Computing exact rational offsets of quadratic triangular Bézier surface patches

CAD Computer Aided Design

Volumn 40, Issue 2, 2008, Pages 197-209

  Bastl, Bohumír ^a   Jüttler, Bert ^b   Kosinka, Jiří ^b   Lávička, Miroslav ^a  

^a UNIVERSITY OF WEST BOHEMIA   (Czech Republic)

^b JOHANNES KEPLER UNIVERSITY LINZ   (Austria)

Author keywords

Convolution surfaces; Offsets; Quadratic B zier triangular surface patches; Steiner surfaces

Indexed keywords

APPROXIMATION ALGORITHMS; COMPUTATION THEORY; CONVOLUTION; PARAMETERIZATION; PROBLEM SOLVING;

CONVOLUTION SURFACES; QUADRATIC BEZIER TRIANGULAR SURFACE PATCHES; STEINER SURFACES;

COMPUTER AIDED DESIGN;

EID: 38849185940     PISSN: 00104485     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cad.2007.10.008     Document Type: Article

References (44)

1. Elber G., Lee I.-K., and Kim M.-S. Comparing offset curve approximation methods. IEEE Computer Graphics and Applications 17 (1998) 62-71
2. Šír Z., Feichtinger R., and Jüttler B. Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics. Computer-Aided Design 38 (2006) 608-618
3. Zhao H.-Y., and Wang G.-J. Error analysis of reparametrization based approaches for curve offsetting. Computer-Aided Design 39 (2007) 142-148
4. Pham B. Offset curves and surfaces: A brief survey. Computer-Aided Design 24 (1992) 223-229
5. Maekawa T. An overview of offset curves and surfaces. Computer-Aided Design 31 (1999) 165-173
6. Farouki R.T. The approximation of non-degenerate offset surfaces. Computer Aided Geometric Design 3 (1986) 15-43
7. Pottmann H., and Leopoldseder S. A concept for parametric surface fitting which avoids the parametrization problem. Computer Aided Geometric Design 20 (2003) 343-362
8. Sarkar B., and Menq C.-H. Parameter optimization in approximating curves and surfaces to measurement data. Computer Aided Geometric Design 8 (1991) 267-290
9. Hoschek J., Schneider F.-J., and Wassum P. Optimal approximate conversion of spline surfaces. Computer Aided Geometric Design 6 (1989) 293-306
10. Elber G., and Cohen E. Error bounded variable distance offset operator for free form curves and surfaces. International Journal of Computational Geometry & Applications 1 (1991) 67-78
11. Kimmel R., and Bruckstein A.M. Shape offsets via level sets. Computer-Aided Design 25 (1993) 154-162
12. Piegl L., and Tiller W. Computing offsets of NURBS curves and surfaces. Computer-Aided Design 31 (1999) 147-156
13. Elber G., and Kim M.-S. Special issue on offsets, sweeps, and Minkowski sums. Computer-Aided Design 31 (1999) 163
14. Ravi Kumar G.V.V., Shastry K.G., and Prakash B.G. Computing non-self-intersecting offsets of NURBS surfaces. Computer-Aided Design 34 (2002) 209-228
15. Ravi Kumar G.V.V., Shastry K.G., and Prakash B.G. Computing offsets of trimmed NURBS surfaces. Computer-Aided Design 35 (2003) 411-420
16. Ravi Kumar G.V.V., Shastry K.G., and Prakash B.G. Computing constant offsets of a NURBS B-Rep. Computer-Aided Design 35 (2003) 935-944
17. Kulczycka M., and Nachman LJ. Qualitative and quantitative comparisons of B-spline offset surface approximation methods. Computer-Aided Design 34 (2002) 19-26
18. Sun V.F., Nee A.Y.C., and Lee K.S. Modifying free-formed NURBS curves and surfaces for offsetting without local self-intersection. Computer-Aided Design 36 (2004) 1161-1169
19. Seong J.-K., Elber G., and Kim M.-S. Trimming local and global self-intersections in offset curves/surfaces using distance maps. Computer-Aided Design 38 (2006) 183-193
