The unit simplex in $\mathbb{R}^3$ is
$$\Delta^3 = \left\{(t_1,t_2,t_3)\in\mathbb{R}^{3}\mid t_1+t_2+t_3 = 1 \mbox{ and } t_i \ge 0 \mbox{ for all } i\right\}$$ When trying to describe it parametrically, an obvious choice is $$ (x,y,1-x-y)$$ over an approriate two-dimensional domain. This particular parameterization is not symmetric, as the third coordinate plays a different role than the first two.
Are there any other parameterizations of the unit simplex $\Delta^3$ which are more natural? For example, ones in which the distance to the edge can be easily read. If so, I would appreciate some examples. Thank you!
Take
$$\begin{aligned} t_1 &= \sin^2\theta\cos^2\phi \\ t_2 &= \sin^2\theta\sin^2\phi \\ t_3 &= \cos^2\theta \end{aligned}$$