Energy-minimized interpolatory B-spline lofting for sequences of closed contours

Abstract

B-spline lofting through closed cross-sections is common in reverse engineering, but interpolating each section on its own knot vector can produce an unnecessarily large control net. We construct a common-knot method for closed section curves of odd and even degree by minimizing the strain energy of each section subject to its interpolation constraints. Feasibility for closed section data and uniqueness of the resulting saddle-point solution require full row rank of the periodic collocation matrices, a property that a previously proposed periodic Schoenberg-Whitney window condition does not always guarantee. We establish the exact parity distinction: the condition is sufficient at odd point counts but admits singular configurations at every admissible even point count. For the odd and even degrees used here, the natural and shifting anchor knots are nonsingular; at odd point counts the window condition certifies the selected candidate, whereas at even point counts a rank test with an anchor fallback is applied. The lofting method therefore retains a nonsingular square subsystem for each section and uses nestedness to transfer full row rank to the rectangular common-knot system. Across six closed-contour models the common-knot method uses 83-98% fewer control points than traditional interpolating lofting, with section energies no larger than those of a previous knot-selection method within numerical precision.