This Article 
 Bibliographic References 
 Add to: 
On NURBS: a Survey
January/February 1991 (vol. 11 no. 1)
pp. 55-71

Nonuniform rational B-spline (NURBS) curves and surfaces, which are based on rational and B-splines, are defined. The important characteristics of NURBS that have contributed to their wide acceptance as standard tools for geometry representation and design are summarized. Their application to representing conic sections and commonly used surfaces, designing curves and surfaces, and modifying shapes is examined.

1. S.A. Coons, "Surfaces for Computer-Aided Design of Space Figures," Design Division, Dept. of Mechanical Eng., MIT, Cambridge, Mass., Jan. 1964.
2. R.A. Forrest,Curves and Surfaces for Computer-Aided Design, doctoral dissertation, Cambridge Univ., Cambridge, UK, 1968.
3. C. de Boor,A Practical Guide to Splines, Springer-Verlag, New York, 1978.
4. R.F. Riesenfeld,Applications of B-spline Approximation to Geometric Problems of Computer-Aided Design, doctoral dissertation, Syracuse Univ., Syracuse, N.Y., 1973. Also available as Tech. Report UTEC-CSc-73-126, Dept. of Computer Science, Univ. of Utah, 1973.
5. K.J. Versprille,Computer-Aided Design Applications of the Rational B-Spline Approximation Form, doctoral dissertation, Syracuse Univ., Syracuse, N.Y., 1975.
6. R.M. Blomgren, "B-Spline Curves," Class notes, Boeing document B- 7150-BB-WP-2811D-4412, 1981.
7. R.M. Blomgren, "Mathematical Methods for Advanced Geometric Modeling, 3-D Curves and Surfaces," CASE/SME Seminar, SME, Dearborn, Mich., Mar. 1982.
8. E.T.Y. Lee, "A B-Spline Primer," Boeing document, Boeing, Seattle, Wash., 1983.
9. E.T.Y. Lee, "A Treatment of Conies in Parametric Rational Bezier Form," Boeing document, Boeing, Seattle, Wash., Feb. 1981.
10. E.T.Y. Lee, "The Rational Bezier Representation for Conics," Boeing document, Boeing, Seattle, Wash., 1983.
11. R.D. Fuhr, "A Theoretical Introduction to the Rational B-Spline Representation for Curves and Surfaces," Boeing document D6-48379-100, Boeing, Seattle, Wash., 1981.
12. A.L. Klosterman, R.H. Ard, and J.W. Klahs, "Geometric Modeling Speeds Design of Mechanical Assemblies,"Computer Technology Review, Winter 1982, pp. 103-111.
13. A.L. Klosterman, R.H. Ard, and J.W. Klahs, "A Geometric Modeling Program for the System Designer,"Proc. Conf. CAD/CAM Technology in Mechanical Eng., MIT, Cambridge, Mass., 1982.
14. A.L. Klosterman, "A Geometric Modeler Based on a Dual Geometry Representation--Polyhedra and Rational B-Splines,"NASA Symp. Computer-Aided Geometric Modeling, NASA, 1983, pp. 7-13.
15. W. Tiller, "Rational B-Splines for Curve and Surface Representation,"CG&A, Vol. 3, No. 10, Sept. 1983, pp. 61-69.
16. R.F. Riesenfeld et al., "Using the Oslo Algorithm as a Basis for CAD/CAM Geometric Modeling,"Proc. NCGA National Conf., NCGA, Fairfax, Va., 1981, pp. 345-356.
17. G. Maiorino, "Intergraph Approach in Computer Aided Geometric Design Using Free-Form Surfaces,"Proc. Symp. Automotive Technology and Automation, Graz, Austria, Vol. 2, 1985, pp. 401- 421.
18. "Initial Graphics Exchange Specification, Version 3.0," Doc. No. NB-SIR 86-3359, NIST, Gaithersburg, Md., 1986.
19. "PHIGS+ Function Description Revision 3.0,"Computer Graphics(Proc. SIGGRAPH), Vol. 22, No. 3, July 1988, pp. 125-218.
20. L. Piegl and W. Tiller, "Curve and Surface Constructions Using Rational B-Splines,"Computer-Aided Design, Vol. 19, No. 9, Nov. 1987, pp. 485-498.
21. W. Tiller, "Geometric Modeling Using Non-Uniform Rational B-Splines: Mathematical Techniques," Siggraph tutorial notes, ACM, New York, 1986.
