A Method for Constructing Surfaces with Smooth Curvature Variations Based on Discrete Geometry Models

This paper presents a new scheme for construction of a piecewise G2 surface which interpolates a rectangular curvature continuous quintic Bezier curve mesh. For application to practical CAD/CAM, a parametric surface with a controlled curvature distribution can be generated based on a discrete surface with a smooth curvature distribution. First, a discrete surface is constructed by minimizing an integral of square of the 2nd derivative of principal curvatures. The discrete surfaces have good curvature distributions, because they have a high degree of freedom. A G2 quintic Bezier curve mesh is constructed by curves that minimize an arc length integral of square of the 2nd arc length derivative of curvature binormal, based on the discrete surface. After estimating twist and cross-boundary derivatives from geometric properties of the discrete surface, a G2 surface interpolates the curve network by minimizing an energy function that measures the variation of principal curvatures. The generated surface elements are bi-quintic Gregory patches.