atan2x

atan2X is not a standard mathematical function, but a generalized form of the two-argument arctangent used in some software libraries and theoretical contexts to incorporate an extra parameter that modifies the input before applying the arctangent. The specific semantics of atan2X vary by implementation, so a definition should be taken from the accompanying documentation.

One common interpretation is to introduce a scalar scaling parameter s that acts on the x input prior to the arctangent. In this form, atan2X(y, x; s) is defined as atan2(y, s·x). When s equals 1, the function reduces to the ordinary atan2(y, x). This scaling preserves the overall structure of the atan2 result, including quadrant information, for positive s; if s is negative, the effective x-coordinate is flipped, which alters the quadrant in a way analogous to reversing the x sign in the standard function.

Other uses of the notation may describe atan2X as representing an angle relative to a rotated reference axis, for example by introducing a rotation parameter φ so that atan2X(y, x; φ) = atan2(y, x) − φ. In these contexts, the exact transformation depends on the intended coordinate frame.

Implementation notes: in code, the common scalable form can be computed as angle = atan2(y, s*x). Care should be taken with degenerate cases such as x = 0 or y = 0, and with the sign of s which influences the resulting quadrant. See also atan2, polar coordinates, and coordinate transformations.