The smallest difference between 2 Angles

Question

Given 2 angles in the range -PI -  PI around a coordinate  what is the value of the smallest of the 2 angles between them   Taking into account that the difference between PI and -PI is not 2 PI but zero   Example   Imagine a circle  with 2 lines coming out from the center  there are 2 angles between those lines  the angle they make on the inside aka the smaller angle  and the angle they make on the outside  aka the bigger angle  Both angles when added up make a full circle  Given that each angle can fit within a certain range  what is the smaller angles value  taking into account the rollover

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Arithmetical  as opposed to algorithmic  solution   angle   Pi - abs abs a1 - a2  - Pi

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For UnityEngine users  the easy way is just to use Mathf DeltaAngle

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If your two angles are x and y  then one of the angles between them is abs x - y   The other angle is  2   PI  - abs x - y   So the value of the smallest of the 2 angles is   min  2   PI  - abs x - y   abs x - y     This gives you the absolute value of the angle  and it assumes the inputs are normalized  ie  within the range  0  2p      If you want to preserve the sign  ie  direction  of the angle and also accept angles outside the range  0  2p  you can generalize the above  Here s Python code for the generalized version   PI   math pi TAU   2 PI def smallestSignedAngleBetween x  y       a    x - y    TAU     b    y - x    TAU     return -a if a  lt  b else b   Note that the   operator does not behave the same in all languages  particularly when negative values are involved  so if porting some sign adjustments may be necessary

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x is the target angle  y is the source or starting angle   atan2 sin x-y   cos x-y     It returns the signed delta angle  Note that depending on your API the order of the parameters for the atan2   function might be different

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I rise to the challenge of providing the signed answer   def f x y     import math   return min y-x  y-x 2 math pi  y-x-2 math pi  key abs

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There is no need to compute trigonometric functions  The simple code in C language is    include  lt math h gt   define PIV2 M PI M PI  define C360 360 0000000000000000000 double difangrad double x  double y    double arg   arg   fmod y-x  PIV2   if  arg  lt  0    arg    arg   PIV2  if  arg  gt  M PI  arg    arg - PIV2   return  -arg     double difangdeg double x  double y    double arg  arg   fmod y-x  C360   if  arg  lt  0    arg    arg   C360  if  arg  gt  180  arg    arg - C360  return  -arg       let dif   a - b   in radians  dif   difangrad a b     let dif   a - b   in degrees  dif   difangdeg a b    difangdeg 180 000000   -180 000000    0 000000 difangdeg -180 000000   180 000000    -0 000000 difangdeg 359 000000   1 000000    -2 000000 difangdeg 1 000000   359 000000    2 000000   No sin  no cos  no tan      only geometry

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This gives a signed angle for any angles   a   targetA - sourceA a    a   180    360 - 180   Beware in many languages the modulo operation returns a value with the same sign as the dividend  like C  C    C   JavaScript  full list here   This requires a custom mod function like so   mod    a  n  - gt  a - floor a n    n   Or so   mod    a  n  - gt   a   n   n    n   If angles are within  -180  180  this also works   a   targetA - sourceA a     a gt 180    -360    a lt -180    360   0   In a more verbose way   a   targetA - sourceA a -  360 if a  gt  180 a    360 if a  lt  -180

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An efficient code in C   that works for any angle and in both  radians and degrees is   inline double getAbsoluteDiff2Angles const double x  const double y  const double c           c can be PI  for radians  or 180 0  for degrees       return c - fabs fmod fabs x - y   2 c  - c

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A simple method  which I use in C   is  double deltaOrientation   angle1 - angle2  double delta    remainder deltaOrientation  2 M PI

[language-agnostic] The smallest difference between 2 Angles

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