2 Projective Geometry and Transformations of 2D 1 2.1 Planar geometry The classical approach of Euclid studied geometry from a “geometric” or coordinate- free viewpoint. Since Descartes, it has been seen that geometry may be algebraicized. In the algebraic approach, geometric entities are described in terms of coordinates and algebraic entities. A significant advantage of the algebraic approach to geometry is that results derived in this way may more easily be used to derive algorithms and practical computational methods. 2.2 The 2D projective plane 2.2.1 Points and lines Homogeneous representation of lines. A line in the plane is represented by an equation such as ax + by + c = 0 or the vector (a, b, c)⊤. Since (a, b, c) and k(a, b, c) (k = 0) represent the same line, an equivalence class ⊤ ⊤ ∕ of vectors is known as a homogeneous vector. The set of equivalence classes of vectors in R3 (0, 0, 0) forms the projective space P2. − ⊤ Homogeneous representation of points. Since (x, y, 1)⊤ and k(x, y, 1)⊤ (k = 0) rep- resent the same point, points as homogeneous vectors are also elements of ∕P2. We distinguish between the homogeneous coordinates x = (x1, x2, x3)⊤ of a point and the inhomogeneous coordinates (x, y)⊤. Result 2.1. The point x lies on the line l if and only if x⊤l = 0. Degrees of freedom (dof). In order to specify a point (or a line), two values must be provided. Namely it has two degrees of freedom. Result 2.2. The intersection of two lines l and l is the point x = l l . ′ × ′ Given x = l l , we have × ′ x⊤l = l⊤x = l⊤l l′ = 0 × Result 2.4. The line through two points x and x is l = x x . ′ × ′ Given l = x x , we have × ′ x⊤l = x⊤x x′ = 0 × 1The notes in this document are taken by Wufei Ma from Multiple View Geometry in Computer Vision by Richard Hartley and Andrew Zisserman. I added some comments and calculations to those steps that are not quite “obvious” to me. 1 x 2 ideal point l O x x π 3 x 1 Figure 1: A model of the projective plane. 2.2.2 Ideal points and the line at infinity Intersection of parallel lines. Consider two lines l = (a, b, c)⊤ and l′ = (a, b, c′)⊤. Using Result 2.2, the intersection is l l′ = (bc′ cb, ca ac′, ab ab) = (c′ c)(b, a, 0)⊤ × − − − − Ideal points and the line at infinity. One may augment R2 with last coordinate x3 = 0. The resulting space is the set of all homogeneous 3-vectors, namely the 2 projective space P . The points with last coordinate x3 = 0 are known as ideal points, or points at infinity. The set of ideal points, given by (x1, x2, 0)⊤, lies on a single line, the line at infinity, denoted by l∞ = (0, 0, 1)⊤. In the projective plane P2, one may state without qualification that two distinct lines meet in a single point. The study of the geometry of P2 is known as projective geometry. A model for the projective plane. A fruitful way of thinking of P2 is as a set of rays 3 3 in R , as shown in Figure 1. A ray in R , k(x1, x2, x3)⊤, represent a single point in P2. The lines in P2 are planes passing through the origin. Result 2.6. Duality principle. To any theorem of 2-dimensional projective geometry there corresponds to a dual theorem, which may be derived by interchanging the roles of points and lines in the original theorem. 2.2.3 Conics and dual conics A conic is a curve described by a second-degree equation in the plane. In Euclidean geometry conics are of three main types: hyperbola, ellipse, and parabola (apart from degenerate conics). 2 The equation of a conic in inhomogeneous coordinates is ax2 + bxy + cy2 + dx + ey + f = 0 “Homogenizing” by the replacements: x x /x and y x /x gives → 1 3 → 2 3 2 2 2 ax1 + bx1x2 + cx2 + dx1x3 + ex2x3 + f x3 = 0 or in matrix form x⊤Cx = 0 where a b/2 d/2 C = b/2 c e/2 ! # d/2 e/2 f " $ The conic has five degress of freedom. Five points define a conic. Each point (xi, yi) on the conic gives a constraint 2 2 axi + bxiyi + cyi + dxi + eyi + f = 0 which can be written as 2 2 xi xiyi yi xi yi 1 C = 0 Stacking the constraints we% obtain & AC = 0 and the conic is the null vector of the matrix A. With five points, A is a 5 6 matrix. This shows that a conic is determined uniquely (up to scale) by five po×ints. Result 2.7. The line l tangent to C at a point x