Generating points uniformly distributed over the SURFACE of a standard simplex

Question

I would like to generate points that are uniformly distributed over the SURFACE of a standard $k$-simplex ($k$ dimensions, $k+1$ vertices).

One way to efficiently generate points that are uniformly distributed over the ENTIRETY of the $k$-simplex is to generate samples of $k$ iid exponential random variables, $E_i$, $i\in \left[1,k\right]$, then normalize, and $\vec{X}=\frac{\left\langle E_1,E_2,\dots,E_k\right\rangle}{\sum_{i=1}^k E_i}$, will be uniformly distributed over the entire $k$-simplex.

My problem is that the probability that one of these points lands on the surface (i.e. at a vertex, or along a hyper-edge, or on hyper-face, etc.) is nearly $0$, so that in practice all of the points end up being in the INTERIOR of the $k$-simplex.

In an attempt to sample the SURFACE of the $k$-simplex I have considered each hyper-edge, hyper-face, etc. separately and generated uniformly distributed samples on each (since they are themselves lower dimensional simplices). However, the problem becomes computationally unfeasible as $k$ increases because the number of hyper-edges, hyper-faces, etc. blows up.

Does anyone know of a method to explicitly generate samples that are uniformly distributed over the SURFACE (or BOUNDARY) of a standard $k$-simplex?

Answer 1

I created an algorithm for that. for detail see http://www.sciencedirect.com/science/article/pii/0098300493900457

briefly speaking, you can use following recursive formula to generate:

x_1=1-R_1^{1/n-1}
x_k=(1-\sum_{i=1}^{k} x_i)(1-R_k^{1/n-k}),    k=2,…,n-1
x_n=1-\sum_{i=1}^{n-1} x_i

Best wishes