20. Farouki RT. Pythagorean hodograph curves. In: Farin G., Hoschek J., and Kim M. (Eds). Handbook of computer aided geometric design (2002), North-Holland, Amsterdam 405-427
21. Choi H-I, Farouki RT, Kwon S-H, Moon HP. Topological criterion for selection of quintic pythagorean hodograph hermite interpolants. Computer Aided Geometric Design [in press]
22. Pottmann H. Rational curves and surfaces with rational offsets. Computer Aided Geometric Design 12 (1995) 175-192
23. Peternell M., and Pottmann H. A Laguerre geometric approach to rational offsets. Computer Aided Geometric Design 15 (1998) 223-249
24. Hoschek J. Dual Bézier curves and surfaces. In: Barnhill R.E., and Boehm W. (Eds). Surfaces in computer aided geometric design (1983), North-Holland, Amsterdam 147-156
25. Jüttler B., and Sampoli M.L. Hermite interpolation by piecewise polynomial surfaces with rational offsets. Computer Aided Geometric Design 17 (2000) 361-385
26. Sampoli M.L., Peternell M., and Jüttler B. Rational surfaces with linear normals and their convolutions with rational surfaces. Computer Aided Geometric Design 23 (2006) 179-192
27. Keyser J., Cilver T., Foskey M., Krishnan S., and Manocha D. ESOLID-a system for exact boundary evaluation. Computer-Aided Design 36 (2004) 175-193
28. Fogel E., and Halperin D. Exact and efficient construction of Minkowski sums of convex polyhedra with applications. Computer-Aided Design 39 (2007) 929-940
29. Dokken T. The GAIA project on intersection and implicitization. In: Jüttler B., and Piene R. (Eds). Geometric modeling and algebraic geometry (2008), Springer 5-28
30. Coffmann A., Schwartz A.J., and Stanton C. The algebra and geometry of Steiner and other quadratically parameterizable surfaces. Computer Aided Geometric Design 13 (1996) 257-286
31. Degen WLF. The types of triangular Bézier surfaces. In: Mullineux G. (Ed). The mathematics of surfaces VI (1996), Oxford University Press 153-170
32. Peters J., and Reif U. The 42 equivalence classes of quadratic surfaces in affine n-space. Computer Aided Geometric Design 15 (1998) 459-473
33. Powell M.J.D., and Sabin M.A. Piecewise quadratic approximations on triangles. ACM Transactions on Mathematical Software 3 (1977) 316-325
34. Hoschek J., and Lasser D. Fundamentals of computer aided geometric design (1993), AK Peters, Wellesley
35. Prautzsch H., Boehm W., and Paluszny M. Bézier and B-spline techniques (2002), Springer, Berlin
36. Lávička M., and Bastl B. Rational hypersurfaces with rational convolutions. Computer Aided Geometric Design 24 (2007) 410-426
37. Peternell M, Odehnal B. Convolution surfaces of quadratic triangular Bézier surfaces. Computer Aided Geometric Design [in press]
38. Kreyszig E. Differential geometry (1991), Dover, New York
39. Bloomenthal J., and Shoemake K. Convolution surfaces. Computer Graphics 25 (1991) 251-256
40. Cox D., Little H., and O'Shea D. Ideals, varieties, and algorithms (1992), Springer, New York
41. Sederberg T, Chen F. Implicitization using moving curves and surfaces. In: Computer graphics proceedings, annual conference series. ACM SIGGRAPH. 1995. p. 301-8
42. Wang R.-H. Multivariate spline functions and their applications (2001), Kluwer, Dordrecht
43. Nürnberger G., and Zeilfelder F. Developments in bivariate spline interpolation. Journal of Computational and Applied Mathematics 121 (2000) 125-152
44. Shalaby M.F., Thomassen J.B., Wurm E.M., Dokken T., and Jüttler B. Piecewise approximate implicitization: Experiments using industrial data. In: Mourrain B., Elkadi M., and Piene R. (Eds). Algebraic geometry and geometric modelling (2006), Springer 37-52