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projective geometry

Projective geometry is the branch of geometry that deals with properties of geometric figures that remain unchanged under projection.

A mathematical theory of perspective grew out of the studies of Renaissance architects and painters who asked themselves how to best represent a three-dimensional object on a two-dimensional surface. The Greeks had done some early work on perspective, and the great geometer Pappus of Alexandria is credited with the first theorem in projective geometry. However, the subject reached mathematical maturity through the efforts first of Girard Desargues, and then, much later through the work of Jean Poncelet and by Karl von Staudt (1798–1867).

The basic elements of projective geometry are points, lines, and planes. These elements retain their character under projection; for example, the projection of a line is another line, and the point of intersection of two lines is projected into another point that is the intersection of the projections of the two original lines. However, lengths and ratios of lengths are not invariant under projection, nor are angles or the shapes of figures. The concept of parallelism doesn't appear at all in projective geometry; any pair of distinct lines intersects in a point, and if these lines are parallel in the sense of Euclidean geometry, then their point of intersection is at infinity. The plane that includes the ideal line, or line at infinity, consisting of all such ideal points, is called the projective plane.

Two properties that are invariant under projection are the order of three or more points on a line and the harmonic relationship, or cross ratio, among four points, A, B, C, D, i.e., AC/BC : AD/BD. The most remarkable concept in projective geometry is that of duality. In the plane, the terms point and line are dual and can be interchanged in any valid statement to yield another valid statement; in space, the terms plane, line, and point are interchanged with point, line, and plane, respectively, to yield dual statements. Entire theorems also occur in dual pairs, so that one can be instantly transformed into the other. For example, Pascal's theorem (given a hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line) is the dual of Brianchon's theorem (given a hexagon circumscribed on a conic section, the lines joining opposite diagonals meet in a single point). In fact, all the propositions in projective geometry occur in dual pairs.