BetaTurns

Betaturns are a theoretical class of planar curves defined by a single controlling parameter, beta, that governs how sharply a path turns relative to a reference trajectory. The concept is used in geometry and motion planning to describe a continuum of shapes that interpolate between straight lines and curved routes. Different authors implement the idea with different turning rules, but all share the same basic notion: beta tunes the amount of turning along the arc length.

Construction and definition: A betaturn is generated by applying a beta-controlled turning rule to a seed path. Practically, at each step along the curve the heading changes by an amount proportional to beta times a prescribed turning function of the local context (for example a local estimate of curvature). When beta is zero, the resulting curve coincides with the seed path; as beta increases, bends accumulate and the curve becomes more curved.

Properties: Betaturns form a continuous family under beta; they are typically smooth and can maintain bounded curvature for finite beta. They are invariant under rigid motions (translation and rotation). With appropriate boundary conditions, large beta values can produce loops or tight bends, whereas small beta values yield nearly straight segments.

Applications and related terms: The concept is used in robotics and computer animation to design smooth, easily controllable trajectories. It also appears in discussions of curve design and path planning. See also curvature, differential geometry, and beta-turn in protein chemistry for a linguistically related term.