Download PDF by Derrick Norman Lehmer: An Elementary Course in Synthetic Projective Geometry

By Derrick Norman Lehmer

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Additional resources for An Elementary Course in Synthetic Projective Geometry

Example text

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.

To get a clear conception of their relative motion we may fix the points L and M of the quadrangle K, L, M, N (Fig. 6). Then, as B describes the point-row AC, the point N describes the point-row AM perspective to it. Projecting N again from C, we get a pointrow K on AL perspective to the point-row N and thus projective to the point-row B. Project the point-row K from M and we get a point-row D on AC again, which is projective to the point-row B. For every point B we have thus one and only one point D, and conversely.

As noted above, we can find self-corresponding points arbitrarily close to P, and it appears, then, that we can take a point D as close to P as we wish, and yet the distance between the corresponding points D' and P' approaches d as a limit, and not zero, which contradicts the postulate of continuity. 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.

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An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer


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