ABSTRACT
The eighth chapter is dedicated to conics: the ellipse, the parabola, and the hyperbola. These curves have been a subject of mathematical inquiry for many centuries, and there are many elegant theorems about them. There are four different definitions of conics: as a cross-section of a plane and a cone (hence the name), a metric definition, an algebraic definition, and a definition via a focus and a directrix. Since all four definitions apply to the same three types of curves, the definitions must be equivalent. Several sections of this chapter prove this equivalency; this includes the classic and elegant result by Dandelin. Another section presents equations for conic curves in polar coordinates. Finally, this chapter discusses the ray properties for conics (with another beautiful proof) and explains the ingenious way these properties are used in the design of X-ray telescopes.
