atan2x2

atan2x2 is a nonstandard term used in some numerical computing contexts to denote a variant or extension of the two-argument arctangent function applied to 2-by-2 data. Because there is no universally accepted definition, its precise meaning depends on the implementation. In practice, there are two common interpretations.

One interpretation is a vectorized arctangent on a 2x2 matrix: treat each column as a 2D vector (x1, y1) and (x2, y2), and compute theta1 = atan2(y1, x1) and theta2 = atan2(y2, x2), returning a 2-element vector [theta1, theta2]. This provides orientation angles for two embedded 2D vectors and is convenient when handling paired 2D data stored in a matrix.

A second interpretation appears in the context of 2D linear transformations: when extracting the rotation component from a 2x2 transformation, atan2x2 may denote theta = atan2(R21, R11) where R is the rotation part of the matrix obtained via polar decomposition. This yields the in-plane rotation angle corresponding to the transformation.

Properties and considerations: Each angle follows the standard atan2 convention and lies in (-pi, pi]. The exact result depends on the chosen interpretation and input layout. Degenerate or zero-vector columns can lead to undefined results or special-case handling.

Applications: The concept is used informally in computer graphics, robotics, computer vision, and signal processing to obtain orientation information from 2D data or transformations. It is not a standard mathematical function, so definitions vary by library or domain.

See also: atan2, polar decomposition, 2D rotation matrices.