In hyperbolic Geometry the area of a circle is given by the equation
$$a=4{\pi}\left(Rsinh\left(\frac{r}{2R}\right)\right)^2$$
with $a$ being the area and $r$ being the radius of the circle, and $R$ being a constant for any hyperbolic plane.
In order to generate random points inside the circle that follow a uniform distribution, I can first generate random areas for smaller circles within this circle that follow a uniform distribution. Next I can convert the randomly selected areas into randomly selected distances from the center of the circle by manipulating the above formula to get it in the form
$$r=2Rarcsinh\left(\frac{\sqrt{a}}{R\sqrt{4\pi}}\right)$$
Next I can generate random numbers between $0$ and $2\pi$ that follow the uniform distribution.
Now in Lorentzian Geometry there are two sheeted hyperboloids, in which every point on each sheet is of equal spacetime distance from the center of the two sheeted hyperboloid. Each sheet of this type of two sheeted hyperboloid has constant negative curvature, and each point on this sheet is indistinguishable from every other point on the sheet, and all directions along this sheet are also equivalent. This sheet is equivalent to the hyperbolic plane. This means that one way to get the u, v coordinates for each of the randomly selected points I can treat the hyperbolic plane as a sheet of a two sheeted hyperboloid embedded in a 3d Lorentzian Spacetime, and convert from the coordinates that I get for the Lorentzian Spacetime to the u, v coordinates seeing as Lorentzian Geometry is flat.
Now in 3d Lorentzian Geometry I could use the distance formula $$Distance=\sqrt{{\Delta}x^2+{\Delta}y^2-{\Delta}z^2}$$
I could also say that $R$ in the previous equations is the spacetime distance between all the points in the hyperbolic plane and the center of the two sheeted hyperboloid that it is a part of.
The parametric equation of the hyperbolic plane embedded in 3d Lorentzian Spacetime could, that would also follows from the coordinate system that I wrote about in my question could be given as
$$x=Rsinh\left(\frac{u}{R}\right)cosh\left(\frac{v}{R}\right)$$
$$y=Rsinh\left(\frac{v}{R}\right)$$
$$z=Rcosh\left(\frac{u}{R}\right)cosh\left(\frac{v}{R}\right)$$
Now given a distance from the point with u, v coordinates $(0,0)$ it is possible to find the z value of that point using, in 3d Lorentzian Spacetime using the formula
$$z=Rcosh\left(\frac{r}{R}\right)$$
Next the value of $x^2+y^2$ can be found using the formula $$R^2+z^2=x^2+y^2$$
If the random numbers between $0$, and $2\pi$ I described earlier are given by the variable $\theta$ then I can convert them to x, and y using the formulas $$x=\sqrt{R^2+z^2}cos{\theta}$$ $$y=\sqrt{R^2+z^2}sin{\theta}$$
So now I would have the x, y, z coordinates of the points in a Lorentzian Geometry, and just need to convert them to the u, v coordinates I described earlier. The v coordinates of the points could be found using the equation $$v=Rarcsinh\left(\frac{y}{R}\right)$$ and the u coordinates could be found using the equation $$u=Rarcsinh\left(\frac{x}{Rcosh\left(\frac{v}{R}\right)}\right)$$