parametric equation

The elliptical arc can be seen as a parametric curve, each point is defined as an explicit vectorial function of one parameter $E(\eta)$ giving the two coordinates $x$ and $y$ of the point. We will use as a parameter the $\eta$ angle computed such that:

$$\left\{\begin{aligned}r\cos\lambda &= a\cos\eta\\ r\sin\lambda &= b\sin\eta \end{aligned}\right.$$

$$\Rightarrow \left\{\begin{aligned}\eta &= \arctan_2\left(\frac{\sin\lambda}{b... ...da}{a}\right)\\ \lambda &= \arctan_2(b\sin\eta, a\cos\eta) \end{aligned}\right.$$

where $\lambda$ is the geometrical angle considered between one end of the semi-major axis and the current point, counted from the center of the ellipse (see $\lambda_1$ and $\lambda_2$ in the figure). This $\eta$ angle is a theoretical angle, it has no representation on the preceding figure (except if the ellipse is really a circle, of course).

Using this angle, the ellipse parametric equation is:

$$E(\eta)\left\{\begin{gathered}c_x + a\cos\theta\cos\eta - b\sin... ... a\sin\theta\cos\eta + b\cos\theta\sin\eta \end{gathered} \right.$$ (3)

The derivatives of the parametric curve are:

$$E'(\eta)\left\{\begin{gathered}-a\cos\theta\sin\eta - b\sin\the... ...-a\sin\theta\sin\eta + b\cos\theta\cos\eta \end{gathered} \right.$$ (4)

and

$$E''(\eta)\left\{\begin{gathered}-a\cos\theta\cos\eta + b\sin\th... ...-a\sin\theta\cos\eta - b\cos\theta\sin\eta \end{gathered} \right.$$ (5)