Ellipse

PD Image
Fig. 1. Ellipse (black closed curve). The sum of the lengths of the two red line segments (ending in P₁) is equal to the sum of the lengths of the two blue line segments (ending in P₂).

In mathematics, an ellipse is a planar locus of points characterized by having an equal sum of distances to two given fixed points in the plane. In figure 1 two fixed points F₁ and F₂ are shown, these are the foci of the ellipse. An arbitrary point P₁ on the ellipse has distance |F₁P₁| to F₁ and distance |F₂P₁| to F₂. Let d be the sum of distances of P₁ to the foci,

$d\equiv |\mathrm {F} _{1}\mathrm {P} _{1}|+|\mathrm {F} _{2}\mathrm {P} _{1}|,$

then all points of the ellipse have the sum of distances d. Thus, another arbitrary point P₂ on the ellipse has distance |F₁P₂| to F₁ and distance |F₂P₂| to F₂. By definition the sum of distances of P₂ to the foci is equal to d,

$|\mathrm {F} _{1}\mathrm {P} _{2}|+|\mathrm {F} _{2}\mathrm {P} _{2}|=d.\,$

The horizontal line segment S₁–S₂ in figure 1, going through the foci, is known as the major axis of the ellipse. Traditionally, the length of the major axis is indicated by 2a. The vertical dashed line segment, drawn halfway the foci perpendicular to the major axis, is referred to as the minor axis of the ellipse; its length is usually indicated by 2b. The major and minor axes are distinguished by a ≥ b. When a = b the ellipse is a circle—a special case of an ellipse. Clearly both axes are symmetry axes, reflection in either of them transforms the ellipse into itself. Basically, this a consequence of the fact that a reflection conserves (sums of) distances.

Clearly, the two foci and the points S₁ and S₂ are connected by reflection in the minor axis. Hence the distance |S₂F₂| ≡ p is by symmetry equal to the distance |S₁F₁|. The distance of S₂ to F₁ is equal to 2a − p. The distance of S₂ to F₂ is equal to p. By the definition of the ellipse the sum is equal to d, hence

$d=2a-p+p=2a.\;$

The sum d of distances from any point on the ellipse to the foci is equal to the length of the major axis.