New PDF release: An Elementary Course in Synthetic Projective Geometry
By Derrick Norman Lehmer
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Additional info for An Elementary Course in Synthetic Projective Geometry
Discuss the question given in the last problem when the two lines revolve in opposite directions. Can you recognize the locus? 40 An Elementary Course in Synthetic Projective Geometry FIG. 10 PROBLEMS 41 FIG. 11  CHAPTER IV - POINT-ROWS OF THE SECOND ORDER 60. Point-row of the second order defined. We have seen that two fundamental forms in one-to-one correspondence may sometimes generate a form of higher order. Thus, two point-rows (§ 55) generate a system of rays of the second order, and two pencils of rays (§ 57), a system of points of the second order.
Lines joining four points of the locus to a fifth. Suppose that the points S, S', B, C, and D are fixed, and that four points, A, A1, A2, and A3, are taken on the locus at the intersection with it of any four harmonic rays through B. , on the fixed ray SD. , on the fixed line DS'. These last four harmonic points give four harmonic rays CA, CA1, CA2, CA3. Therefore the four points A which project to B in four harmonic rays also project to C in four harmonic rays. But C may be any point on the locus, and so we have the very important theorem, Four points which are on the locus, and which project to a fifth point of the locus in four harmonic rays, project to any point of the locus in four harmonic rays.
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.
An Elementary Course in Synthetic Projective Geometry by Derrick Norman Lehmer