Geometric Algorithm for Optimal Quadratic Uniform Trigonometric B-Spline Interpolation Curves

A geometric algorithm is developed for quadratic uniform trigonometric B-spline interpolation with segment-specific shape parameters. The interpolation conditions generate all control points recursively from an initial vertex. For prescribed parameters, the complete derivative calculations give a weighted initial vertex that uniquely minimizes each of the three piecewise derivative penalties. The segment-dependent weights are retained in the subsequent constrained parameter optimization. Individual and normalized weighted objectives define different choices of curve, while a consistent squared graph-error functional is used for function approximation. The geometric interpretation of the derivative penalties is distinguished from arc length, bending energy, and curvature continuity. Recomputed spatial, closed planar, and function-graph examples report objective values, interpolation and graph errors, and join diagnostics, with comparisons against polynomial constructions. The results exhibit objective-dependent trade-offs and do not establish uniform superiority or general shape preservation.