on C is given by l = Cx. To show that the point x is on l, we have x⊤l = x⊤Cx = 0 To show that the line l is tangent to C, assume that l intersects with C at another point y (besides x). We have y⊤l = y⊤Cx = 0 x⊤Cx = 0 Since C = C⊤, it follows that 2 (y + αx)⊤C(y + αx) = y⊤Cy + 2y⊤Cx + α x⊤Cx = 0, α ∀ 3 Figure 2: Conic envelope: the lines l satisfying l⊤C∗l = 0. This means the whole line l is on the conic C, and C is degenerate. Dual Conics. The conic defined above is known as a point conic. By duality principle, there is a conic that defines an equation on lines, known as a dual conic, or line conic. The dual conic is given by C∗, which is the adjoint matrix of C. 1 From Result 2.7, we have l = Cx, so x = C− l. Since x⊤Cx = 0, we have 1 1 (C− l)⊤C(C− l) = 0 which is l⊤C−⊤l = l⊤C∗l = 0 Dual conics are also known as conic envelopes, as illustrated in Figure 2. Cofactor and adjoing matrix Given a square matrix M, the cofactor matrix M∗ is given by i+j M∗ = ( 1) det(Mˆ ) ij − ij where Mˆ ij is the (i, j) minor of M. The adjoint matrix adj(M) is the transpose of M∗. It is known that 1 adj(M) = det(M)M− Degenerate conics. Degenerate line conics include two lines (rank 2), and a repeated line (rank 1). For example, C = lm⊤ + ml⊤. Degenerate point conics include two points (rank 2), and a repeated point (rank 1). For example, C = xy⊤ + yx⊤. 2.3 Projective transformations Definition 2.9. 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. A projectivity is also called a collinearation, a projective transformation, or a homography. 4 Theorem 2.10. A mapping h : P2 P2 is a projectivity if and only if there exists a → non-singular 3 3 matrix H such that for any point in P2 represented by a vector x × it is true that h(x) = Hx. Definition 2.11. Projective transformation. A planar projective transformation is a linear transformation on homogeneous 3-vector represented by a non-singular 3 3 matrix: × x′ = Hx Note that the matrix H is a homogeneous matrix, and it follows that the projective transformation has eight degrees of freedom. 2.3.1 Transformations of lines and conics Transformation of lines. Under the point transformation x′ = Hx, we have 1 1 0 = l⊤x = l⊤H− Hx = l⊤H− Hx = (H−⊤l)⊤Hx Hence a line transforms as l′ = H−⊤l 1 Points transform according to H, whereas lines transform according to H− . One says that points transform contravariantly and lines transform covariantly. Transformation of conics. Under the point transform x′ = Hx, we have 1 0 = x⊤Cx = x′⊤H−⊤CH− x′ Result 2.13. Under a point transformation x′ = Hx, a conic C transforms to C′ = 1 H−⊤CH− . 1 The presence of H− in this equation may be express by saying that a conic trans- forms covariantly. Result 2.14. Under a point transform x′ = Hx, a dual conic C∗ transforms to C∗′ = HC∗H⊤. 2.4 A hierarchy of transformations The projective transformations form a group, known as the projective linear group. The group of invertible n n matrices with real elements is the (real) general × linear group on n dimensions, or GL(n). The projective linear group is identified by a scalar multiplier, giving PL(n) (this is a quotient group of GL(n)). 5 2.4.1 Class I: Isometries Isometries are transformations of the plane R2 that preserve Euclidean distance, rep- resented as x ε cos θ sin θ t x ′ − x y = ε sin θ cos θ t y ' ′) ! y# ' ) 1 0 0 1 1 ( * " $ ( * where ε = 1. ± If ε = 1, the isometry is orientation-preserving and is a Euclidean transformation. If ε = 1, the isometry is orientation-reversing. A plan−ar Euclidean transformation can be written more precisely as R t x = H x = x ′ E 0 1 + ⊤ , A planar Euclidean transformation has three degrees of freedom and can be com- puted from two point correspondences. 2.4.2 Class II: Similarity transformations A similarity transformation (or similarity) is an isometry composed with an isotropic scaling, also known as equi-form transformation. With no reflection, the similarity has matrix representation x s cos θ s sin θ t x ′ − x y = s sin θ s cos θ t y ' ′) ! y# ' ) 1 0 − 0 1 1 ( * " $ ( * which can be written in block form as sR t x = H x = x ′ S 0 1 + ⊤ , A planar similarity has four degrees of freedom and can be computed from two point correspondences.