angleDifferencea

angleDifferencea is a mathematical concept and a common programming construct used to measure the angular difference between two directions in a plane. Given two angles a and b, expressed in radians or degrees, angleDifferencea(a, b) yields the signed smallest difference from b to a. The result indicates both how far one must rotate to align with the other and the direction of rotation: a positive value corresponds to a counterclockwise rotation, a negative value to a clockwise rotation.

Let d = a - b. The value of angleDifferencea(a, b) is d normalized to the principal interval (-π, π] if angles are in radians, or (-180°, 180°] if in degrees. A common implementation uses wrap to the principal value: d' = (d + π) mod 2π - π; similarly for degrees with 360°. This ensures the result always lies within the chosen interval and is periodic with period 2π.

Properties of angleDifferencea include antisymmetry: angleDifferencea(a, b) = -angleDifferencea(b, a), and the fact that the absolute value abs(angleDifferencea(a, b)) gives the smallest angular distance between the directions, in [0, π]. Examples: angleDifferencea(30°, 350°) = 40°; angleDifferencea(π/4, −π/4) = π/2. The function is invariant to adding multiples of 360° (or 2π) to either angle.

Applications of angleDifferencea span robotics, navigation, computer graphics, and animation, where a consistent angular difference is needed for interpolation, orientation control, or state estimation. In three-dimensional contexts, angleDifferencea must be replaced by orientation differences using quaternions or rotation matrices, since planar differences do not capture full 3D orientation.