Projective Geometry Lecture Notes
نویسنده
چکیده
2 Vector Spaces and Projective Spaces 3 2.1 Vector spaces and their duals . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Projective spaces and homogeneous coordinates . . . . . . . . . . . . . . . 5 2.2.1 Visualizing projective space . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2 Homogeneous coordinates . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Linear subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.1 Two points determine a line . . . . . . . . . . . . . . . . . . . . . . 7 2.3.2 Two planar lines intersect at a point . . . . . . . . . . . . . . . . . 8 2.4 Projective transformations and the Erlangen Program . . . . . . . . . . . . 8 2.4.1 Erlangen Program . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.2 Projective versus linear . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.4.3 Examples of projective transformations . . . . . . . . . . . . . . . . 11 2.4.4 Direct sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4.5 General position . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.5 Classical Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5.1 Desargues’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.5.2 Pappus’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
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