Geometric design of uniform developable B-spline surfaces: constraints and degrees of freedom

Abstract

This study investigates the geometric design of B-spline surfaces constructed by two boundary curves. The developability constraints are geometrically derived from the de Boor algorithm and expressed as a set of equations that must be fulfilled by the B-spline control points. These equations specify the number of degrees of freedom (DOFs) for the surface design. For a cubic B-spline surface with a freely selected first boundary curve, five more DOFs are available for a second boundary curve when both curves are defined by four control points. There remain (7–2m) DOFs for designing a cubic surface that consists of m consecutive patches. The results are consistent with previous findings for equivalent composite Bézier surfaces. A test example demonstrates design methods that fully use all of the DOFs without generating over-constrained systems in the solution process. This work provides a preliminary foundation for applications of developable B-spline surfaces in product design and manufacture.