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Discussion on relationship between minimal energy and curve shape.

Applied Mathematics(A Journal of Chinese Universities) 2014年第4期29卷 379-390页

作者： LI Xue-mei ZHANG Yong-xia MA Long ZHOU Yuan-feng ZHANG Cai-ming School of Computer Science and Technology Shandong University Shandong Province Key Lab of Digital Media Technology Shandong University of Finance and Economics

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 shape., 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 shape. of cubic Hermite curves could be determined for about 60 percent of all cases, some of which have unsatisfactory shape., however. Based on strain energy model and analysis, a new model is presented for constructing cubic Hermite curves with satisfactory shape., 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 shape. 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.

关键词： computer simulation cubic Hermite curve minimum strain energy curve shape.

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