22. D. F. Rogers and J. A. Adams,Mathemational Elements for Computer Graphics, 2nd ed. New York: McGraw-Hill, 1990.
23. C. de Boor, "On Calculating with B-Splines,"J. Approximation Theory, Vol. 6, No. 1, July 1972, pp. 50-62.
24. M.G. Cox, "The Numerical Evaluation of B-Splines,"J. Inst. Mathematics and Applications, Vol. 10, 1972, pp. 134-149.
25. H.S.M. Coxeter,Projective Geometry, Univ. of Toronto Press, Toronto, 1974.
26. M. A. Penna and R. R. Patterson,Projective Geometry and its Applications to Computer Projective Graphics. Englewood Cliffs, NJ: Prentice-Hall, 1986.
27. R.F. Riesenfeld, "Homogeneous Coordinates and Projective Planes in Computer Graphics,"CG&A, Vol. 1, No. 1, Jan. 1981, pp. 50-55.
28. L.G. Roberts, "Homogeneous Matrix Representation and Manipulation ofN-Dimensional Constructs," Tech. Report MS-1405, Lincoln Lab, MIT, Cambridge, Mass., 1965.
29. C. de Boor, "B(asic)-Spline Basics," inFundamental Developments in CADCAM Geometric Modeling, L. Piegl, ed., Butterworth-Heinemann, Guildford, UK, 1991.
30. E.T.Y. Lee, "Rational Quadratic Bezier Representation for Conics," inGeometric Modeling: Algorithms and New Trends, G. Farin, ed., SIAM, Philadelphia, 1987, pp. 3-19.
31. L. Piegl, "A Geometric Investigation of the Rational Bezier Scheme of Computer-Aided Design,"Computers in Industry, Vol. 7, No. 5, Oct. 1986, pp. 401-410.
32. L. Piegl, "Modifying the Shape of Rational B-Splines. Part 1: Curves,"Computer-Aided Design, Vol. 21, No. 8, Oct. 1989, pp. 509-518.
33. L. Piegl, "Modifying the Shape of Rational B-Splines. Part 2: Surfaces,"Computer-Aided Design, Vol. 21, No. 9, Nov. 1989, pp. 538- 546.
34. A.R. Forrest, "The Twisted Cubic Curve: A Computer-Aided Geometric Design Approach,"Computer-Aided Design, Vol. 12, No. 4, July 1980, pp. 165-172.
35. L. Piegl, "Infinite Control Points--A Method for Representing Surfaces of Revolution Using Boundary Data,"CG&A, Vol. 7, No. 3, Mar. 1987, pp. 45-55.
36. L. Piegl, "Algorithms for Computing Conic Splines,"ASCE J. Computing in Civil Engineering, Vol. 4, No. 3, July 1990, pp. 180-198.
37. L. Piegl and W. Tiller, "A Menagerie of Rational B-Spline Circles," CG&A, Vol. 9, No. 5, Sept. 1989, pp. 48-56.
38. D. Hildebrand,Rationale Polynomialkurven und -flaechen 2. Grades fur CAD [Rational Polynomial Curves and Surfaces of Degree Two for CAD], Diplomarbeit, Darmstadt Technical Univ., Darmstadt, West Germany, 1986.
39. D. Hildebrand, "Darstellung von Flaechen 2. Ordnung als rationale Bezierflechen 2. Grades" [Description of Surfaces of Second Degree as Rational Bezier Surfaces], tech. report, Darmstadt Technical Univ., 1986.
40. H. Prautzsch, "Degree Elevation of B-Spline Curves,"Computer Aided Geometric Design, Vol. 1, No. 2, Nov. 1984, pp. 193-198.
41. E. Cohen, T. Lyche, and L.L. Schumaker, "Algorithms for Degree-Raising of Splines,"ACM Trans. Graphics, Vol. 4, No. 3, July 1985, pp. 171-181.
42. W. Boehm, "Inserting New Knots into B-Spline Curves,"Computer-Aided Design, Vol. 12, No. 4, July 1980, pp. 199-201.
43. E. Cohen, T. Lyche, and R.F. Riesenfeld, "Discrete B-Splines and Subdivision Techniques in Computer-Aided Geometric Design and Computer Graphics,"Computer Graphics and Image Processing, Vol. 14, No. 2, Oct. 1980, pp. 87-111.
44. G. Farin, B. Piper, and A.J. Worsey, "The Octant of a Sphere as a Non-Degenerate Triangular Bezier Patch,"Computer Aided Geometric Design, Vol. 4, No. 4, Dec. 1987, pp. 329-332.
