How can I efficiently calculate the clockwise angle between two vectors in both 2D and 3D space?

Calculating Clockwise Angles Between Vectors Efficiently

Traditionally, computing the angle between two vectors required utilizing the dot product, which determines the inner angle within the range of 0 to 180 degrees. However, this approach presents challenges when determining the appropriate result between the angle and its complement.

Is there a more direct method for computing the clockwise angle?

2D Case

Similar to how the dot product is related to the cosine of the angle, the determinant is proportional to its sine. By incorporating this relationship, we can calculate the angle as follows:

dot = x1 * x2 + y1 * y2  # Dot product between [x1, y1] and [x2, y2]
det = x1 * y2 - y1 * x2  # Determinant
angle = atan2(det, dot)  # atan2(y, x) or atan2(sin, cos)

The orientation of the calculated angle aligns with that of the coordinate system. In a left-handed coordinate system where x points right and y points down, clockwise angles will yield positive values. Conversely, in a mathematical coordinate system where y points up, the result reflects counterclockwise angles as is customary in mathematics. Swapping the input vectors' order changes the sign, providing the flexibility to modify the result's sign.

3D Case

In three dimensions, arbitrary vectors define their own axis of rotation perpendicular to both. Since this axis does not have a fixed orientation, it's not possible to uniquely determine the direction of the angle of rotation. A common convention is to assign positive angles and align the axis to accommodate this convention. In this context, the dot product of normalized vectors suffices for angle computation:

dot = x1 * x2 + y1 * y2 + z1 * z2  # Between [x1, y1, z1] and [x2, y2, z2]
lenSq1 = x1 * x1 + y1 * y1 + z1 * z1
lenSq2 = x2 * x2 + y2 * y2 + z2 * z2
angle = acos(dot / sqrt(lenSq1 * lenSq2))

Plane Embedded in 3D

For vectors constrained within a plane with a known normal vector n, there's a specific case to consider. The axis of rotation coincides with n, and the orientation of n fixes the orientation of the axis. In this scenario, we can modify the 2D computation above to include n in the determinant, transforming it into a 3x3 matrix:

dot = x1 * x2 + y1 * y2 + z1 * z2
det = x1 * y2 * zn + x2 * yn * z1 + xn * y1 * z2 - z1 * y2 * xn - z2 * yn * x1 - zn * y1 * x2
angle = atan2(det, dot)

For this computation to be valid, the normal vector n must be normalized to unit length.

Alternatively, the determinant can be expressed as a triple product:

det = n · (v1 × v2)

This approach may be easier to implement in some APIs and provides insight into the underlying mechanics: The cross product is proportional to the angle's sine and lies perpendicular to the plane, meaning it is a multiple of n. The dot product, therefore, essentially measures the length of that vector with the correct sign applied.

Range 0 – 360°

Most atan2 implementations return angles within the range [-π, π] in radians or [-180°, 180°] in degrees. To obtain positive angles within [0, 2π] or [0°, 360°], the following transformation can be applied:

dot = x1 * x2 + y1 * y2  # Dot product between [x1, y1] and [x2, y2]
det = x1 * y2 - y1 * x2  # Determinant
angle = atan2(det, dot)  # atan2(y, x) or atan2(sin, cos)

Alternatively, the following expression avoids case distinction:

dot = x1 * x2 + y1 * y2 + z1 * z2  # Between [x1, y1, z1] and [x2, y2, z2]
lenSq1 = x1 * x1 + y1 * y1 + z1 * z1
lenSq2 = x2 * x2 + y2 * y2 + z2 * z2
angle = acos(dot / sqrt(lenSq1 * lenSq2))

This correction technique is not limited to this specific problem but can be applied to any scenario involving atan2.

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