By Derrick Norman Lehmer
Meant to provide, As easily As attainable, The necessities of man-made Projective Geometry - Chapters: One-To-One Correspondence - kinfolk among basic types In One-To-One Correspondence With one another - mixture of 2 Projectively similar primary 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 background of artificial Projective Geometry - Index
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Additional resources for An Elementary Course In Synthetic Projective Geometry
This happens, for example, when two projective point-rows are projected to the same point. Similarly, two projective axial pencils may have the same axis. We examine now the possibility of two forms related in this way, having an element or elements that correspond to themselves. We have seen, indeed, that if B and D are harmonic conjugates with respect to A and C, then the point-row described by B is projective to the point-row described by D, and that A and C are self-corresponding points. Consider more generally the case of two pencils perspective to each other with axis of perspectivity u' (Fig.
Thus, let a point P lie between A and B. Construct now D, the fourth harmonic  34  An Elementary Course in Synthetic Projective Geometry of C with respect to A and B. D may coincide with P, in which case the sequence is closed; otherwise P lies in the stretch AD or in the stretch DB. If it lies in the stretch DB, construct the fourth harmonic of C with respect to D and B. This point D' may coincide with P, in which case, as before, the sequence is closed. If P lies in the stretch DD', we construct the fourth harmonic of C with respect to DD', etc.
Construct the fourth harmonic of PS with respect to the two given lines. Draw through P a line parallel to this line. 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.
An Elementary Course In Synthetic Projective Geometry by Derrick Norman Lehmer