Ellipse

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. \,