45. J.E. Cobb, "Tiling the Sphere with Rational Bezier Patches," Tech. Report TR UUCS-88-009, Computer Science Dept., Univ. of Utah, Salt Lake City, 1988.
46. M.G. Cox, "Algorithms for Spline Curves and Surfaces," inFundamental Developments in CADCAM Geometric Modeling, L. Piegl, ed., Butterworth-Heinemann, Guildford, UK, 1991.
47. D.F. Rogers, "B-Spline Curves and Surfaces for Ship Hull Definition," Soc. of Naval Architects and Marine Engineers, New York, 1977, pp. 79-96.
48. E.T.Y. Lee, "Choosing the Nodes in Parametric Curve Interpolation,"Computer-Aided Design, Vol. 21, No. 6, July/Aug. 1989, pp. 363-370.
49. L. Piegl, "Interactive Data Interpolation by Rational Bezier Curves,"CG&A, Vol. 7, No. 4, Apr. 1987, pp. 45-58.
50. H. Akima, "A New Method of Interpolation and Smooth Curve Fitting Based on Local Procedures,"J. ACM, Vol. 17, No. 4, Oct. 1970, pp. 589-602.
51. G. Renner, "A Method of Shape Description for Mechanical Engineering Practice,"Computers in Industry, Vol. 3, 1982, pp. 137- 142.
52. D.F. Rogers, "Constrained B-Spline Curve and Surface Fitting,"Computer-Aided Design, Vol. 21, No. 10, Dec. 1989, pp. 641-648.
53. D.F. Rogers and S.G. Satterfield, "Dynamic B-Spline Surfaces,"Proc. 4th Conf Computer Applications in the Automation of Shipyard Operation and Ship Design, North-Holland, Amsterdam, 1982, pp. 1-8.
54. L. Piegl, "A Technique for Smoothing Scattered Data with Conic Sections,"Computers in Industry, Vol. 9, No. 3, Nov. 1987, pp. 223-237.
55. B.A. Barsky and D.P. Greenberg, "Determining a Set of B-Spline Vertices to Generate an Interpolating Surface,"Computer Graphics and Image Processing, Vol. 14, No. 3, Nov. 1980, pp. 203-226.
56. I. Faux and M. Pratt,Computational Geometry for Design and Manufacture. Ellis Horwood, 1979.
57. J.K. Hinds and L.P. Kuan, "Surfaces Defined by Curve Transformation,"Proc. 15th Numerical Control Sot. Ann. Meeting and Technical Conf, 1978, pp. 325-340.
58. H.P. Moreton and R.D. Bergeron, "SUDS--Surface Description System,"Proc. Eurographics 86, North-Holland, Amsterdam, 1986, pp. 221-236.
59. C.D. Woodward, "Cross-Sectional Design of B-Spline Surfaces,"Computers and Graphics, Vol. 11, No. 2, 1987, pp. 193-201.
60. C.D. Woodward, "Skinning Techniques for Interactive B-Spline Surface Interpolation,"Computer-Aided Design, Vol. 20, No. 8, Oct. 1988, pp. 441-451.
61. R.E. Parent, "A System for Sculpting 3-D Data,"Computer Graphics(Proc. Siggraph), Vol. 11, No. 2, July 1977, pp. 138-147.
62. W.E. Carlson,Techniques for the Generation of Three Dimensional Data for Use in Complex Image Synthesis, doctoral dissertation, Ohio State Univ., Columbus, Ohio, 1982.
63. J.P. Duncan and G.W. Vickers, "Simplified Method for Interactive Adjustment of Surfaces,"Computer-Aided Design, Vol. 12, No. 6, Nov. 1980, pp. 305-308.
64. R.F. Riesenfeld, "Aspects of Modeling in Computer-Aided Geometric Design,"Proc. Nat'l Computer Conf., AFIPS, Reston, Va., 1975, pp. 597-602.
65. E.S. Cobb,Design of Sculptured Surfaces Using the B-Spline Representation, doctoral dissertation, Univ. of Utah, Salt Lake City, 1984.
66. A.H. Barr, "Global and Local Deformations of Solid Primitives," Siggraph seminar notes,State of the Art of Image Synthesis, ACM, New York, 1983.

Les Piegl, "On NURBS: a Survey," IEEE Computer Graphics and Applications, vol. 11, no. 1, pp. 55-71, Jan.-Feb. 1991, doi:10.1109/38.67702
Usage of this product signifies your acceptance of the Terms of Use.