Conic Sections in Polar Coordinates

The equations for conic sections can be expressed rather elegantly in polar coordinates. As shown in Fig. 5.9, the origin is defined as the focus and a line corresponding to x = d serves as the directrix. Recall the relations between the Cartesian and polar coordinates x = rcos 9, y = rsin6 . The point Fwill trace out the conic section, moving in such a way that the ratio of its distance to the focus r to its distance to the directrix d — x = d—rcos9 is a constant. This ratio is called the eccentricity, e (not to be confused with Euler s 6 = 2.718...)  

FiGURE 5.9 Coordinates used to represent conic sections. The point P traces out a conic section as r is varied, keeping a constant value of the eccentricity. 

Newton showed that, under the inverse-square attraction of gravitational forces, the motion of a celestial object follows the trajectory of a conic section. The stable orbits of the planets around the sun are ellipses, as found by Kepler s many years of observation of planetary motions. A parabolic or hyperbolic trajectory would represent a single pass through the solar system, possibly that of a comet. The better known comets have large elliptical orbits with eccentricities close to 1 and thus have long intervals between appearances. Halley s comet has e = 0.967 and a period of 76 years. 

Ellipses and hyperbolas clearly have two distinct foci. The same ellipse or hyperbola can be constmcted using its other focus and a corresponding directrix. In terms of their semimajor and semiminor axes, the eccentricities of ellipses and hyperbolas are given by 

Another way of constmcting ellipses and hyperbolas makes use of their two foci, labeled A and B. An ellipse is the locus of points the sum of whose distances to the two foci, -h rb, has a constant value. Different values of the sum generate a family of ellipses. Analogously, a hyperbola is the locus of points such that the difference va — rb is constant. This is shown in Fig. 5.11. The two families of confocal ellipses and hyperbolas are mutually orthogonal—that is, every intersection between an ellipse and a hyperbola meets at a 90° angle. 

---