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