G ² Interpolating Spline with Local Maximum Curvature

We introduce a novel class of G 2 continuous splines constructed using an innovative blending method, which guarantees precise interpolation of given control points. These splines are designed to achieve local curvature maxima specifically at these control points and possess compact local support, thereby eliminating the need for global optimization processes. The formulation ensures the splines are free from cusps and self-intersections and, notably, prevents adjacent segments from intersecting—a significant improvement over prior blending-based curve techniques. This framework utilizes quadratic Bézier splines in conjunction with quartic Bézier blending functions. A constructive algorithm is presented that generates these curvature-controlled curves without relying on global optimization. Through parametric adjustments of curvatures, the curve's geometry near control points can be tuned to create features ranging from smooth to sharp, thus broadening the design possibilities. Rigorous mathematical proofs and visual demonstrations validate all claimed properties of the framework.