Curvi

Curvi is a term used in geometry and computational design to describe a family of smooth, differentiable curves with adjustable curvature profiles. In this context, curvi curves are defined by a parametric curve r(t) mapping an interval into two- or three-dimensional space, with a curvature function κ(t) that can be prescribed or controlled during design. The curvature κ(t) measures how quickly the curve's direction changes with respect to arc length.

Construction approaches: One method specifies κ(t) directly and integrates the curve using the Frenet-Serret frame to recover r(t). Another common approach uses curvature-constrained splines: Bezier- or B-spline-like schemes augmented with local curvature control at control points. Numerical methods often enforce bounds on κ(t to prevent sharp bends and ensure smooth transitions. In practice, designers seek curvature continuity and stability across segments, balancing aesthetic and functional constraints.

Applications: Curvi concepts appear in computer-aided design, typography, animation, and robotics path planning, where smooth trajectories and predictable bending behavior are desirable. They can be combined with constraints such as obstacle avoidance, endpoint tangents, or curvature continuity (G1/G2 continuity) to produce aesthetically pleasing or physically feasible curves. In animation and manufacturing, curvi trajectories help ensure predictable motion and manufacturability.

Etymology and status: The term curvi derives from curvus, Latin for "bent," and is used informally in some design and mathematics communities as a descriptive label rather than a formally standardized notion. As a result, definitions and implementations of curvi may vary between sources. This article treats curvi as a conceptual framework within mathematical design rather than a universally adopted standard.