Ellipse: foci and axes

An ellipse is the set of all points of a plane whose distances from two fixed points, the foci, add up to the same constant. That constant must be greater than the distance between the foci.

The mid-point of the foci is the centre. The major axis is the segment through the foci; the minor axis is the segment through the centre perpendicular to it. The ends of the major axis are the vertices.

Write 2a for the length of the major axis, 2b for the minor axis and 2c for the distance between the foci, so a and b are the semi-major and semi-minor axes and each focus is c from the centre.

At a vertex P, PF₁ + PF₂ = (a + c) + (a − c) = 2a, so the constant sum is always 2a. At an end Q of the minor axis both distances are √(b² + c²), so 2√(b² + c²) = 2a.

Hence a² = b² + c², or c = √(a² − b²). The focus, the centre and an end of the minor axis form a right triangle with hypotenuse a.

The eccentricity of an ellipse is e = c/a, the ratio of the centre-to-focus distance to the centre-to-vertex distance. Each focus is therefore ae from the centre.

Since c < a, e < 1. As c shrinks towards 0 the foci merge, b approaches a and the ellipse rounds off towards a circle; as c approaches a it flattens.