I have a 3d normal vector $N$ which is adjusted to $N^\prime=normalize({N+\Delta N})$ with $\Delta N=(0,a,b)$.
$N$ was rotated by a matrix $M$ to give $X=NM$. I also want the normal vector $X^\prime=N^\prime M$.
For computational reasons, $M$ itself is not easily available, but $a$, $b$, and $X$ are.
Can I efficiently compute $X^\prime$ from $\{N, a, b, X\}$, without knowing $M$, and ideally without computing $\Delta N$?
Or, is there an approximation which works for "small" $\Delta N$?