karradius

Karradius is a geometric descriptor used to quantify how uniformly a curve bends by measuring the dispersion of its local radius of curvature. For a smooth plane or space curve γ(t), t in [0,1], the radius of curvature at t is R(t) = 1/κ(t), where κ(t) is the curvature. The karradius is defined as the coefficient of variation of R over the curve: K = σ_R / μ_R, with μ_R the mean of R(t) and σ_R its standard deviation. In practice, curvature is evaluated at discrete samples along the curve or its arc length, and R is regularized where κ approaches zero to avoid infinities.

Properties of karradius include scale invariance: if the curve is uniformly scaled, R and its statistics scale proportionally, leaving K unchanged. A small karradius indicates nearly constant curvature, while a large value signals substantial variation in bending. For a perfect circle, where κ is constant, karradius is zero. For a straight line, κ is zero everywhere and K is undefined; practical implementations use a lower curvature threshold or regularization.

Calculation considerations involve sampling density, noise, and handling inflection points where curvature changes sign. Robust implementations may use arc-length parametrization, smoothing, or robust statistics to mitigate outliers in R(t).

Applications span analysis of smoothing quality in spline fitting, characterization of handwriting or gesture trajectories, and evaluation of road or flight paths where uniform curvature is desirable. Limitations include sensitivity to measurement noise and the need for consistent parameterization. See also curvature, radius of curvature, and coefficient of variation.