Lecture 14: Projective Space vs. the Universal Space of Mass-PointsYet there shall be a space € KJoshua 3:41. Algebra and GeometryGeometry guides our intuitions; algebra conducts our computations. So far, in order to facilitate calculations on a computer, our coordinate representations for geometry have been driven largely by linear algebra: matrix representations for affine transformations pointed us to affine coordinates; matrix representations for projective transformations steered us to homogeneous coordinates.Algebra expedites computations, but in the wake of all this matrix algebra many geometric questions have been left unanswered or even suppressed. The goal of this lecture is to confront these geometric issues head on and to provide some straightforward answers to help guide our intuition.Here are a few basic questions that we have avoided till now, but should have been bothering you all along:1. For vectors v, we know the geometric meaning of € cv for any constant c. But for points P, we have a geometric interpretation of € cP only for € c = 0,1. What is the geometric meaning of € cP when € c ≠ 0,1?2. In affine coordinates each vector v and each point P has a unique representation: € (x, y,z,0) represents a unique vector v, and € (x, y,z,1) represents a unique point P. But in homogeneous coordinates, the 4-tuple € (wx, wy,wz,w) represents the point € P = (x, y, z) for any € w ≠ 0. Thus there are infinitely many w’s for the same point P. What is the geometric interpretation of all the homogeneous 4-tuples € (wx, wy,wz,w) that represent the same point € P = (x, y, z)?3. We have been using two distinct, complementary models of space: points and vectors. Is there a single consistent model of space that contains both points and vectors?4. Perspective projection does not map vectors to vectors. Nevertheless, we can still multiply a vector € v = (v1,v2,v3,0) by the € 4 × 4 matrix M that represents perspective projection. How should we interpret the result of perspective projection applied to a vector?There are two sets of answers to each of these questions: one invokes projective space, the other the universal space of mass-points. Below we shall investigate these two approaches to geometry, along with their advantages and disadvantages for Computer Graphics.2. Projective Space: The Standard ModelProjective space consists of two types of points: affine points and points at infinity. An affine point is the same as an ordinary point in Euclidean space; a point at infinity is a new kind of point, defined by a direction in Euclidean space (see Figure 1). Points at infinity are represented by equivalence classes of vectors: two vectors that point in the same direction or in diametrically opposite directions represent the same point at infinity. Thus € v and € cv represent the same point at infinity for any constant € c ≠ 0. We write the equivalence class of the vector v using homogeneous coordinates as € [v,0]. Thus in projective space € [v,0] ≡ [cv,0] ≡ c[v,0] for any € c ≠ 0. Similarly, in projective space we represent affine points P using equivalence classes of homogeneous coordinates, so € [P,1] ≡[cP, c] ≡ c[P,1] for any € c ≠ 0. Notice that the zero vector does not correspond to a point in projective space, since the zero vector does not define a direction in Euclidean space.Affine PointsPoints at Infinity Affine PointsPoints at InfinityPoints at Infinity€ •Figure 1: Projective space. The affine points are the same as the standard points in Euclidean space. Points at infinity correspond to directions in Euclidean space and are glued onto the affine points at infinity. In this figure we picture the projective plane (2-dimensional projective space), but the construction is valid in any dimension..Projective space is the standard model for geometry adopted in most textbooks on Computer Graphics. Here is how projective space provides answers to the four questions listed in the introduction to this lecture. 1 What is the geometric meaning of € cP when € c ≠ 0,1?If € P = [x, y, z,1], then € cP = [cx,cy,cz,c] is just another member of the equivalence class representing the same affine point P.2. What is the geometric interpretation of all the homogeneous 4-tuples € (wx, wy,wz,w) that represent the same affine point € P = (x, y, z)?The 4-tuples € (wx, wy,wz,w) represent the points on the line through the origin in the direction € (x, y,z,1). Thus each line through the origin represent a point in projective space.23. Is there a single consistent model of space that contains both points and vectors?Yes. Projective space includes all the affine points and incorporates the vectors as points at infinity.4. How should we interpret the result of perspective projection applied to a vector?Matrix multiplication commutes with scalar multiplication. Hence if M is a € 4 × 4 matrix and € v is a vector, then for any € c ≠ 0€ c(v,0)( )∗ M = c (v,0)∗ M( ).Thus matrix multiplication is well-defined in projective space, since matrix multiplication maps equivalence classes to equivalence classes -- that is, matrix multiplication maps constant multiples to constant multiples. Notice, however, that points at infinity are not necessarily mapped to points at infinity by matrix multiplication, since the last component of € (v,0)∗ M need no longer be zero when the last column of M is not € (0,0,0,1)T. This result accounts for our observation that vectors are not mapped to vectors by perspective projection. In fact, under perspective projection, the point at infinity corresponding to the vector v is mapped to the intersection with the projection plane S of the line determined by the eye point E and the point at infinity corresponding to the vector v (see Figure 2). •••E(eye)R(point )••€ Rnew€ S (Projective Plane)€ ( perspective point )€ ( perspective point)€ P€ v€ Pnew€ vnew€ ( point at infinity)€ ( point at infinity)Figure 2: Perspective projection maps points at infinity to the intersection of the line through the eye point E and the point at infinity v with the projection plane S. Notice too that points R on a line through E parallel to S are