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    <p>Note that the inversion of matrix is not that trivial at all! Firstly, all n (where n is the dimension of your space) points must be in general position (i.e. no individual point can be expressed as a linear combination of rest of the points [caveat: this may seem to be a simple requirement indeed, but in the realm of numerical linear algebra, it's nontrivial; final decison wheter such configuration really exist or not, will eventually be based on the 'actual domain' specific knowledge]). </p> <p>Also the 'correspondence' of the new and old points may not be exact (and then you should utilize the best possible approximator of the 'true correspondence', i.e.:). Pseudo inverse (instead of trying to utilize the plain inverse) is recommend allways when your lib provides it. </p> <p>The pseudo inverse has the advantage that you'll be able to use more points for your transformation, hence increasing the probability that at least n points will be in general position. </p> <p>Here is an example, rotation of unit square 90 deg. ccw in 2D (but obviously this determination works in any dim), with <code>numpy</code>:</p> <pre><code>In []: P= matrix([[0, 0, 1, 1], [0, 1, 1, 0]]) In []: Pn= matrix([[0, -1, -1, 0], [0, 0, 1, 1]]) In []: T= Pn* pinv(P) In []: (T* P).round() Out[]: matrix([[ 0., -1., -1., 0.], [ 0., 0., 1., 1.]]) </code></pre> <p>P.S. <code>numpy</code> is also fast. Transformation of 1 million points in my modest computer:</p> <pre><code>In []: P= matrix(rand(2, 1e6)) In []: %timeit T* P 10 loops, best of 3: 37.7 ms per loop </code></pre>
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    1. POrotating coordinate system via a quaternion
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    1. COthank you for this, numpy looks like a very attractive alternative to euclid. my main roadblock was simply figuring out the math without a background in linear algebra. [xyz]*[i'j'k']^-1=[x'y'z'] was the missing part.
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