Location of Focal Points of Ellipses

Abstract

O in talks or published articles (1,2), one notes gross errors in the location of the focal points of ellipses. Through some psychological orientation toward circular symmetry, engineers and scientists alike are apt to place the focus near the center of curvature despite a distance error that may amount to 100 per cent. I t is the intention here to suggest simple checks, which the editor, or any reader, can make on the spot. Scientists sometimes excuse their carelessness in locating the focus on the grounds that it does not upset the validity of the theory they are expounding. While the pure mathematician is entitled to a measure of poetic license, it would be straining the bounds of tolerance, for example, to accept an error of 100 per cent in the perigee distance of an orbit. Engineers can fall into the same error, e.g., in a blackboard talk concerning parabolic reflector-type antennas. In this case, the slip is usually corrected before it gets into print, thanks to the shop drawings made for the design and test of the prototype. Without going into the mathematics of ellipses, let us consider methods of locating the focus. One of the properties of the ellipse is illustrated in Fig. 1, where FB, the distance from the focus to an extremity B of the minor diameter, equals the semi-major axis PO. Location of the focal point F may be made simply, as follows. Placing the edge of a strip of paper along the line of apsides (major diameter), mark off on the strip the perigee and apogee points P and A. Fold the paper in two, superposing P and A. Now crease to locate the center 0 of the ellipse. With the paper still folded, superpose the P and A points on B, holding in place by the point of a pencil. Rotate the other end 0 until it crosses the axis at the point F. Mark the focal point F. Thus, in a few seconds, one has checked whether the author calculated or guessed at the position of F. A further useful check is as follows. Draw a vertical line through F intersecting the ellipse at E. The line segment FE is the semilatus rectum P, which is the radius of curvature of