JOURNAL BROWSE
Search
Advanced SearchSearch Tips
MULTI-DEGREE REDUCTION OF BÉZIER CURVES WITH CONSTRAINTS OF ENDPOINTS USING LAGRANGE MULTIPLIERS
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
MULTI-DEGREE REDUCTION OF BÉZIER CURVES WITH CONSTRAINTS OF ENDPOINTS USING LAGRANGE MULTIPLIERS
Sunwoo, Hasik;
  PDF(new window)
 Abstract
In this paper, we consider multi-degree reduction of curves with continuity of any (r, s) order with respect to norm. With help of matrix theory about generalized inverses we can use Lagrange multipliers to obtain the degree reduction matrix in a very simple form as well as the degree reduced control points. Also error analysis comparing with the least squares degree reduction without constraints is given. The advantage of our method is that the relationship between the optimal multi-degree reductions with and without constraints of continuity can be derived explicitly.
 Keywords
curve;degree reduction;Lagrange multipliers;generalized inverse;
 Language
English
 Cited by
 References
1.
S. L. Campbell and C. D. Meyer, Jr., Generalized inverses of linear transformations, Dover Publications, Inc., New York, 1979.

2.
G.-D. Chen, G.-J. Wang, Optimal multi-degree reduction of Bezier curves with constraints of endpoints continuity, Computer Aided Geometric Design bf 19 (2002), 365-377. crossref(new window)

3.
M. Eck, Least squares degree reduction of Bezier curves, Comput. Aided Design 27 (1995), 845-851. crossref(new window)

4.
G. Farin, Algorithms for rational Bezier curves, Comput. Aided Design 15 (1983), 73-77. crossref(new window)

5.
R. T. Farouki, Legendre-Bernstein basis transformations, J. Comput. Appl. Math. 119 (2000), 145-160. crossref(new window)

6.
A. R. Forrest, Interactive interpolation and approximation by Bezier polynomials, Computer J. 15 (1972), 71-79. crossref(new window)

7.
B. G. Lee and Y. Park, Distance for Bezier curves and degree reduction, Bull. Austral. Math. Soc. 56 (1997), 507-515. crossref(new window)

8.
B. G. Lee, Y. Park, and J. Yoo, Application of Legendre-Bernstein basis transformations to degree elevation and degree reduction, Comput. Aided Geom. Design 19 (2002), 709-718. crossref(new window)

9.
D. Lutterkort, J. Peters, and U. Reif, Polynomial degree reduction in the $L_2$ norm equals best euclidean approximation of Bezier coefficients, Comput. Aided Geom. Design 16 (1999), 607-612. crossref(new window)

10.
R. M. Pringle and A. A. Rayner, Generalized inverse matrices with applications to Statistics, Charles Griffin & Co. Ltd., London, 1971.

11.
A. Rababah, B. G. Lee, and J. Yoo, A simple matrix form for degree reduction of Bezier curves using Chebyshev-Bernstein basis transformations, Appl. Math. Comput. 181 (2006), 310-318. crossref(new window)

12.
A. Rababah, B. G. Lee, and J. Yoo, Multiple degree reduction and elevation of Bezier curves using Jacobi-Bernstein basis transformations, Numer. Funct. Anal. Optim. 28:9-10 (2007), 1170-1196.

13.
S. R. Searle, Matrix Algebra Useful for Statistics, Wiley-Interscience, New York, 2006.

14.
H. Sunwoo, Matrix presentation for multi-degree reduction of Bezier curves, Comput. Aided Geom. Design 22 (2005), 261-273. crossref(new window)

15.
H. Sunwoo, Multi-degree reduction of Bezier curves for fixed endpoints using Lagrange multipliers, Comp. Appl. Math. 32 (2013), 331-341. crossref(new window)