A QUADRATIC MAPPING IN PROJECTIVE FOUR-DIMENSIONAL SPACE

Oeme Bottema
1979 Demonstratio Mathematica
has recently studied the following mapping [1]. In projective 4 -space P^ two planes t^ and r^ are given, intersecting at the point T. A projecting plane 9 is defined as a plane through T intersecting both r^ and z^ in a straight line. It is shown that through a point A of P^ passes, in general, one projecting plane 9(A). A projection plane ir is introduced, not passing through T, and intersecting r1 at T^' (i=1,2). The projection A' of A onto ir is defined as the intersection of 9(A) and jr.
more » ... veral properties of the mapping MjP^-» jt with M( A) = A', are derived, the most important being that M is quadratic: a line 1 in P^ corresponds to a conic 1;' in jt passing through T^ and T2. The method is purely synthetic. In this note we discuss M by analytical means; the properties obtained by J.Bohdanowicz-Grochowska are verified and some further theorems are added. 2_. We introduce in P^ homogeneous projective point coordinates (i=1,2,3,4,5). Let r1 and x^ be given by the equations = x2 = 0 and x^ = x^ = 0 respectively, and jt by x 2 -x^ = x^ = 0. Then T = (0,0,0,0,1), T1 = = (0,0,1,0,0), T2 = (1,0,0,0,0). The t'wo-dimensional set of projecting planes {9J is represented by -1023 -Unauthenticated Download Date | 2/25/20 5:16 PM