An Elementary Course In Synthetic Projective Geometry by Derrick Norman Lehmer

By Derrick Norman Lehmer

Meant to offer, As easily As attainable, The necessities of artificial Projective Geometry - Chapters: One-To-One Correspondence - family among basic varieties In One-To-One Correspondence With one another - mixture of 2 Projectively comparable basic types - Point-Rows Of the second one Order - Pencils Of Rays Of the second one Order - Poles And Polars - Metrical homes Of The Conic Sections - Involution - Metrical houses Of Involutions - at the historical past of artificial Projective Geometry - Index

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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.

Two of the points may be taken as centers of two projective pencils, and the three others will determine three pairs of corresponding rays of the pencils, and therefore all pairs. If four points of the locus are given, together with the tangent at one of them, the locus is likewise completely determined. For if the point at which the tangent is given be taken as the center S of one pencil, and any other of the points for S', then, besides the two pairs of corresponding rays determined by the remaining two points, we have one more pair, consisting of the tangent at S and the ray SS'.

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. In other words, we have set up a one-to-one correspondence between the points of a single point-row, which is also a projective correspondence because four harmonic points B correspond to four harmonic points D.

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