The equation of an ellipse is:
Where (h,k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by (x,y).
The semi-major axis is the mean value of the maximum and minimum distances and of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis. In astronomy these extreme points are called apsides.
The semi-minor axis of an ellipse is the geometric mean of these distances:
The eccentricity of an ellipse is defined as
- so .
Now consider the equation in polar coordinates, with one focus at the origin and the other on the direction,
The mean value of and , for and is
In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix.
The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge.
The semi-minor axis is related to the semi-major axis through the eccentricity and the semi-latus rectum , as follows:
A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping fixed. Thus and tend to infinity, faster than .
The length of the semi-minor axis could also be found using the following formula,
where is the distance between the foci, and are the distances from each focus to any point in the ellipse.