Pole/polar• consider projective geometry again• we may want to compute the tangents of a

curve that pass through a point (e.g., visibility)

• let C be a conic and x a point• if x lies on the conic, we know that Cx is the

tangent at x• if x does not lie on the conic, Cx is a line

called the polar of x with respect to C• x is the pole of the polar• the polar of x passes through the 2 points of

tangency of the 2 tangents of C through x– if tangent line at y, Cy, contains x, then x^t

Cy=0 or (Cx)^t y = 0 so y lies on the polar• the polar is the generalization of the tangent;

it degenerates to the tangent when x lies on C

• HZ58

Conjugacy

• points x and y are conjugate if x^t C y = y^t C x = 0

• if y lies on the polar Cx, then x and y are conjugate

• conjugacy is symmetric(!): if y lies on polar of x, then x lies on polar of y

• HZ59

Pole/polar for quadrics

• quadric Q

• polar plane of X = QX

• if X lies on Q, QX = tangent plane

• if X lies outside Q, QX passes through the points of tangency of the tangent cone through X

• HZ73-74

Conic fitting

• fitting point data is an important task in shape design– we shall explore in future, but let’s see what projective space has to say first

• fitting a conic is equivalent to finding the null space of a matrix• each point (x,y) on a conic imposes a constraint on the conic coefficients

– ax^2 + bxy + ... + f = 0– or, separating the unknowns from the knowns,– (x^2, xy, y^2, x, y, 1) . C = 0, where C = (a,b,c,d,e,f)

• 5 points define a 5 x 6 matrix M: MC = 0• Q: how do you solve for conic C?• A: 1d null space of M defines the coefficients C of the conic

• null space computation is also used to find fundamental matrix, epipoles (from F) and camera center (from camera matrix)

• HZ31

Projective 3-space

• 2D projective geometry was introduced through the duality of lines and points; 3D is motivated through the duality of planes and points; lines are self-dual

• plane = 4-vector• point X lies on plane π iff πtX = 0• projective transformation is represented by a 4x4

nonsingular matrix H: X’ = HX• homographies preserve lines, incidence and order of

contact

• HZ65-66

3 points plane

• consider 3 points Xi that lie on the plane π• build the 3x4 matrix M with ith row = Xi

t

• Mπ = 0 for the plane π• if Xi are in general position, M is rank 3 and its null vector is the plane• if the three points are collinear, M is rank 2 and its 2D null space defines a pencil of

planes• i.e., plane = null space of point matrix• the plane can also be expressed through determinants• π = (D234, - D134, D124, - D123)t, where

– Dijk = determinant of rows ijk of the 4x3 matrix with ith column = Xi (note that this is transpose of above matrix)

– proof: build 4x4 matrix [X,X1,X2,X3], whose determinant = 0 if X is on plane– 3D analogue of line = cross product of two points

• equivalently, π = ((X1’ – X3’) x (X2’ – X3’), -X3’t (X1’ x X2’))t where Xi = (Xi’ 1)

• 3 planes point is a dual result: null vector of the plane matrix• HZ66-67

Lines

• how do you represent a line in 3-space?• line = join of n-1 points or intersection of n-1 planes

– shows duality

• line has 4dof– can define line by intersection with 2 orthogonal planes (2

parameters per plane)

• awkward to represent lines in 3-space since the natural representation by homogeneous 5-vector does not play well with the point and plane’s 4-vectors

• see HZ68-70 for representation of lines as null spaces (nice implementation consistency with other results)

Lines as Plücker matrices• the most famous representation of a line is by Plücker• represent the line through points A and B as:

– L = AB^t – BA^t– 4x4 skew-symmetric matrix called the Plücker matrix

• L is independent of points!• L is rank 2• L has 4 dof (6 in skew-symm – 1 homogeneous – 1 det 0)• L generalizes 2d line = point x point• valency-2 tensor: if point transforms as X HX, then Plücker matrix transforms as

L HLH^t• dual Plücker matrix from two planes: L* = PQ^t – QP^t• point X and line L plane L* X

– and L*X = 0 reflects X lying on L• plane X and line L point L X

– and LX=0 reflects line L lying in the plane P

• HZ70-71

Plücker coordinates and Klein quadric

• Plücker coordinates are simply the 6 nonzero elements extracted from the Plücker matrix: {12,13,14,23,42 !,34}– from upper triangle with fifth element reversed

• since Plücker matrix is rank 2, det=0 imposes a quadratic constraint on these coordinates:– 12*34 + 13*42 + 14*23 = 0– dot product of 1st half of L with reversed second half of L– this codimension-1 surface in projective 5-space is called the Klein

quadric

• two lines L and L’ intersect if and only if L . reversed(L’) = 0– Klein quadric encodes the fact that a line must intersect itself: L . reversed

L = 0

• HZ71-72