I am attempting to find the inverse of the function:
$f:S^2 \to S^2$, defined as:
$f(x,y,z)=(x\cos(z)+y\sin(z), x\sin(z)-y\cos(z),z)$
My approach so far has been to use linearity and express the function as a sum of scaled basis vectors:
$f(x,y,z)=x(\cos(z),\ sin(z),0)+y(\sin(z), -\cos(z),0)+z(0,0,1)$
However, I'm unsure how to proceed from this point. Any guidance or assistance would be greatly appreciated! Thanks.
Best Answer
Your transformation can be written under a matrix form :
$$\pmatrix{x'\\y'\\z'}=\pmatrix{\cos(z) & \sin(z) & 0\\ \sin(z) & -\cos(z) & 0\\0&0&1}\pmatrix{x\\y\\z}$$
Its inverse transformation is therefore given by the inverse matrix (the inverse being obtained by inverting the diagonal blocks) :
$$\pmatrix{x\\y\\z}=\pmatrix{\cos(z) & \sin(z) & 0\\ \sin(z) & -\cos(z) & 0\\0&0&1}\pmatrix{x'\\y'\\z'}$$