Specifically,
we can never tell where the world points
were to begin with. Adding one to every x coordinate in P and then subtracting 1 in every tx is undetectable.
So, wlog
we can assume that sum(P(k,:)) = 0 for k
from 1 to 3, ie., sum(x1 … xn) = 0, sum(y1…yn)
= 0, sum(z1 … zn) = 0.
Rotation
doesn’t move the origin, which is now the
center of mass. Neither does scaled orthographic projection. So, this only moves from
translation.
Explicitly,
we assume sum(p) = (0,0,0)^T.
Then: sum(s*R(p)) = s*R(sum(p)) = s*R(0,0,0)^T = (0,0,0)^T.
(^T means transpose).