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.
In computational implementations, care must be taken to avoid division by zero when dy approaches zero, which