TORO Curves: G^2 Interpolating Curve with Maximum Curvature Control

We introduce TORO curves, a new family of G 2 -continuous interpolating splines that provide precise local curvature control without global optimization. Our construction assembles two types of primitive segments, each attaining curvature maxima strictly on the endpoint circular arcs: the trigonometric offset (TO) segment for inflection transitions, and the rational Bézier offset (RO) segment for constant-sign curvature spans. A core pillar of our framework is an offset-based decoupling strategy: by exploiting properties of offset curves, we decouple the geometry from the curvature magnitude, thereby enabling direct curvature prescription at control points. The construction is deterministic and local; each segment is solved via simple quadratic equations, and the rational Bézier weights are directly determined by geometric quantities. This provides strict local support and enables realtime responsiveness during interactive manipulation. Finally, we introduce an adaptive curvature configuration that guarantees robustness and visual continuity under deformation. Our results demonstrate that TORO curves can represent complex shapes while accurately reproducing basic primitives such as circular arcs and ellipses, offering an efficient, versatile tool for modern geometric design.