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Check whether an angle is between two other angles

Weirdly, I’ve had to do this a lot recently: given an angle, determine whether it’s bounded by two other angles. Technically, this is always true — just go the other direction around the circle — but usually I care about going in a particular direction.

Step one is to normalize all the angles so that they fit into the range [0, 2π] (or [0, 360] if you prefer degrees). We’ll use TypeScript to demonstrate:

const TWO_PI = Math.PI * 2;
export function normalize(angle: number) {
  const clamped = angle % TWO_PI, // normalize into range [-2π, 2π]
    positive = clamped + TWO_PI, // translate into range [0, 4π]
    result = positive % TWO_PI; // normalize into range [0, 2π]

  return result;
}

Three steps here:

  1. Modulo the angle by 2π. If it’s positive, this compresses it into the range [0, 2π]; if it’s negative, this compresses it into the range [-2π, 0].
  2. Add 2π. This increases the range to [0, 4π], ensuring it’s positive.
  3. Modulo the angle by 2π again, compressing it into the range [0, 2π].

Once all the angles are within the range [0, 2π], they can be checked:

export function between(min: number, theta: number, max: number) {
  if (min < max) return min <= theta && theta < max;
  else return min <= theta || theta < max;
}
  1. If min is less than max, we can just ensure that theta is between them.
  2. Otherwise, it means the angles wrap around 0 (for example, min might be 7π / 4 and max might be π / 4). In that case, we need to check whether theta is greater than min (and implicitly less than 2π, which is the top of the range) or less than max (and implicitly greater than 0, which is the bottom of the range).

This function assumes that min is inclusive and max is exclusive.