Kurveskiftet

Kurveskiftet is a concept in differential geometry that describes a controlled modification of a plane curve’s curvature profile along its arc length. It is used to analyze how local changes in turning behavior affect the global shape of a curve and to design curves with prescribed curvature properties.

Definition and construction: Let gamma(s) be a regular plane curve parameterized by arc length s in [0, L], with curvature kappa(s) and tangent angle theta(s). A kurveskiftet by a shift function h(s) yields a new curve gamma_h(s) whose curvature is kappa_h(s) = kappa(s) + h(s) and whose tangent angle is theta_h(s) = theta(s) + ∫_0^s h(u) du. The new curve is obtained by integrating the Frenet equations: gamma_h'(s) = (cos theta_h(s), sin theta_h(s)). This framework allows prescribing how the turning rate changes along the curve.

Existence, regularity and constraints: Existence and uniqueness of gamma_h follow from standard ordinary differential equation theory up to a rigid motion of the plane. Regularity requires the shift function h to be integrable on [0, L], and to avoid singularities one typically imposes that kappa_h(s) stays away from zero on the domain or applies suitable boundary conditions when the curve is closed.

Applications and relations: A constant shift h(s) = c changes the overall turning rate uniformly, affecting the curvature integral and, for closed curves, the total curvature. In practice, kurveskiftet provides a design tool in computer graphics, CAD, and path planning, and connects to broader topics in curve evolution and the study of curvature spaces.

See also: curvature, Frenet-Serret formulas, plane curves, curve evolution, offset curves. Etymology: derived from the Swedish words for curve and shift; used in some geometric discussions to describe curvature modification.