Coxeter- Section 9.5 (1)
- Author:
- Pinyi Yao
- Topic:
- Geometry
Degenerate Conics
First, let's think about how to obtain a conic section. A conic section is a curve obtained as the intersection of a cone with a plane. When you use a plane to slice straight through a cone (parallel to the bottom), you get a circle; When you use a plane to slice through a cone with a slight angel, you get an ellipse; When you slice through a cone, where the slice is parallel to the side/edge of the cone, you get a parabola; When you use a plane to slice through a cone with a steep angle (even perpendicular to the bottom), you get a hyperbola.
Second, thinking about you are moving this cutting plane, when you slice through the cone (parallel to the bottom or with slight angle), what will happen when you move the plane right to the center of the cone? You get a point. When your slice is parallel to the edge, what will happen when you move the plane right on the edge? You get a line. When you slice through a cone with a steep angle, what will happen when you move the plane right to the axis? You get two crossed lines. Those (point, a line, two crossed lines) are degenerate conics in Euclidean geometry.
Third, in projective geometry, we say the degenerate conics are two lines.
In the construction below, it is a projectivity. Line a, b, c are the ranges, and point C, D, E, F are pencil points. K is a point used to build line x', and we'll keep it there in order to move line x' around. O=x'·y'. (We turn trace on for point O) We have the projectivity: x' is projective to line y'. Move point K around (same as move line x' around), and see the trace/locus of point O. What do you see?