Nice article! I'm not a graphics programmer but mathematically it makes full sense that cross-product would be a vast optimization over using `sin()`. From a complexity perspective, the computation of a cross-product reduces to calculating a formal determinant, a fixed number of arithmetic operations, and hence resolves to O(1) complexity. By contrast, computing `sin()` is O(M(n)log(n)) (even though faster algorithms are often possible in practice). See Brent, Fast multiple-precision evaluation of elementary functions (1976).