Linear Algebra and Geometry

CHAPTER IV. PROJECTIVE GEOMETRY

CHAPTER IV PROJECTIVEGEOMETRY 511. Projective Space A. Points at infinity Let A and A' be two distinct planes in the ordinary space, and let 0be a point which is neither on A nor onA'.฀The central projection p of A intoA' with respect to the centre of projection 0is defined as follows: for any Q on A we set p(Q) =฀Q' if the points Q, Q' and 0 are collinear (i.e. on a straight line). If A and A'are parallel planes, then p is an affinity of the 2-dimensional affine space A onto the 2-dimensional affine space A'. In particular, p is a bijective mapping taking lines into lines and intersecting lines into intersecting lines and parallel lines into parallel lines. Fig. 8฀ 118฀ 81฀1฀ PROJECTIVE SPACE 119 Consider now the case where A and A' are not parallel. Here two lines, one on each plane, deserve our special attention. The plane which passes through 0 and parallel to A' intersects A in a straight line L and the plane which passes 'through 0 and pvallel to A intersects A' in a straight line L'. It is clear that the points on L have no image points on A' under p and the points on L' are not image points of points on A under p. Therefore we have to exclude these exceptional points; in order to obtain a well-defined bijective mapping p: A \L +฀ At\L'. The situation in relation to lines is equally unsatisfactory. Take a line G on A and suppose G #฀L. Then the image points of G will lie on a line G' of A' different form L' since G and G' are on one and the same plane passing through 0. Here the set of exceptional points is G n฀ L on L and G' n฀ L' on L'. For G =฀L, we do not have a corresponding line G' on A'. Fig. 9 It is now no more true that intersecting lines correspond to intersecting lines and parallel lines correspond to parallel lines. To see 120 IV PROJECTIVEGEOMETRY this, consider two lines G, and G2 of A, neither of which is parallel to the line L. If G, and G2 intersect at a point of L, then the corresponding Cl1฀and Gt2 will be parallel; if G1 and G2 are parallel, then G', and Gt2will intersect at a point of L'. In order to have a concise theory without all these awkward exceptions , we can -฀ and this is a crucial step towards projective geometry -฀ extend the plane A (and similarly the plane A') by the adjunction of a set of new points called points at infinity. More precisely, we understand by a point at infinity of A the direction of a straight line ofA and by the projective extension P of A the set of all points of A together with all points at infinity of A. For convenience, we refer to elements of P as POINTS.Furthermore we define a LINE as either a subset of P consisting of all points of a straight line of A together with its direction or the subset of all points at infinity of A. The LINEconsisting of points at infinity is called the line at infinity of A. Thus the projective extension of a plane is obtained by adjoining to the plane the line at infinity of the plane. We have no difficulty in proving that in the projective extension P of the plane A the followingrules are true without exception: (a) Through any two distinct POINTS there is exactly one LINE. (p) Any two distinct. LINES have exactly one POINT in common. We observe that (a) holds also if one or both POINTSare at infinity and (p) holds also if one of LINESis at infinity. These two incidence properties of the projective extension P stand out in conciseness and simplicity when they are compared with their counterparts (A, )฀(A, )฀and (B) of the plane A: (A,) Through any two points there is exactly one line. (A2) Through any point there is just one line of any complete set of mutually parallel lines. (B) Two distinct lines either have exactly one point in common or they are parallel in which case they have no point in common. We can now also extend the mapping p to a bijective mapping a: P +฀P' by requiring (i) n...