Projection of points on the spherical triangle $[\text{sphericalTriangle}(x,y,z) \Longleftrightarrow \text{planarTriangle}(a, b)]$

Question

I'm looking for a formula to project $(x,y,z)$ point of a spherical triangle $(v_1, v_2, v_3)$ onto the point $(a,b)$ of planar triangle defined by same vertices $(v1, v2, v3)$ and vice versa

$$\text{sphericalTriangle}(x,y,z) \Longleftrightarrow \text{planarTriangle}(a, b)$$

For the sake of argument lets assume that $(a,b)$ origin is at point $v_1$, $\bf{a}$ goes along $(v_3 - v_1)$ vector and $\bf{b}$ is orthogonal to $\bf{a}$.

Answer 1

Done here: WebGL shader implementation