PROJECTIVE SPACE AND THE PINHOLE CAMERA

Transcription

1 PROJECTIVE SPACE AND THE PINHOLE CAMERA MOUSA REBOUH Abstract. Here we provide (linear algebra) proofs of the ancient theorem of Pappus and the contemporary theorem of Desargues on collineations. We also discuss the pinhole camera as a geometric engine to project from 3-D to 2-D. 1. Projective Geometry Figure 1. Man drawing a lute Albrecht Durer (1525) 1.1. Introduction. Cameras make it easy to keep record of a scene today. But it was not always so. The task of faithfully rendering scenes on canvas used to be difficult, but it became easy once artists took a closer look at the geometry of perspective drawing. The etching in Figure 1 shows how to draw a lute in perspective: project from an optical center (the nail on the wall) onto the real projective plane (the door held by the man on the left). Albrecht Durer was one of the painters who laid the foundations for projective geometry. Later, mathematicians developed efficient ways to study projective space. Here we prove two famous theorems on collineations using linear algebra methods. We also introduce the pinhole camera as a geometric machine Projective Space. Definition 1. [?] Let V be a vector space over a field F. Let n + 1 = dim(v ). Write P n F (V ) (or simply Pn ) for the set of lines of V through the origin. A point P P n is an equivalence class of vectors v V, where v v v = λv, λ 0. Note that projective space as defined above amounts to taking the quotient of a vector space (less the origin) over. This suggests that the topology of the real projective plane RP 2 is the quotient topology of R 3 \ {0}. We can also use the unit sphere in R 3, and observe that every line through the origin pierces the sphere through antipodal points to show that RP 2 is compact (a major difference with affine space), but that is beyond the scope of this paper. Date: December 14,

2 2 MOUSA REBOUH Example 1. The projective space of dimension 0 is a point. The projective spaces of dimension 1,2,3,n, are lines, planes, solids, hyperplanes, respectively Projective linear subspaces. Since P n inherits its structure from V, the projective linear subspaces come from the subspaces of V. Lemma 1. For V a finite dimensional vector space, U V P(U) P(V ) Proof. Let U V. Then P(U) is the set of lines of U through 0. But every line of U is a line of V. Hence the result follows. Example 2. [?] Let P = [v], Q = [w] P(V ) be distinct.the line PQ = P, Q is the set of all linear combinations λv + µw such that (λ, µ) (0, 0) Dimension Formula. Theorem 1. Given two projective linear subspaces U, W P n, we have dim(u W ) = dim(u) + dim(w ) dim U, W Proof. U and W are associated with some vector subspaces U, W. So U W is associated with the vector subspace U W, and U, W with the vector subspace U + W. Using the well-known dimension formula for vector spaces yields dim(u W ) = dim(u ) + dim(w ) dim(u + W ) dim(u W ) = dim(u W ) 1 = dim(u) + dim(w ) dim U, W. Example 3. Suppose P(V ) = RP 2. Consider two distinct lines L, L RP 2. Then dim( L) + dim( L ) = 2 = dim(p(v )). These two lines span the real projective plane, so dim( L L ) = 0. Thus, in projective space two distinct lines always intersect in a point. Similarly, two distinct planes always intersect in a line Projective Independence. Theorem 2. [?] n points x i P n are projectively independent if and only if for some choice of n vectors x i V the vectors are linearly independent in V. Proof. Suppose the points x i P n are projectively independent. Choose some vectors x i V such that P x i = x i. Then ( ) ai x i = 0 P ai x i = a i x i = 0 a i = 0 So the vectors x i are linearly independent in V. Conversely, choose n linearly independent vectors x i V. Consider the canonical projection P x i = x i. Then ai x i = 0 a i P x i = 0 a i x i = 0 a i = 0 i So, the points x i are projectively independent in P n. This completes the proof.

