A simple way to pre-compute sin/cos tables

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\sin a and\cos a within a certain interval, using infinitesimal rotations.We start to work in two dimensions with a clockwise rotation matrix:

R = \begin{pmatrix}\cos\alpha\ && -\sin\alpha\  \\ \sin\alpha\ && \cos\alpha\ \end{pmatrix}

For very small rotation angles\alpha\ \to 0 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:

R' = \begin{pmatrix}1  && -\alpha\  \\ \alpha\  && 1 \end{pmatrix}

Now we can generate our values on an interval\left [ s, t \right ) recursively applyingR' on a unit vector\hat{u} = ( \cos s, \sin s ) .

\hat{u'} = \hat{u}R'

=( u_x, u_y ) \begin{pmatrix}1  && -\alpha\  \\ \alpha\ &&  1 \end{pmatrix}

= \begin{cases} u'_x = u_x +  u_y\alpha\ \\ u'_y = -u_x\alpha\ + u_y \end{cases}

Supposing that we don’t have at hand any code or library that can compute\hat{u} for an arbitrary value ofs , we start with s = 0 \to \hat{u} = ( 1, 0 ). Read the rest of this entry »

Yes, yet another blog

Hello World!

I thought about having a blog or some kind of personal web site since the first time I came to use The Almighty Internet.So please, bear with me if at the beginning I’m going to be a tad slow at filling it with new ramblings.
Thank you, and enjoy.