The angle measure i acos(x) is the logarithm of the rotation x + i√(1 − x²). If you want to compute the product of two rotations, you can take logarithms and r

The angle measure i acos(x) is the logarithm of the rotation x + i√(1 − x²). If you want to compute the product of two rotations, you can take logarithms and rewrite it as a sum of angle measures: [x + i√(1 − x²)][y + i√(1 − y²)] = exp(i acos(x) + i acos(y)) [x + i√(1 − x²)][y − i√(1 − y²)] = exp(i acos(x) − i acos(y)) Now if you just look at the real parts, you can write: xy − √(1 − x²)√(1 − y²) = cos(acos(x) + acos(y)) xy + √(1 − x²)√(1 − y²) = cos(acos(x) − acos(y)) Taking the sum on each side: 2xy = cos(acos(x) + acos(y)) + cos(acos(x) − acos(y)) This method is rather more cumbersome (due to rescaling and then converting the factors to rotations as an intermediate step) but fundamentally based on the same concept as: xy = exp(log(x) + log(y))