ellipse

First, lets introduce some notations. Figure 1 shows an arc and the ellipse it belongs to. The ellipse $\mathcal{E}$ is defined by its center ($c_x$, $c_y$), its semi-major axis ($a$), its semi-minor axis ($b$) and its orientation ($\theta$). The arc is defined by its start and end angles ($\lambda_1$ and $\lambda_2$, assuming $\lambda_1<\lambda_2\le\lambda_1+2\pi$). If $a=b$, then the ellipse is a circle and the $\theta$ direction is irrelevant.

Figure 1: notations

The two points $F_1$ and $F_2$ are the focii of the ellipse. The distance between these points and the center of the ellipse is $\sqrt{a^2-b^2}$. This means that with our notation, the coordinates of these points are:

$$F_1 \left\{\begin{gathered}c_x -\sqrt{a^2-b^2}\cos\theta\\ c_y -\... ...\sqrt{a^2-b^2}\cos\theta\\ c_y +\sqrt{a^2-b^2}\sin\theta \end{gathered} \right.$$

If the ellipse is a circle, then the two points $F_1$ and $F_2$ are both at the center. These points have the following interesting property:

$$d(P,F_1)+d(P,F_2) = 2a\quad\forall P \in \mathcal{E}$$ (2)

This property is one classical way to define the ellipse.