Construction of Bézier surfaces with minimal quadratic energy for given diagonal curves

Abstract

Diagonal curve is one of the most important shape measurements of tensor-product Bézier surfaces. An approach to construct Bézier surfaces with energy-minimizing from two prescribed diagonal curves is presented in this paper. Firstly, a general second order functional energy is formulated with several parameters. This functional includes many common functionals as special cases, such as the Dirichlet energy, the biharmonic functional and the quasi-harmonic functional etc. Secondly, the necessary and sufficient conditions for tensor-product Bézier surface with minimal energy from prescribed diagonal curves are derived. Besides prescribing the diagonal curves, other related problems are considered, those where boundary curves are also prescribed. Finally, the effectiveness of the proposed method is illustrated by several surface modeling examples.