By Derrick Norman Lehmer
Meant to offer, As easily As attainable, The necessities of artificial Projective Geometry - Chapters: One-To-One Correspondence - family members among basic varieties In One-To-One Correspondence With one another - mix of 2 Projectively similar basic kinds - Point-Rows Of the second one Order - Pencils Of Rays Of the second one Order - Poles And Polars - Metrical houses Of The Conic Sections - Involution - Metrical houses Of Involutions - at the historical past of artificial Projective Geometry - Index
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Extra resources for An Elementary Course In Synthetic Projective Geometry
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. As a system of points is more familiar to most students of geometry than a system of lines, we study first the point-row of the second order. 61. Tangent line. We have shown in the last chapter (§ 55) that the locus of intersection of corresponding rays of two projective pencils is a point-row of the second order; that is, it has at most two points in common with any line in the plane.
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.
The converse is also true, and we have the very important theorem: 52. If in two protective point-rows, the point of intersection corresponds to itself, then the point-rows are in perspective position. Let the two point-rows be u and u' (Fig. 11). Let A and A', B and B', be corresponding points, and let also the point M of intersection of u and u' correspond to itself. Let AA' and BB' meet in the point S. Take S as the center of two pencils, one perspective to u and the other perspective to u'.
An Elementary Course In Synthetic Projective Geometry by Derrick Norman Lehmer