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Index
Finish reading Chapter 1 of The Geometry of Incidence by Harold L. Dorwart, and then read Chapter 2.
Supplemental reading: pages 106 - 118 of Plane Algebraic Curves by Egbert Brieskorn and Horst Knorrer.
While reading, please keep in mind these key terms. Be sure you can define each upon completion of your readings.
- Real projective plane
- Dot product/ inner product/ cross product
- Homogeneous form (coordinates and equations)
- Inconsistent equations
- Duality principle
- Self dual
- Generalized polygon
- Plane configuration
- Cartesian coordinates
- Collineations/ projectivity
- Line at infinity
- Projective line set
After completing these readings, please answer the following questions:
20 (a) Describe what is meant by homogeneous coordinates and equations.
20 (b) What does it mean to have an inconsistent equation on the Cartesian plane. What do points of the form (a,b,0) such that a and b are real numbers, mean in the projective plane?
20 (c) Describe the incidence between points and lines in terms of dot products and cross products.
20 (d) Why is it unneccesary to have ideal points when using the extended Euclidean plane model for the projective plane?
20 (e) What is a plane configuration?
20 (f) Why can a 73 configuration never exist in ordinary plane geometry?
20 (g) Why can the homogenous coordinates of a point (x0, x1, x2) never be equal to (0,0,0)?
20 (h) Briefly describe the completion of an affine plane to a projective plane via a mapping between homogeneous coordinates.
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