# Thread: Is my approach to achieve the rotation affine transformation formula right?

1. ## Is my approach to achieve the rotation affine transformation formula right?

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)

2. ## Re: Is my approach to achieve the rotation affine transformation formula right?

is there a question here?

3. ## Re: Is my approach to achieve the rotation affine transformation formula right?

Originally Posted by romsek
is there a question here?

So that's a "yup it's perfectly correct"?

4. ## Re: Is my approach to achieve the rotation affine transformation formula right?

Originally Posted by romsek
is there a question here?
The question was in the title of the thread. I have not dealt with affine transformations and don't feel like looking up how they differ from linear ones.