Chebychev polynomial nodes on a triangle

Question

I am trying to find the Chebychev approximation nodes on a triangle (or tetrahedron) with arbitrary edge lengths.

For example

Suppose I have a $D$-variate function, $f(x_{1},...,x_{D})$, with range in $\mathbb{R}^1$.

The domain of $f$ is a tetrahedron, with constraints as follows:

$x_{1}+x_{2}\leq a_1$

$x_{3}+x_{4}\leq a_2$

$\vdots$

$x_{D-1}+x_{D}\leq a_D$

and

$x_i\geq 0 \;\forall \; i={1,...,D}$

(Alternatively a simpler tetrahedron I'd also be interested in would be $\sum_{i=1}^Dx_{i}<a$)

For example, suppose $D=2$ and $a_{1}=25$, how would I find the Chebychev nodes on this space below?

The chebychev nodes on the full rectangle are:

Rectangle chebychev nodes

As you can see, the edges are oversampled to minimize approximation error. If I now just take the domain restrictions, we end up with seemingly too few points along the hypotenuse!

triangle nodes, incorrect