Ellipse

PD Image
Fig. 1. Ellipse. 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 that have the same sum of distances to two given points. 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,

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

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 d,

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