arctan2dy

Arctan2dy is a mathematical function that extends the well‑known arctangent function to accommodate differential or infinitesimal changes in a two‑dimensional vector. In classical vector calculus the arctangent function, often implemented as atan2, returns the angle between the positive x‑axis and a point (x, y) in the plane, taking into account the signs of both coordinates to determine the correct quadrant. Arctan2dy modifies this concept by treating the y component as a differential element dy, allowing the calculation of angle changes in response to infinitesimal perturbations. This makes it especially useful in contexts where vectors evolve continuously, such as in differential geometry, robotics path planning, or dynamic systems modeling.

Mathematically, arctan2dy may be expressed as:

\[ \theta = \mathrm{atan2}\bigl(x,\; dy\bigr) \]

where \(x\) is a finite horizontal component and \(dy\) represents a small displacement in the vertical direction. The function outputs the angle \(\theta\) that describes the orientation of the resulting vector after the differential change. Because dy can be arbitrarily small, the function naturally aligns with the definition of a derivative, providing a direct link between angular response and rate of change.

In computational implementations, care must be taken to avoid division by zero when dy approaches zero, which can be handled by limiting procedures or by using the full atan2 function with a tiny epsilon adjustment. Libraries in numerical analysis software such as MATLAB, NumPy, and other scientific computing environments may provide direct support for this variant, or users may derive it from the standard atan2 by passing the appropriate arguments. The arctan2dy function thus offers a compact tool for quantifying instantaneous rotational changes in systems where one component evolves differentially.