The eccentric anomaly E in terms of these coordinates is given by:

The true anomaly and the eccentric anomaly are related as follows.

With this result the eccentric anomaly can be determined from the true anomaly as shown next.

The eccentric anomaly is related to the mean anomaly by the formula:

The relation between the true anomaly and the eccentric anomaly E is:

Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ.

For a point P orbiting in an ellipse, the eccentric anomaly is the angle E in the figure to the right.

Compute the eccentric anomaly E by solving Kepler's equation:

In celestial mechanics, eccentric anomaly is an angular parameter that defines the position of a body that is moving along an elliptic Kepler orbit.

The eccentric anomaly is one of three angular parameters ("anomalies") that define a position along an orbit, the other two being the true anomaly and the mean anomaly.

The mean anomaly M can be computed from the eccentric anomaly E and the eccentricity e with Kepler's Equation:

Instead of the mean anomaly at epoch, the mean anomaly , mean longitude, true anomaly , or (rarely) the eccentric anomaly might be used.

For constant u, that is on the ellipse which is the intercept with a constant z plane, v then plays the role of the eccentric anomaly for that ellipse.

To find the position of the object in an elliptic Kepler orbit at a given time t, the mean anomaly is found by multiplying the time and the mean motion, then it is used to find the eccentric anomaly by solving Kepler's equation.

Although equations similar to Kepler's equation can be derived for parabolic and hyperbolic orbits, it is more convenient to introduce a new independent variable to take the place of the eccentric anomaly E, and having a single equation that can be solved regardless of the eccentricity of the orbit.