To achieve the formula that rotates a point theta degrees around an arbitrary rotation axis, We take arbitrary point *c* on the rotation axis(we know that it's a fixed point), Now $\displaystyle T(x)$ being the function that rotates the point **x** theta degrees around the rotation axis, we can find a linear transformation $\displaystyle L(x)$ such that $\displaystyle T\left(x\right)-c=L\left(x-c\right) $, So **A** being the corresponding transformation matrix for $\displaystyle L(x)$ we have$\displaystyle T\left(x\right)=c+A\left(x-c\right)$

$\displaystyle =c+A\left(x\right)+A\left(-c\right)$

$\displaystyle =c+A\left(x\right)-A\left(c\right)$

$\displaystyle =A\left(x\right)+\left(c-A\left(c\right)\right)$

This also proves that rotation is an affine transformation(a linear transformation followed by a translation) , and BTW similar approach can be used to achieve formulas for other affine transformations that have a fixed point(like shear, reflection and scaling)