# ellipse

An ellipse is a closed curve in the plane that is defined as the set of all points the sum of whose distances from two fixed points is a constant. This constant is also the distance between the vertices of the ellipse and it is called the major axis. Half the distance is called the semi-major axis and this is denoted by $$a$$. The distance $$b$$ (shown in the figure) is called the semi-minor axis.

Algebraically, an ellipse in the Cartesian plane with horizontal major axis is a relation defined by an equation of the form

$\displaystyle\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=r^2$

where the point $$(h,k)$$ is the center of the ellipse and $$a$$ and $$b$$ are the lengths of the semi-major and semi-minor axes. If the major axis of the ellipse is vertical we swap the roles of $$a$$ and $$b$$. If $$a=b$$ then the ellipse is a circle. Figure 1: An ellipse in the Cartesian plane.

An ellipse also has directices and important reflection properties. See the article on conics for an exposition. 