Given an n-1 dimensional simplex in n-dimensional space (at an unknown rotation), I am looking for a way align the object with a given plane, i.e. rotate it in such a way that the coordinates for that plane are all 0.
For example, a triangle in 3d might have the following vertices:
\begin{align*} A &= (1,0,0) \\ B &= (0,1,0) \\ C &= (0,0,1) \end{align*}Given the y-plane for instance, I would like to rotate this triangle so that the y-coordinates of all vertices are 0, while maintaining shape.
The concept extends to higher dimensions, for instance aligning a tetrahedron in 4d. Unfortunately, I have not found a method to achieve this, or even a way to determine the original rotation of a lower-dimensional object from the vertices alone.
If anyone knows of how to do this or has a general outline it would be very much appreciated.