New PDF release: An Elementary Course In Synthetic Projective Geometry

By Derrick Norman Lehmer

Meant to provide, As easily As attainable, The necessities of man-made Projective Geometry - Chapters: One-To-One Correspondence - relatives among basic kinds In One-To-One Correspondence With one another - mix of 2 Projectively similar primary varieties - 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 homes Of Involutions - at the background of man-made Projective Geometry - Index

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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.

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. We may note also that the correspondence is here characterized by a feature which does not always appear in projective correspondences: namely, the same process that carries one from B to D will carry one back from D to B again. This special property will receive further study in the chapter on Involution.

A similar theorem may be stated for two axial pencils of which the axes intersect. Very frequent use will be made of these fundamental theorems. [34] 55. Point-row of the second order. The question naturally arises, What is the locus of points of intersection of corresponding rays of two projective pencils which are not in perspective position? This locus, which will be discussed in detail in subsequent chapters, is easily seen to have at most two points in common with any line in the plane, and on account of this fundamental property will be called a point-row of the second order.

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An Elementary Course In Synthetic Projective Geometry by Derrick Norman Lehmer


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