For a ellipse $E(\theta)$, which owns the parametric equation as follows:

$\begin{cases}
 x = a \sin\theta + b \cos\theta + c \\
 y = d \sin\theta + e \cos\theta + f \\
\end{cases}$

Now, I have a point $P=\{x_p,y_p\}$, to know the relationship between $E(\theta)$ and $P=\{x_p,y_p\}$, I using this method:

$$
\begin{cases}
\sin \theta =\frac{(c-x_p)e+b(y_p-f)}{b d-a e}\\
\cos \theta =\frac{(-c+x_p)d+a(f-y_p)}{b d-a e}
\end{cases} 
$$

compute $m=\sin^2 \theta+\cos^2 \theta$

- if $m=1$, the point $P$ on the ellipse $E(\theta)$
- if $m>1$, the point $P$ outside the ellipse $E(\theta)$
- if $m<1$, the point $P$ inside the ellipse $E(\theta)$

However, for the compound curve, for example

- ellipse segment $E_1(\theta) \qquad \theta \in [0, 2.4798],[5.8629, 2 \pi]$

- ellipse segment $E_2(\theta) \qquad \theta \in [3.1275, 6.1325]$

- line segment $L(E_1(2.4798),E_2(3.1275))$

---

 mat1 = {{0., -5., 0}, {-5.2203, 0., 1.7945}}; 
 mat2 = {{-0.8583, -4.9384, 0.1765}, {-5.4189, 0.7822, 2.3088}};
 θ1 = 2.4798;
 θ2 = 3.1275;
 Show[
 {ParametricPlot[
 mat1.{Sin[θ], Cos[θ], 1}, {θ, 0, 2.4798}, PlotStyle -> Red], 
 ParametricPlot[
 mat1.{Sin[θ], Cos[θ], 1}, {θ, 5.8629, 2 Pi}, PlotStyle -> Red],
 ParametricPlot[
 mat2.{Sin[θ], Cos[θ], 1}, {θ, 3.1275, 6.1325}],
 Graphics[
 {Line[{mat1.{Sin[θ1], Cos[θ1], 1}, mat2.{Sin[θ2], Cos[θ2], 1}}]}]}, 
 PlotRange -> All]
![enter image description here][1]

Now, I have two points $P_1,P_2$, whose coordinate are `{{-4.50722, 0.455707}, {5.2748, 2.68502}}`,respectively.So I would like to know is there a **simple** method to determine whether $P$ **inside** the boundary?

![enter image description here][2]


 [1]: https://i.sstatic.net/0SP9R.png
 [2]: https://i.sstatic.net/UxwZe.png