Identify a conic in polar form. InKepler believed that the orbit of Mars was oval ; he later discovered that it was an ellipse with the Sun at one focus. Identifying a Conic in Polar Form Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph.
Reflections not passing through a focus will be tangent to a confocal hyperbola or ellipse, depending on whether the ray passes between the foci or not. Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.
The focus and conic section directrix of an ellipse were considered by Pappus. Each of these orbits can be modeled by a conic section in the polar coordinate system.
An ellipse rotated about its minor axis gives an oblate spheroidwhile an ellipse rotated about its major axis gives a prolate spheroid.
As a result, we tend to use polar coordinates to represent these orbits. The ellipse was first studied by Menaechmus, investigated by Euclid, and named by Apollonius.