Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider tranformation which turns one aligned rectangle to another:

enter image description here

This tranformation can be written in matrix form in the following way

enter image description here






i.e. transform matrix is diagonal.

My question is: what about arbitrary matrix? What 2 geometric figures defines it? Is this a pair of two arbitrary quadrangles?

If "yes", then what is the formula, relating angle coordinates to matrix coefficients?

How is it extrapolated to multidimensional?


I realized, that the number of equations in each transformation is $N$, the dimension of space, and the number of unknowns is $N^2+N=N(N+1)$ where $N$ is the number of translation column elements and $N^2$ is the number of linear matrix elements. Hence to make the system complete, we need $N(N+1)/N = (N+1)$ sample pairs of points.

So in the case of 2D we need to provide 2 arbitrary triangles to define transform.

Where to find a formula?

share|cite|improve this question

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.