Surface area of transformed sphere

Question

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface area of the newly transformed sphere? Is there a general algorithm to use that would work here?

Answer 1

As some of the comments suggested, rotations and translations leave the surface area fixed, and a dilation by a factor of $a$ changes the surface area by a factor of $a^2$, but the general case is more difficult.

Consider any affine transformation $T$. Since translations don't affect the area, we may as well postcompose our transformation with a translation and hence assume that $T$ fixes the origin. If the transformation produces an ellipsoid with semi-axis lengths $a, b, c$, we can readily set up the surface integral, but unless $a = b = c$, the resulting integral is elliptic, which means that it can't be evaluated except perhaps for some special values.

Good approximations of the surface area $S$ are, however, available: For example, the approximation $$S \approx 4\pi\left(\frac{(bc)^p + (ca)^p + (ab)^p}{3}\right)^{\frac{1}{p}}$$ is reasonably accurate for certain values of $p$: For $p = \frac{8}{5}$ the error is $< 1.2\%$.

If you have more information, e.g., if $a, b \gg c$, or if $a, b, c$ are all very close, then other good approximations are available.