By Derrick Norman Lehmer
Meant to offer, As easily As attainable, The necessities of man-made Projective Geometry - Chapters: One-To-One Correspondence - relatives among primary varieties In One-To-One Correspondence With one another - mixture of 2 Projectively comparable primary types - 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 heritage of man-made Projective Geometry - Index
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Extra resources for An Elementary Course In Synthetic Projective Geometry
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. For any line u in the plane of the two pencils will be cut by them in two projective point-rows which have at most two self-corresponding points.
This is the required line. 3. Given a parallelogram in the same plane with a given segment AC, to construct linearly the middle point of AC. 4. Given four harmonic lines, of which one pair are at right angles to each other, show that the other pair make equal angles with them. This is a theorem of which frequent use will be made. 5. Given the middle point of a line segment, to draw a line parallel to the segment and passing through a given point. 6. A line is drawn cutting the sides of a triangle ABC in the points A', B', C' the point A' lying on the side BC, etc.
S', and by the same argument S, is then a point where corresponding rays meet. Any ray through S will meet it in one point besides S, namely, the point P where it meets its corresponding ray. Now, by choosing the ray through S sufficiently close to the ray SS', the point P may be made to 62. Determination of the locus 43 approach arbitrarily close to S', and the ray S'P may be made to differ in position from the tangent line at S' by as little as we please. We have, then, the important theorem The ray at S' which corresponds to the common ray SS' is tangent to the locus at S'.