G.1 Preliminaries

It is well known that when a body is in motion under the action of an attractive central force that varies as the inverse square of the distance, the path described will be a conic whose focus is at the center of attraction. The particular conic (ellipse, hyperbola, or parabola) is determined solely by the velocity and the distance from the center of force. In this appendix, we will consider the purely geometric problem of determining the various conic paths that connect two fixed points and that have a focus coinciding with a fixed center of force. Specifically, in this appendix we will discuss the geometric and analytic properties as applied to ballistic missile trajectories.

There are many equivalent definitions of conics; however, we shall find the following ones most convenient for our purposes ^[1], ^[2], ^[3]:

Ellipse:

The locus of points the sum of whose distances from two fixed points (i.e., foci) is constant.

Hyperbola:

The locus of points the difference of whose distances from two fixed points (i.e., foci) is constant.

Parabola:

The locus of points equally distant from a fixed point (i.e., the focus) and a fixed straight line (i.e., the directrix).

The familiar elements of these conics are shown in Figures G-1, G-2, and G-3. In Section 6.2, equation (6.1), the general equation of a conic in Cartesian coordinates was given as a second-degree equation of the form ^[4], ^[5]

Figure G.1: Geometry of the ellipse.

Figure G.2: Geometry of...