# Calculating the semi-major axis with one point, the eccentricity and the center.

Given is one "random" point on the ellipse, the eccentricity and the center of the ellipse. How can I calculate the semi-major axis?

I'm not very math-savvy, so I hope the problem is easy to solve. Thanks a lot.

Edit: In some cases the ellipse is rotated on the y-axis.

Edit 2: So thanks to Cadenza I can calculate the semi-major axis if the ellipse is in a 2D space but what can I do if the ellipse is in a 3D space and rotated on the y-axis?

On a german Q&A website somebody said that it's impossible with these given informations. Is that true? If yes which informations except the semi-minor axis could be helpful?

• Are you given any further information? In some elementary exercises, it is assumed that the axes of the ellipse are horizontal and vertical in the $(x,y)$-plane rather than slanted. I don't think there's enough in what you've posted to answer the question without either that or some other additional information. – Michael Hardy May 25 '14 at 18:37
• @MichaelHardy Sorry, I just realized that there could be cases where the ellipse is slanted(z-Axis). – Vider7CC May 25 '14 at 19:15
• @MichaelHardy With that I mean a rotation on the Y-Axis. – Vider7CC May 25 '14 at 19:36

Any "random" (a better word in this case is arbitrary) point $(x,y)$ on an ellipse whose axes are parallel to the $x$- and $y$-axes satisfy the equation

$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

where $(h,k)$ denotes the center of an ellipse, and $a$ and $b$ denote the semi-major and semi-minor axes ($a$ is semi-major if the ellipse is horizontal).

The eccentricity of an ellipse the ratio of $c$ to the semi-major axis (so if the ellipse is horizontal, the eccentricity is $\frac{c}{a}$). Here, $c$ is the focus of the radius, given by $a^2 = b^2 + c^2$ (again, horizontal case).

You are given $(x,y)$, $(h,k)$, and $\frac{c}{a}$ (or $\frac{c}{b}$). Can you use these equations to recover $a$ (or $b$)?

Note that there is also the case in which the ellipse is rotated (with respect to the standard axes), in which case you would have to consider a more complicated equation (see: Rotation of axes).

• Your first sentence is not true. An ellipse in the $(x,y)$-plane satisfies that equation if its axes are parallel to the $x$- and $y$-axes, but not if they are not. – Michael Hardy May 25 '14 at 18:26
• @MichaelHardy Thanks for catching that. I was jumping too quick with my assumptions of the question. – White Shirt May 25 '14 at 18:39
• Thanks a lot. This will help in most cases. – Vider7CC May 25 '14 at 19:30