臺灣博碩士論文加值系統

當雲形曲線(B-spline)在不同系統之間轉換時，藉由次數提升(Degree Raising)及次數縮減(Degree Reduction)來轉換雲形曲線的次數是很常見的。本研究的主要目的在於使用放蕊步驟(Blossoming Approach)對貝茲曲線(Bézier Curve)的次數縮減作進一步的探討。本研究提出一個次數縮減的演算法使得能夠找出次數縮減後貝茲曲線的控制點(Control Points)。此演算法比舊往的演算法更容易瞭解與程式化並且能夠表達出控制點的幾何關係。此外，本研究提出矩陣的運算於放蕊演算法來改良放蕊演算法且利用此改良演算法來建構貝茲曲線。

The conversion of a B-spline curve in different domains of degree through the degree raising or degree reduction is common when transferring the B-spline from one computer system to another. This study explores further degree reduction in Bézier curve by use of with blossoming approach. I present a degree reduction algorithm to establish the control points of the reduced Bézier curve of degree from . This algorithm is easier to understand than the previous one and is simpler to program and can express the geometric relationship of control points before degree reduction. In addition, I present a modified blossoming algorithm, implemented with matrix operations, to enhance the original work and to construct the degree reduced Bézier curve.

封面內頁
簽名頁
授權書 iii
中文摘要 iv
ABSTRACT v
ACKNOWLEDGEMEMTS vi
TABLE OF CONTENTS vii
LIST OF FIGURES ix
LIST OF TABLES x

Chapter 1. INTRODUCTION 1
1.1 Objectives of Research 1
1.2 Literature Review 2
1.2.1 Cox and De-Boor Algorithm 2
1.2.2 Blossom Algorithm 5
1.2.3 Degree Elevation 9
1.2.4 Bézier Curve 14
1.3 Structure of the Thesis 18
Chapter 2. METHODOLOGIES 19
2.1 Degree Reduction 19
2.2 Example of Inverse Matrix 25
2.3 The Geometric Relationship between Control Points 30
Chapter 3. CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH 32
3.1 Conclusions 32
3.2 Suggestions for further research 33
References 34

Appendix 36

1.Carl de Boor, “On calculating with B-Splines”, Journal of Approximation Theory 6, 50-62, 1970.
2.Cox, M.G., “The numerical evaluation of B-splines,” J. Inst. Maths. Applies. 10, 134-149, 1972.
3.Deng, J. J., “Right-angled Triangle Property in Inversion of General Tridiagonal Matrices,” Journal of the Chinese Institute of Engineers, 27(1), 79-90, 2004.
4.Deng, J. J., “Theory of a B-spline basis function,” International Journal of Computer Mathematics, 80(5), 649-664, 2003.
5.Farin, Gerald , “Curves and Surfaces for CAGD: A Practical Guide, 5th ed.”, Academic Press, San Diego, 2002.
6.Hartmut, Prautzsch , Degree elevation of B-spline curves. In: Computer- Aided Geom. Des. 1(2), 193–198, 1984.
7.Hartmut, Prautzsch and Wolfgang, Boehm and Marco, Paluszny, “Bézier and B-spline Techniques”, 2000.
8.Lutterkort, J. Peters and U. Reif, “Polynomial degree reduction in the L2-norm equals best Euclidean approximation of Bézier coefficients.” Computer Aided Geometric Design 16, 607–612, 1999.
9.Lyle Ramshaw, “Blossoming: A Connect-the-Dots Approach to Splines”, 1987.
10.Matthias Eck, “Degree reduction of Bézier curves.” Computer Aided Geometric Design 10(3-4), 237-251, 1993.
11.Ron Goldman “Blossoming and Divided Difference.” Geometric Modeling, Computing (Suppl), 14, 155-184, 2001.
12.Schoenberg, I. J., “Contribution to the problem of approximation of equidistant data by analytical functions,” Quarterly Applied Mathematics, 4, 45-99, 112-141, 1946.
13.Seidel, H., “Computing B-spline control points, in: Strasser, W. and Seidel, H., eds., Theory and Practice of Geometric Modeling, Springer, Berlin, 17-32, 1989.
14.Tony deRose, Ronald Goldman, “A Tutorial Introduction to Blossoming.” Geometric Modeling, Springer-Verlag, 267-286, 1991.
15.Wayne Liu, “A simple, efficient degree raising algorithm for B-spline curves.” Computer-Aided Design 14, 693–698, 1997.