This is a kind of an old school trick, rarely useful nowadays, but it’s nonetheless interesting due to its simplicity and elegancy.
The main idea is about filling a table with pre-computed values of
within a certain interval, using infinitesimal rotations.We start to work in two dimensions with a clockwise rotation matrix:
For very small rotation angles
thus we can expand our rotation matrix elements with a Taylor series around zero.
Ignoring all the non linear terms we get something like this:
Now we can generate our values on an interval
on a unit vector
Supposing that we don’t have at hand any code or library that can compute
for an arbitrary value of
, we start with
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