There are two common ways to characterize ellipses. One is three-dimensional, as the intersection of a plane and a single nappe of a cone.
The second way to characterize ellipses is by using the two-dimensional locus definition; This definition is the set of points in a plane (locus) such that the sum of the distances of each point from two given points (the foci - pronounced 'foe - sigh') is a constant.
This is a two-dimensional locus definition since the picture is 2 dimensions; and the locus is the purple line; The set of points (in the white plane) has the relationship that when you pick one point on the purple ellipse, the distance from that point to the first focus added to the distance from that same point to the second focus is the same for ANY other point on the ellipse:
(By way of reminder, the expression P1F1 means the segment with length whatever from P1 to F1. P1F2 means the segment with length whatever from P1 to F2 etc...)
In other words, P1F1 + P1F2 = P2F1 + P2F2 = P3F1 + P3F2 = ...
To further explain, Let F1 and F2 be two given points in a plane and k be a positive real number with k > `F1F2. Then the ellipse with foci F1 and F2 and focal constant k is the set of all points in the plane which satisfy