arctangrade

Arctangrade is a proposed unit of angular measure used in theoretical and some fictional contexts to express slope-to-angle relationships via the arctangent function. It is not a standard physical unit, but a convenient way to compare slopes on a fixed, non-linear scale.

Definition and scale. Let s be the slope, defined as rise divided by run (a dimensionless ratio). Denote arctan(s) as the angle in radians whose tangent is s. If one adopts a fixed reference slope s0, commonly s0 = 1, then one arctangrade is defined as the angle arctan(s0) in radians, i.e., arctan(1) = pi/4 radians = 45 degrees. The arctangrade value A for any slope s is then A = arctan(s) / arctan(s0). With s0 = 1, A ranges from 0 to 2 (corresponding to 0 to 90 degrees). Thus 1 arctangrade equals 45 degrees, and a slope of infinity (vertical) maps to 2 arctangrades.

Properties and use. Arctangrade provides a nonlinear, bounded scale for representing slopes, highlighting the diminishing angular increase as slopes become steeper. It is mainly used for theoretical discussions, data visualizations, or fiction where a fixed, compact slope scale is desirable. Because it is defined via arctangent, small slopes change slowly in arctangrade terms, while very steep slopes approach the upper bound rapidly.

Conversion. To degrees, multiply by 45 when s0 = 1. To convert back, use s = tan(A arctan(s0)). See also arctangent, slope, and grade (angle measure).

Note. Arctangrade is not an established scientific unit and should be understood as a conceptual tool rather than a standard.