomposition of a projective transformation H • A is a non-singular matrix given byA=sRK+tvT • And K is an upper triangular matrix normalized as • Det K =1

46. An example

48. The projective Geometry of 1D

49. A few comments on the cross ratio

50. Cross ratio: alternative definition • Cross ratio expressed as ratio of ratios of lengths: • Let A, B, C, and D be 4 collinear points. The basic invariant of any homology is the cross ratio • (A, B, C, D) = • The points at infinity