This book was created by students at Westminster College in Salt Lake City, UT, for the May Term 2014 course Projective Geometry (Math 300CC-01). In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this concept. We will later see that this theorem is special in several respects. For example, parallel and nonparallel lines need not be treated as separate cases; rather an arbitrary projective plane is singled out as the ideal plane and located "at infinity" using homogeneous coordinates. The topics get more sophisticated during the second half of the course as we study the principle of duality, line-wise conics, and conclude with an in- Projective geometry can also be seen as a geometry of constructions with a straight-edge alone. The projective axioms may be supplemented by further axioms postulating limits on the dimension of the space. We then join the 2 points of intersection between B and C. This principle of duality allowed new theorems to be discovered simply by interchanging points and lines. (L3) at least dimension 2 if it has at least 3 non-collinear points (or two lines, or a line and a point not on the line). The following animations show the application of the above to transformation of a plane, in these examples lines being transformed by means of two measures on two sides of the invariant triangle. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. We briefly recap Pascal's fascinating `Hexagrammum Mysticum' Theorem, and then introduce the important dual of this result, which is Brianchon's Theorem. The diagram illustrates DESARGUES THEOREM, which says that if corresponding sides of two triangles meet in three points lying on a straight … The parallel properties of elliptic, Euclidean and hyperbolic geometries contrast as follows: The parallel property of elliptic geometry is the key idea that leads to the principle of projective duality, possibly the most important property that all projective geometries have in common. This book was created by students at Westminster College in Salt Lake City, UT, for the May Term 2014 course Projective Geometry (Math 300CC-01). The group PΓP2g(K) clearly acts on T P2g(K).The following theorem will be proved in §3. P is the intersection of external tangents to ! Under the projective transformations, the incidence structure and the relation of projective harmonic conjugates are preserved. 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