parametric formula for rotatopes

[bo198214]

There was just asked about deriving the parametric formula in this thread. But a derivation is not that difficult.
First we need a parametric formula of the d-Spheres.
For sake of simplicity, I dont give them here, but only mention "general polar coordinates" (edit: as I now see they are anyway already given in this thread). Let us denote by H_i^d the i-th component of a parametrization of the d-Sphere. H depends on a d-dimensional parameter vector and i runs from 1 to d+1. But H depends also on the radius R. Let me write it this way H[R]. Then the signature is roughly H^d[R]: R^d -> R^d+1.
Now we can easily give the parameter formula of (A₁...A_n1...1) with d 1's. It is
P[R] = P₁[R₁+H^n+d-1₁[R]] x ... x P_n[R_n+H^n+d-1_n[R]] x (H^n+d-1_n+1[R],...,H^n+d-1_n+d[R])
Where the P_i's are (recursively) the parametric formula of the A_i, R_i is the radius parameter of P_i, and the cartesian product x of parametric formulas is self explaining.

As trivial example consider (11). Its parametric formula is by the above formula P[R] = (H¹₁[R],H¹₂[R]) = H¹[R]. Which is of course trivially indeed the 1-sphere. Or more specifically H¹[R](t)=(Rcos(t),Rsin(t))

Having this representation we can continue with the torus ((11)1) as a second example. We apply the above formula an gain:
P[R] = P₁[R₁+H¹₁[R]] x (H¹₂[R])
= H¹[R₁+H¹₁[R]] x (H¹₂[R])
P[R](s,t)= H¹[R₁+Rcos(t)](s) x (Rsin(t))
= ((R₁+Rcos(t))cos(s),(R₁+Rcos(t))sin(s),Rsin(t))
which is indeed the torus parametric formula.

This procedure can be similarly applied to the rectangular bracket [] and the lozenge/tegum angles <> by exchanging H^d correspondingly.