Projective Geometry- 3D

1. Projective Geometry- 3D Points, planes, lines and quadrics

2. Points in Homogeneous coordinates • X in 3-space is a 4-vector X= (x1, x2, x3, x4) T with x4 not 0 represents the point ( x, y, z)T where x = x1/ x4 , y = x1/ x4 z= x1/ x4 For example X = ( x, y, z, 1)

3. Projective transformation in p3 • A projective transformation H acting on p3 is a linear transformation on homogeneous 4-vectors and is a non-singular 4x4 matrix: • X’ =HX It has 15 dof • 2.2.1 Planes with 4 coefficients: • p =( p1, p2, p3, p4 )

4. Planes • The plane: A plane in 3-space may be written as • p1x1 + p2x2 + p3x3 + p4x4 = 0 • pT X = 0 • In inhomogeneous coordinates in 3-vector notation • Where n =( p1, p2, p3 ), x4 =1 • d= p4 , , d /||n|| is the distance of the origin.

5. Joins and incidence relation(1)A plane is uniquely define by three points, or the join of a line and a point in general position. (2) Two planes meet at a line, three planes meet at a point

8. Three planes define a point

10. Lines in 3 space • A line is defined by the join of two points or the intersection of two planes. A line has 4 dof in 3 space. It is a 5 –vector in homogenous coordinates, and is awkward.

11. Null space and point representation • A and B are 2 space points. Then the line joining these points is represented by the span of the row space of the 2x4 matrix W. • (i)The Span of W is the pencil of points lA+mB on the line.

12. (ii) The the span of the 2D right null space of W is the pencil of planes with the line as axis

13. The dual representation of a line as the intersection of two planes P and Q

14. Examples

15. Join and incidence relations from null-space

16. Plucker matrices

17. Properties of L

18. Properties of L 2

19. Examples(Plucker matrices)where the point A and B are the origin and the ideal point in x direction

20. A dual Plucker representation L*

21. Join and incidence properties

22. Examples 2

24. Two lines

25. The bilinear product (L !L^)

26. Quadrics and dual quadrics • A quadric Q is a surface in p3 defined by the equation • XT Q X = 0 • Q is a 4 x 4 matrix • (i) A quadric has 9 degree of freedom. These corresponds to 10 independent elements of a 4x4 symmetric matrix less one for scale. Nine points in general position define a quadric

27. Properties of Q • (ii) If the matrix Q is singular, the quadric degenerates • (iii) A quadric defines a polarity between a point and plane. The plane p= QX is the polar plane of X w.r.t. Q • (iv) The intersection of a plane p with a quadric Q is a conic C

28. Dual quadric • (v) Under the point transformation X’ =HX, a point quadric transforms as • Q’ = H-T Q H • The dual of a quadric is a quadric on planes pTQ* p =0 where Q* = adjoint Q or Q-1 if Q is invertible A dual quadirc transform as Q*’ = H-T Q* HT

29. Classification of quadrics • Decomposition Q = UT D U • Where U is a real orthogonal matrix and D is a real diagonal matrix. • By scaling the rows of U, one may write Q=HTDH where D is a diagonal with entries 0,1, or –1. • H is equivalent to a projective transform. Then up to a projective equivalence, the quadric is represented by D

30. Classification of quadrics 2 • Signature of D denoted by s(D) = Number of 1 entries minus number of –1 entries • A quadric with diag(d1,, d2,, d3,, d4 ,) corresponds to a set of point given by d1x2 + d2y2 + d3z2 + d4T2 =0

31. Categorization of point quadrics

32. Some examples of quadrics • The sphere, ellipsoid, hyperboloid of two sheets and paraboloid are allprojectively equivalent. • The two examples of ruled quadrics are also projectively equivalent. Their equations are • x2 + y2 = z2 + 1 • xy = z

33. Non-ruled quadrics: asphere and an ellipsoid

34. Non ruled quadrics: a hyperboloid of two sheets and a paraboloid

35. Ruledquadrics: Two examples of hyperboloid of one sheet are given. A surface is made up of two sets of disjoint straight lines

36. Degenerate quadrics

37. The twisted cubic is a 3D analogue of a 2D conic

38. Various views of the twisted cubic(t3, t2, t)T

40. The screw decomposition • Any particular translation and rotation is equivalent to a rotation about a screw axis together with a translation along the screw axis. The screw axis is parallel to the original rotation axis. • In the case of a translation and an orthogonal rotation axis ( termed planar motion), the motion is equivalent to a rotation about the screw axis.

41. 2D Euclidean motion and a screw axis

42. 3D Euclidean motion and the screw decomposition. • Since t can be decomposed into tll and (components parallel to the rotation axis and perpendicular to the rotation axis). • Then a rotation about the screw axis is equivalent to a rotation about the original and a translation

43. 3D Euclidean motion and the screw decomposition2

44. The plane at infinity • p2  linf,circular points I,J on linf • p3  pinf, absolute conic Winf on pinf The canonical form of pinf = (0,0,0,1)T in affine space. It contains the directions D = (x1, x2, x3, 0)T

45. The plane at infinity 2 • Two planes are parallel if and only if , their line of intersection is on pinf • A line is parallel to another line, or to a plane if the point of intersection is on pinf • The plane pinf has 3 dof and is a fixed plane under affine transformation but is moved by a general projective transform

46. The plane at infinity 3 • Result 2.7 The plane at infinity pinf, is fixed under the projective transformation H, if and only if H is an affinity. • Consider a Euclidean transformation

47. The plane at infinity 4 • The fixed plane of H are the eigenvectors of HT . • The eigenvalues are ( eiq, e –iq, 1, 1) and the corresponding eigenvectors of HT are

48. The plane at infinity 5 • E1 and E2 are not real planes. • E3 and E4 are degenerate. Thus there is a pencil of fixed planes which is spanned by these eigenvectors. The axis of this pencil is the line of intersection of the planes with pinf

49. The absolute conic • The absolute conic, Winf is a point conic on pinf. In a metric frame , pinf = (0,0,0,1)T and points on Winf satisfy • x12 + x22 + x32 = 0 • x4 = 0 • The conic Winf is a geometric representation of the 5 additional dof required to specify metric properties in an affine coordinate frame.

50. The absolute conic 2 The absolute conic Winf is fixed under the projective transformationH if and only if H is a similarity transformation. In a metric frame, Winf = I3 x 3 and is fixed by HA. One has A-T I A-1 = I (up to scale) Taking inverse gives AAT =I implying A is orthogonal