3 PROJECTIVE SPACE 3 2. Perspectivities and Collineations The pinhole model describes perspectivities well as these are the projective transformations performed by a camera or a perspective machine such as the one shown in Figure 1. The pinhole model requires two planes Π, Π and an optical center O off those two planes. Pick some point P Π. Define f : Π Π, f(p ) = P where we obtain P by intersecting the ray OP with Π. f is a perspectivity. An important fact which is not proved here is that the cross-ratio of four points on a line is invariant under perspectivities. Note that the focal length is the distance d(o, Π ). If we wished to scale the image, we would just need to change the focal length. Figure 2. The Desargues configuration Theorem 3. (Desargues Theorem.) If two triangles are perspective from a point, they are perspective from a line. Proof. Let ABC and A B C be two triangles in P n, n 2. Suppose these two triangles are in perspective from a point O as in Figure 2. Consider the projective linear subspaces U = O, A, C, A, C and W = O, B, C, B, C. The dimension formula yields dim( U, W ) = 3. So, the points in our configuration either lie in some P 2 or span some P 3. Let E be the point where AC and A C meet. Note that this makes sense since A and A are in perspective from O, and the same goes for C and C. Similarly, let D = AB A B and F = BC B C. First, assume: dim( U, W ) = 3. By assumption A, B, C and A, B, C are two distinct planes, which must meet in a line by the dimension formula. By construction, D,E,F belong to that line. So, we re done. Second, assume: dim( U, W ) = 2. By assumption the vertices of the two triangles and O lie in some plane Π P 2. As in [?], picking some point O P 3 Π c, and lifting two points, say A, A will do the trick. Pick two points G, G in perspective from O with G and A in perspective from O, and G and A in perspective from O. The triangles GBC and G B C are like the triangles in the previous case. Let Ẽ = GC G C and D = GB G B. The points D, Ẽ, F are collinear. But D, Ẽ are lifts of D, E in perspective from O. Thus, we conclude that D, E, F are collinear.

4 4 MOUSA REBOUH Remark 1. Apply the principle of projective duality to state the dual of Desargues theorem: if two triangles are perspective from a line, they are perspective from a point Basis and linear transformations. Definition 2. A projective basis is a set B of n+2 points, no n+1 are linearly dependent. Naturally, the standard basis comes from the projected vector space, to which we add a calibration point. Since V R n+1 we can take n+1 basis elements e i of R n+1 and project them to obtain a set {P 0 = P e 0,..., P n = P e n } to which we add P n+1 = P e n+1 where n e n+1 = λ i e i The following theorem will be needed in the proof of Pappus theorem, so it is stated here, but without proof. Theorem 4. Given a projective basis of P n, there is a one-to-one correspondence between projective transformations and bases (plural). Definition 3. Write P v P n for the point represented by v V. T is a projective transformation if there is an invertible matrix A such that T (P v ) = P Av for all v V. i= The theorem of Pappus. Figure 3. The Pappus configuration Theorem 5. (Pappus Theorem.) If alternate vertices of a hexagon lie on two lines, the three pairs of opposite sides meet in three collinear points. Proof. Take two distinct lines L, L P 2. Pick three distinct points on each line, A, B, C L and D, E, F L as in Fig. 3. Choose a standard projective basis B = {A, B, D, E} with A = (1 : 0 : 0), B = (0 : 1 : 0), D = (0 : 0 : 1), E = (1 : 1 : 1) Then L = AB : {z = 0}, DB : {x = 0}. And L = DE : {x = y}, AE : {y = z}. Thus X = AE DB = (0 : 1 : 1). Let C = (1 : c : 0) and F = (1 : 1 : f). AF : {z = fy}, DC : {y = cx} Y = AF DC = (1 : c : cf) BF : {z = fx}, EC : {(x z)c = y z} Z = BF EC = (1 : (c + f cf) : f) Then c cf = c(1 f) = (c + f cf) f Y, Z {y z = c(1 f)x}. Clearly, X also lies on that line. This completes the proof.

5 PROJECTIVE SPACE 5 3. The pinhole camera Perspective machines from the Renaissance led to the invention of the camera obscura (the dark room), which is an instance of the pinhole camera. In the case of perspective machines, the artist draws the projection by hand. The 3-D to 2-D transformation performed by the pinhole camera involves: a retinal plane Π an optical center O a focal length d(o, Π). OP Π describes the image P of a point P R 3. This is the pinhole model. The optical axis is the line through O perpendicular to Π, and it pierces Π at a principal point O. Take a 2-D orthonormal system of coordinates for Π centered at O, and a 3-D orthonormal system of coordinates called the camera coordinate system centered at O. Thales theorem gives a relationship between the coordinates of P and P. An important property of the pinhole model is that we can use homogeneous coordinates to make the relationship between the world coordinates and the pixel coordinates linear projective [?]. Figure 4. The pinhole model References [1] Onishchik, Arkady L. and Sulanke, Rolf, Projective and Cayley-Klein geometries, Springer- Verlag, Berlin Heidelberg, 2006 [2] Reid, Miles and Szendroi, Balazs, Geometry and Topology,Cambridge University Press, 2005 [3] Luong, Quang Tuan and Faugeras, Olivier, The Geometry of Multiple Images: the laws that govern the formation of multiple images of a scene and some of their applications, The MIT Press, Cambridge, MA, address: mousa@mail.sfsu.edu