Construction of B-spline surface from cubic B-spline asymptotic quadrilateral

Abstract

Asymptote is widely used in astronomy, mechanics, architecture and relevant subjects. In this paper, by analyzing the Frenet frame and the Darboux frame of a curve on the surface, the necessary and sufficient conditions for a quadrilateral boundary being asymptotic of a surface are derived. This quadrilateral is called asymptotic quadrilateral. Given corner data including positions, tangents and curvatures of a cubic B-spline quadrilateral with six control points in each boundary, a family of asymptotic quadrilaterals are constructed after solving the identification conditions of the control points. An optimized one is obtained by minimizing the strain energy of the boundary curves. Then, the transverse tangent vectors along the boundaries of the B-spline surface can be obtained by the asymptotic conditions and the resulting B-spline surface is of bi-quintic degree. Two arrays of control points of the surface along the quadrilateral are obtained from combinations transverse tangent vectors and the boundaries which are elevated from the cubic B-spline curves. For the given inner control points, B-spline surface of bi-quintic degree interpolating the cubic B-spline asymptotic quadrilateral is constructed. The optimized surface is the one with the minimized thin plate spline energy. The method is verified by some representative examples including the boundary curves with lines and inflections. Such interpolation scheme for the construction of the tensor-product B-spline surfaces is compatible with the CAD systems.