Polytope examples in morphometer

Random polygon

This method simulates a random polygon. We choose random angles $\theta_i \in [0,1]$ until $\sum \theta_i > 2\pi$ . The coordinates of points $x_i$ are then given by $R_i \cdot(\cos(\theta_i), \sin(\theta_i))$ where $R_i$ are independent random radii between 0 and 150.
(Note that the intervals of $\theta_i$ and $R_i$ are arbitrarily chosen.)

Uniform distribution of points on the unit sphere

This method simulates $N$ random points on the unit sphere by sampling $N$ random angles theta in $[0,2\pi)$ . The coordinates of points are then given by $(\cos(\theta), \sin(\theta))$ . The parameter $N$ defines the number of points and $R$ the radius of the sphere.

Random beta polygon

This method simulates a beta polygon created by $N$ independent identically distributed random points with density

(1) $\begin{equation*} f_\beta(x) = \frac{\beta + 1}{\pi} (1 - ||x||^2)^\beta \quad , \quad ||x|| < 1 \quad \textnormal{and} \quad x \in \mathrm{R}^2. \end{equation*}$

For more details, see https://arxiv.org/pdf/1805.01338.pdf.

Here a short description of the simulation of the model:
* Choose the number of points $N > 2$
* Choose a parameter $\beta > -1$
* Sample $N$ random angles $\theta \in [0,2pi)$
* Sample $N$ radii $r \in [0,1)$ that follow the probability density: $2 (\beta+1) r (1 - r^2)^\beta$
by using the inverse transform sampling:
– simply draw a random number $y$ uniformly from $[0,1)$
– then $r = \sqrt{1-(1-y)^{(1/(\beta+1))}}$
* Coordinates of points are then given by r*(cos(theta), sin(theta))