Construction of Bézier surfaces with energy-minimizing diagonal curves from given boundary

Diagonal curve is one of the most important shape measurements of tensor-product Bézier surfaces. An approach to construct Bézier surfaces with energy-minimizing diagonal curves from four input boundary curves is presented in this paper. Firstly, the expression for the diagonal energy is formulated. Secondly, the necessary and sufficient conditions for tensor-product Bézier surface with minimal diagonal energy are derived. Both stretch energy and bending energy of the diagonal curves are studied. Finally, from the derived conditions, a construction algorithm for Bézier surfaces with minimal diagonal energy by using Lagrange-multiplier method is presented. The effectiveness of the proposed method is illustrated by several surface modeling examples.