An ellipse is defined as the locus of all points such that the sum of the distances from two foci to any point on the ellipse is a constant. Below we see the elliptical orbit of a planet, P, with the Sun, S, at one of the foci. The other focus, F, is often called the empty focus. From the definition of an ellipse, we know that

[NMSU, N. Vogt]

Recall the equation for an ellipse in Cartesian coordinates:

This reduces to the equation of a circle when a = b. We define the following properties of the ellipse.

• major axis - the line AA´, where A and A´ are the vertices of the ellipse.
• semi-major axis - the lines CA and CA´, where C is the center of the ellipse. For each point P on the ellipse at a distance r from focus S, there is a symmetrical point P´ a distance r´ from S - the average of these distances is (r + r´) / 2 = a. This result holds for any arbitrary but symmetrical pair of points. Hence the semi-major axis is equal to the mean distance from the Sun of a planet in an elliptical orbit.
• minor axis - the line BB´.
• semi-minor axis - the lines CB and CB´. If a and b denote the lengths of the semi-major and semi-minor axes, respectively, then using the dashed lines (r = r´ = a) and the Pythagoras theorem, we find b² = a²(1 - e²), where e is the eccentricity of ellipse.
• eccentricity - the ratio CS / CA. The eccentricity gives an idea of how elongated the ellipse is. If the ellipse is a circle, e=0, since S and F are coincident with C. The other limit for e is 1, obtained when the ellipse is so narrow that the empty focus is removed to infinity. The distance from each focus to the center of the ellipse is ae.
• perigee - when the planet, P, is at A. It is then nearest the star since SA = CA - CS = a - ae = a (1 - e). If the gravitating body is the Sun, this point is known as perihelion.
• apogee - when the planet, P, is at A´. It is then farthest from the star since SA´ = CA´ + CS = a + ae = a (1 + e). If the gravitating body is the Sun, this point is known as aphelion.
• true anomaly - the angle ASP.