Finding which is the shorter 2D angle

Calculating the shorter angle of rotation in 2D

Here’s a really simple math problem in 2D games development that has a surprisingly complex solution. If I were to give you angles a and b, how would you calculate 1) the direction — that is, clockwise being positive; and anti-clockwise being negative — and 2) magnitude of the shorter angle of rotation from a to b?

Main angle question
We want the angle in green.

When you visualise it, both values seem so obvious; which was why I was so surprised I couldn’t figure it out. On the surface, it seems really simple — if you just take b – a, doesn’t it give you the solution?

(What looks like) the solution

Here’s an example that illustrates how taking b – a can work:

If we let a = 150° and b = 170°, b – a = 170° – 150° = 20°

The case above demonstrates that for the rotations pictured below, the formula correctly gives us a counter-clockwise rotation of 20° as the shorter angle of rotation.

Example 1
a needs to rotate 20° (i.e. counter-clockwise) to get to b.

Conversely:

If we let a = 150° and b = 140°, b – a = 140° – 150° = -10°

The case above demonstrates that for these other rotations pictured below, the formula correctly gives us a clockwise rotation of -10° as the shorter angle of rotation.

Example 2
a needs to rotate -10° (i.e. clockwise) to get to b.

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The problem with angular values

When you begin to dig deeper, however, you will quickly find that things can get surprisingly complex problem for 2 reasons:

  1. Angles loop around beyond 360°, so if a = 380° and b = 40°, we get 340° as the shorter angle of rotation. This is obviously wrong, because the shorter angle of rotation is never larger than 180°.
  2. In most (i.e. 99%) game engines, the lower half of the circle goes from 0° to -180°, like radians do (refer to the picture below). Hence, if we were to consider cases like a = 160° and b = -160°, we will get 320° as the shorter angle of rotation.
Positive and negative angles
If measured in radians, 180° will be π.

The (actual) solution

Well, a – b actually works. Once you convert the result so that it is between -180° and 180° (or -π and π), that is. Here’s how you convert any angle into a value between -180° and 180°:

  1. Firstly, if your result is already between -180° and 180°, then you ignore the following steps.
  2. Next, convert your result into an angle between -360° and 360°. All you have to do is to modulo the result with 360, i.e. result % 360.
  3. Finally, if your result is less than -180°, add 360° to it; otherwise, if it is more than 180°, subtract 360° from it.

Or, in short, expressed mathematically:

f(x) = { sgn x × 360° – |x| if |x| > 180 x if 0 ≥ |x| ≤ 180

If that is hard to read, here’s a code snippet in C# that expresses the same thing:

float result = b - a; // Shorter angle of rotation.

// If our result's value is more than 180 or less than -180.
if(Mathf.Abs(result) > 180)
	// Covert the value to be between -180 and 180.
	result = -Math.Sign(result) * (360 - Math.Abs(result));

Conclusion

Did the article help you figure out your angles? Or maybe you spotted some errors with our math? Feel free to leave us a comment below.


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There are 3 comments:

  1. Thanks for the article! Minor nit: in step 3, you say “Finally, if your result is negative, add 360° to it; otherwise, subtract 360° from it.”. This should say “if your result is higher lower than 180°, add 360° to it; if it is higher than 180°, subtract 360° from it.”

    This is what the rest of the article does, and is the correct thing to do. It’s just a typo in the prose that threw me off.

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