## Random polygon

This method simulates a random polygon. We choose random angles until . The coordinates of points are then given by where are independent random radii between 0 and 150.

(Note that the intervals of and are arbitrarily chosen.)

## Uniform distribution of points on the unit sphere

This method simulates random points on the unit sphere by sampling random angles theta in . The coordinates of points are then given by . The parameter defines the number of points and the radius of the sphere.

## Random beta polygon

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

(1)

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

* Choose a parameter

* Sample random angles

* Sample radii that follow the probability density:

by using the inverse transform sampling:

– simply draw a random number uniformly from

– then

* Coordinates of points are then given by r*(cos(theta), sin(theta))