By Derrick Norman Lehmer
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Extra resources for An Elementary Course in Synthetic Projective Geometry
49. Fundamental theorem. Postulate of continuity 33 48. We may also give an illustration of a case where two superposed projective point-rows have no self-corresponding points at all. Thus we may take two lines revolving about a fixed point S and always making the same angle a with each other (Fig. 10). They will cut out on any line u in the plane two point-rows which are easily seen to be projective. For, given any four rays SP which are harmonic, the four corresponding rays SP' must also be harmonic, since they make the same angles with each other.
Definition of projectivity. Two fundamental forms are protectively related to each other when a one-to-one correspondence exists between the elements of the two and when four harmonic elements of one correspond to four harmonic elements of the other. FIG. 6 37. Correspondence between harmonic conjugates. Given four harmonic points, A, B, C, D; if we fix A and C, then B and 38. Separation of harmonic conjugates 23 D vary together in a way that should be thoroughly understood. 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.
Four harmonic planes. We also define four harmonic planes as four planes through a line which pass one through each of four harmonic points, and we may show that Four harmonic planes are cut by any plane not passing through their common line in four harmonic lines, and also by any line in four harmonic points. For let the planes ±, ², ³, ´, which all pass through the line g, pass also through the four harmonic points A, B, C, D, so that ± passes through A, etc. Then it is clear that any plane À through A, B, C, D will cut out four harmonic lines from the four planes, for they are lines through the intersection P of g with the plane À, and they pass through the given harmonic points A, B, C, D.