ABSTRACT

Take a flashlight and shine it straight onto a wall. You will see a circle. Tilt the light, and the circle will turn into an ellipse. Tilt further, and the ellipse will become more and more elongated, and will become a parabola eventually. Tilt a little more, and you will have a hyperbola—actually one branch of it. The beam of your flashlight is a cone, and the image it generates on the wall is the intersection of that cone with a plane (i.e., the wall). Thus, we have the name conic section for curves that are the intersections of cones and planes. The three types of conics, ellipses, parabolas, and hyperbolas arise in many situations and are the subject of this chapter. The basic tools for handling them are nothing but the matrix theory developed earlier. This chapter demonstrates applications of affine maps, eigen things, quadratic forms, and symmetric matrices. Finally, we examine the equation of the action ellipse that has been discussed throughout the book. Sketches and figures illustrate the concepts.