Signed angle between two vectors without a reference plane

Question

(In three dimensions) I\'m looking for a way to compute the signed angle between two vectors, given no information other than those vectors. As answered in this question, it is

Answer 1

The relevant mathematical formulas:

  dot_product(a,b) == length(a) * length(b) * cos(angle)
  length(cross_product(a,b)) == length(a) * length(b) * sin(angle)

For a robust angle between 3-D vectors, your actual computation should be:

  s = length(cross_product(a,b))
  c = dot_product(a,b)
  angle = atan2(s, c)

If you use acos(c) alone, you will get severe precision problems for cases when the angle is small. Computing s and using atan2() gives you a robust result for all possible cases.

Since s is always nonnegative, the resulting angle will range from 0 to pi. There will always be an equivalent negative angle (angle - 2*pi), but there is no geometric reason to prefer it.