Determining the average angle

Question

I'm developing an application that involves getting the camera angle in a game. The angle can be anywhere from 0-359. 0 is North, 90 is East, 180 is South, etc. I'm using an API, which has a getAngle() method in Camera class.

How would I find the average between different camera angles. The real average of 0 and 359 is 179.5. As a camera angle, that would be South, but obviously 0 and 359 are both very close to North.

Answer 1

You can think of it in terms of vectors. Let θ1 and θ2 be your two angles expressed in radians. Then we can determine the x and y components of the unit vectors that are at these angles:

x1 = sin(θ1)
y1 = cos(θ1)

x2 = sin(θ2)
y2 = cos(θ2)

You can then add these two vectors, and determine the x and y components of the result:

x* = x1 + x2
y* = y1 + y2

Finally, you can determine the angle of this resulting vector:

θavg = tan-1(y*/x*)

or, even better, use atan2 (a function supported by many languages):

θavg = atan2(y*, x*)

You will probably have to separately handle the cases where y* = 0 and x* = 0, since this means the two vectors are pointing in exactly opposite directions (so what should the 'average' be?).

Answer 2

It depends what you mean by "average". But the normal definition is the bisector of the included acute angle. You must put both within 180 degrees of each other. There are many ways to do this, but a simple one is to increment or decrement one of the angles. If the angles are a and b, then this will do it:

if (a < b)
  while (abs(a - b) > 180) a = a + 360
else
  while (abs(a - b) > 180) a = a - 360

Now you can compute the simple average:

avg = (a + b) / 2

Of course you may want to normalize one more time:

while (avg < 0) avg  = avg + 360
while (avg >= 360) avg = avg - 360

On your example, you'd have a=0, b=359. The first loop would increment a to 360. The average would be 359.5. Of course you could round that to an integer if you like. If you round up to 360, then the final set of loops will decrement to 0.

Note that if your angles are always normalized to [0..360) none of these loops ever execute more than once. But they're probably good practice so that a wild argument doesn't cause your code to fail.