Polycurves

Polycurves are curves defined as the concatenation of a finite sequence of smooth curve segments. In a Euclidean space R^n, a polycurve gamma is obtained by joining a finite sequence of parameterized arcs gamma_i: [t_{i-1}, t_i] -> R^n with gamma_i(t_i) = gamma_{i+1}(t_i). The resulting map gamma: [t_0, t_m] -> R^n is continuous and piecewise smooth, with gamma restricted to each subinterval being smooth. If the tangent directions agree at the junctions, gamma is C^1; more generally, the joints may impose higher-order continuity.

Polycurves generalize polylines, which are the special case where each gamma_i is linear. They also generalize piecewise curves such as circular-arc or Bezier segments, and are central in computer-aided geometric design (CAGD) and robotics for constructing complex paths and shapes.

Properties include that length is additive across segments; curvature is concentrated at the joints; the image is a compact 1-dimensional subset of R^n and can be self-intersecting. The parameterization may be arbitrary, with reparameterization possible to a single interval or to arc length.

Variants and related notions: polycurve is related to piecewise-smooth curves and to piecewise Bezier or B-spline constructions. In computational geometry and CAD, polycurves are used to model complex trajectories and shapes, and to support operations such as subdivision, rendering, collision detection, and path planning.