This is the Fixed Point Theorem of projective geometry. In w 1, we introduce the notions of projective spaces and projectivities. This method of reduction is the key idea in projective geometry, and in that way we shall begin our study of the subject. Looking at geometric con gurations in terms of various geometric transformations often o ers great insight in the problem. Desargues' theorem is one of the most fundamental and beautiful results in projective geometry. A very brief introduction to projective geometry, introducing Desargues Theorem, the Pappus configuration, the extended Euclidean plane and duality, is then followed by an abstract and quite general introduction to projective spaces and axiomatic geometry, centering on the dimension axiom. This process is experimental and the keywords may be updated as the learning algorithm improves. Thus, for 3-dimensional spaces, one needs to show that (1*) every point lies in 3 distinct planes, (2*) every two planes intersect in a unique line and a dual version of (3*) to the effect: if the intersection of plane P and Q is coplanar with the intersection of plane R and S, then so are the respective intersections of planes P and R, Q and S (assuming planes P and S are distinct from Q and R). The spaces satisfying these For these reasons, projective space plays a fundamental role in algebraic geometry. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. 5. Johannes Kepler (1571–1630) and Gérard Desargues (1591–1661) independently developed the concept of the "point at infinity". A projective range is the one-dimensional foundation. (In the later spirit of the Erlangen programme one could point to the way the group of transformations can move any line to the line at infinity).   [5] An algebraic model for doing projective geometry in the style of analytic geometry is given by homogeneous coordinates. G3: If lines AB and CD intersect, then so do lines AC and BD (where it is assumed that A and D are distinct from B and C). The simplest illustration of duality is in the projective plane, where the statements "two distinct points determine a unique line" (i.e. The whole family of circles can be considered as conics passing through two given points on the line at infinity — at the cost of requiring complex coordinates. I shall state what they say, and indicate how they might be proved. There are two approaches to the subject of duality, one through language (§ Principle of duality) and the other a more functional approach through special mappings. 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