Polar Equations of Conics

A conic section is the set of points for which the ratio of the distance to a fixed point, called the focus, and a fixed line, called the directrix, is constant.

This ratio is called eccentricity, and it determines the conic’s shape: an ellipse if between zero and one, a parabola if one, and a hyperbola if greater than one.

When the focus is at the origin in polar coordinates, a single polar equation describes all conics using the radial distance, polar angle, eccentricity, and distance to the directrix.

If the directrix is vertical, the equation uses cosine, which measures horizontal displacement in polar form.

A directrix to the right of the focus gives a plus sign, while one to the left gives a minus sign—indicating its position along the positive or negative horizontal axis.

If the directrix is horizontal, the equation uses sine, which measures vertical displacement in polar form.

A directrix above the focus gives a plus sign, while one below gives a minus sign—indicating if the directrix is on the positive or negative vertical axis.

Polar equations of conics also help design domes by defining curves that evenly distribute structural loads.