MHKurve

MHKurve is a term used to describe a class of planar curves defined by a curvature function that is a polynomial in the arc length parameter. Formally, the curvature κ(s) is given by κ(s) = a0 + a1 s + a2 s^2 + ... + an sn, where s ranges over an interval [0, S]. The curve is constructed by selecting an initial direction θ0 and computing the tangent angle as θ(s) = θ0 + ∫0^s κ(u) du. The spatial coordinates are then obtained by integrating the unit tangent: x(s) = ∫0^s cos θ(u) du and y(s) = ∫0^s sin θ(u) du.

Special cases help illustrate the family. If κ is constant, the resulting curve is a circle. If κ is linear in s, the curve is a clothoid (Cornu spiral), a well-known transition curve used in design and robotics. Higher-degree polynomial curvature functions yield a broader set of smooth transition profiles, allowing more complex shaping of the curve to meet specific path requirements.

Properties of MHKurves include smoothness and controllable curvature distribution. Because κ(s) is a polynomial, the curves are infinitely differentiable, and the coefficients can be chosen to achieve desired turning behavior. Depending on the coefficients, the curves can be arranged to be non-self-intersecting or, in some configurations, closed, though closure is not guaranteed for arbitrary parameter choices.

Applications of MHKurves span trajectory planning for autonomous vehicles and robotics, road and railway engineering for smooth transition curves, and computer graphics where customizable, smooth interpolations between segments are useful.

See also: clothoid, Euler spiral, planar curve, curvature, trajectory planning.