Discussion on relationship between minimal energy and curve shapes

作     者：LI Xue-mei ZHANG Yong-xia MA Long ZHOU Yuan-feng ZHANG Cai-ming 

作者机构：School of Computer Science and Technology Shandong University Shandong Province Key Lab of Digital Media Technology Shandong University of Finance and Economics 

出 版 物：《Applied Mathematics(A Journal of Chinese Universities)》 (高校应用数学学报（英文版）（B辑）)

年 卷 期：2014年第29卷第4期

页      面：379-390页

核心收录：

学科分类：081203[工学-计算机应用技术] 08[工学] 0835[工学-软件工程] 0812[工学-计算机科学与技术（可授工学、理学学位）] 

基　　金：Supported by the National Natural Science Foundation of China(61173174,61103150,61373078) the NSFC Joint Fund with Guangdong under Key Project(U1201258) the National Research Foundation for the Doctoral Program of Higher Education of China(20110131130004) 

主　　题：computer simulation cubic Hermite curve minimum strain energy curve shape. 

摘      要：Energy minimization has been widely used for constructing curve and surface in the fields such as computer-aided geometric design, computer graphics. However, our testing examples show that energy minimization does not optimize the shape of the curve sometimes. This paper studies the relationship between minimizing strain energy and curve shapes, the study is carried out by constructing a cubic Hermite curve with satisfactory shape. The cubic Hermite curve interpolates the positions and tangent vectors of two given endpoints. Computer simulation technique has become one of the methods of scientific discovery, the study process is carried out by numerical computation and computer simulation technique. Our result shows that： （1） cubic Hermite curves cannot be constructed by solely minimizing the strain energy; （2） by adoption of a local minimum value of the strain energy, the shapes of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shapes, however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shapes, which is a modification of strain energy model. The new model uses an explicit formula to compute the magnitudes of the two tangent vectors, and has the properties： （1） it is easy to compute; （2） it makes the cubic Hermite curves have satisfactory shapes while holding the good property of minimizing strain energy for some cases in curve construction. The comparison of the new model with the minimum strain energy model is included.