arctanRiseRun

Arctanriserun is a descriptive term used to refer to the angle obtained by taking the arctangent of the ratio of rise to run for a line segment in a two-dimensional plane. If a segment moves up by Δy (rise) and to the right by Δx (run), then the slope is m = Δy/Δx and a common way to express the angle θ that the line makes with the positive x-axis is θ = arctan(m). The phrase arctanriserun is not a standard mathematical notation, but it captures this basic computation.

In practice, arctan(rise/run) yields a principal value in (-π/2, π/2). This can be insufficient to determine the actual direction of the line when the run value is negative, or when both rise and run are negative. For reliable angle determination, the two-argument variant atan2(Δy, Δx) is used in most programming languages and mathematical contexts, as it returns the angle in (-π, π] corresponding to the vector (Δx, Δy). If Δx = 0, the angle is π/2 if Δy > 0 or -π/2 if Δy < 0. If both Δx and Δy are zero, the direction is undefined.

Example: rise 3, run 4 gives arctan(3/4) ≈ 0.6435 radians (36.87 degrees). If rise 3, run -4, arctan(3/(-4)) ≈ -0.6435 radians, whereas atan2(3, -4) yields approximately 2.498 radians (143.13 degrees), reflecting the correct quadrant.

Applications of this concept appear in geometry, physics, computer graphics, and navigation, where converting a gradient or slope into an angular measure is useful. While arctanriserun is a workable description, the standard approach relies on arctan for a slope and atan2 for full directional angles.