Characterizing degrees of freedom for geometric design of developable composite Bézier surfaces

This paper studies geometric design of developable composite Bézier surfaces from two boundary curves. The number of degrees of freedom (DOF) is characterized for the surface design by deriving and counting the developability constraints imposed on the surface control points. With a first boundary curve freely chosen, (2m+3)(2m+3), (m+4)(m+4), and five DOFs are available for a second boundary curve of a developable composite Bézier surface that is G0G0, G1G1, and G2G2, respectively, and consists of mm consecutive patches, regardless of the surface degree. There remain five and (7-2m)(7-2m) DOFs for the surface with C1C1 and C2C2 continuity. Allowing the end control points to superimpose produces Degenerated triangular patches with four and three DOFs left, when the end ruling vanishes on one and both sides, respectively. Examples are illustrated to demonstrate various design methods for each continuity condition. The construction of a yacht hull with a patterned sheet of paper unrolled from 3D developable surfaces validates practicality of these methods in complex shape design. This work serves as a theoretical foundation for applications of developable composite Bézier surfaces in product design and manufacturing.