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By H.S.M. Coxeter

In Euclidean geometry, structures are made with ruler and compass. Projective geometry is easier: its structures require just a ruler. In projective geometry one by no means measures whatever, as an alternative, one relates one set of issues to a different through a projectivity. the 1st chapters of this booklet introduce the real recommendations of the topic and supply the logical foundations. The 3rd and fourth chapters introduce the recognized theorems of Desargues and Pappus. Chapters five and six utilize projectivities on a line and airplane, repectively. the following 3 chapters advance a self-contained account of von Staudt's method of the idea of conics. the trendy technique utilized in that improvement is exploited in bankruptcy 10, which bargains with the best finite geometry that's wealthy sufficient to demonstrate all of the theorems nontrivially. The concluding chapters convey the connections between projective, Euclidean, and analytic geometry.

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Suppose the sides of a quadrilateral are 4 invariant lines. Then the vertices (where the sides intersect in pairs) are 6 invariant points, 3 on each side. Since the relation between corresponding sides is projective, every point on each side is invariant. Any other ]ine contains invariant points where it meets the sides and is consequently invariant. Thus the collineation must be the identity. The dual argument gives the same result when there is an invariant quadrangle. 13 Given any two complete quadrilaterals (or quadrangles), with their four sides (or vertices) named in a corresponding order, there is just one projective collineation that will transform the first into the second.

Moreover, any two distinct points A and B are the invariant 48 ONE-DIMENSIONAL PROJECTIVITIES points of a unique hyperbolic involution, which is simply the correspondence between harmonic conjugates with respect to A and B. This is naturally denoted by (AA)(BB). The harmonic conjugate of C with respect to any two distinct points A and B may now be redefined as the mate of C in the involution (AA)(BB). 42 Any point is its own harmonic conjugate with respect to itself and any other point. EXERCISES 1.

Regarding ABPCSA1 as a hexagon whose six vertices lie alternately on two lines, what can be said about the intersections of pairs of "opposite" sides of this hexagon? 3. 2A. 4. If H(AB, CD) then ABCD X BACD. 12 tells us that there is only one projectivity ABC n A'B'C' relating three distinct points on one line to three distinct points on the same or any other line. 1A) shows how, when the lines AB and A'B' are distinct, this unique projectivity can be expressed as the product of two perspectivities whose centers may be any pair of corresponding points (in reversed order) of the two related ranges.

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