By Derrick Norman Lehmer

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Extra resources for An Elementary Course in Synthetic Projective Geometry

Sample text

63. This statement is equivalent to the following: Given three pairs of corresponding rays in two projective pencils, it is possible to find a ray of one which corresponds to any ray of the other. 64. We proceed, then, to the solution of the fundamental PROBLEM: Given three pairs of rays, aa', bb', and cc', of two protective pencils, S and S', to find the ray d' of S' which corresponds to any ray d of S. Call A the intersection of aa', B the intersection of bb', and C the intersection of cc' (Fig.

62. Determination of the locus. We now show that it is possible to assign arbitrarily the position of three points, A, B, and C, on the locus (besides the points S and S'); but, these three points being chosen, the locus is completely determined. 63. This statement is equivalent to the following: Given three pairs of corresponding rays in two projective pencils, it is possible to find a ray of one which corresponds to any ray of the other. 64. We proceed, then, to the solution of the fundamental PROBLEM: Given three pairs of rays, aa', bb', and cc', of two protective pencils, S and S', to find the ray d' of S' which corresponds to any ray d of S.

50. It follows also that two projective pencils which have the same center may have no more than two self-corresponding rays, unless the pencils are identical. For if we cut across them by a line, we obtain two projective point-rows superposed on the same straight line, which may have no more than two selfcorresponding points. The same considerations apply to two projective axial pencils which have the same axis. 52. Point-rows in perspective position 35 51. Projective point-rows having a self-corresponding point in common.