Numerical example

The following example show the first iterations of the algorithm for a very flat ellipse:

$$\left\{\begin{aligned}a_e &= 100\\ f &= 0.9 \end{aligned}\right.$$

We consider a test point $A$ having the following coordinates:

$$T\left\{\begin{aligned}\varphi &= 75^\circ\\ h &= 0.1 \end{aligne... ...\begin{aligned}r &= 93.7139699\\ z &= \phantom{0}3.5930796 \end{aligned}\right.$$

We assume we know only $r$ and $z$ and want to compute the unknown $\varphi$ and $h$.

Equations (6) give us the tangent of the half slope:

$$t_0=0.0191634$$

Applying equation (7), we get the signed distance between $A$ and the intersection of the first line and the ellipse:

$$k_0= 0.3419763 \Rightarrow \varphi_0=75.3818944^\circ$$

>From this value of $k$, we deduce the first coefficients $\alpha$, $\beta$, $\gamma$ and $\delta$ of the quartic (we don't compute $\varepsilon$). We reduce it to a cubic using the already known root $t_0$:

$$\tau^3 + a_1 \tau^2 + a_2 \tau + a_3=0 \quad \left\{\begin{aligned}a_1 &= -3.5542558\\ a_2 &= \phantom{-}1.3365805\\ a_3 &= -3.5478057\\ \end{aligned}\right.$$

This polynomial has only one real root:

$$\tilde{t}_0=3.4640705 \Rightarrow \tilde{\zeta}_0=147.7954987^\circ \Rightarrow \tilde{\varphi}_0=74.5916410^\circ$$

The middle point between $\varphi_0$ and $\tilde{\varphi}_0$ is the best approximation we have at the end of this first iteration:

$$\varphi_1=74.9867677^\circ$$

We know an upper bound for the error of this approximation is the half-width of the interval, which is $0.3951267^\circ$ (in fact the error is about 30 times smaller). As we consider this is too much, we compute the starting values for a second iteration:

$$k_1=0.1005980 \quad t_1=0.8577741$$

The steps of the second iteration are similar to the steps of the first one. They lead to the following result:

$$\tilde{\varphi}_1=75.0132026^\circ$$

So at the end of the second iteration, we have an estimate $\varphi_2=74.9999851^\circ$ and an upper bound of its error of $0.0132174^\circ$ (the real error being almost 900 times smaller).