Subscribe
Issue No.05 - May (1985 vol.5)
pp: 51-78
Rida Farouki , General Electric Company
ABSTRACT
This survey of the philosophy underlying the unified parametric polynomial formalism for geometric modling suggests some novel applications and assesses the problems and advantages of the approach.
INDEX TERMS
null
CITATION
Rida Farouki, "A Hierarchy of Geometric Forms", IEEE Computer Graphics and Applications, vol.5, no. 5, pp. 51-78, May 1985, doi:10.1109/MCG.1985.276393
REFERENCES
1. T.Varady, “Surface-Surface Intersections for Double-Quadratic Parametric Patches in a Solid Modeler,” Proc. UK-Hungarian Seminar Computational Geometry for CAD/CAM 1983 Cambridge University
2. H. A.Steinberg, “A Smooth Surface Based on Biquadratic Patches,” IEEE Computer Graphics and Applications Vol. 4, No. 9, pp. 20-23 Sept. 1984
3. A. R.Forrest, “Computational Geometry,” Proc. Royal Society of London 1971 Vol. A321, pp. 187-195
4. I. D.Faux and M. J.Pratt, Computational Geometry for Design and Manufacture , Ellis Horwood Ltd. 1979
5. J. G.Semple and L.Roth, Introduction to Algebraic Geometry , Oxford University Press 1949
6. T. W.Sederberg, “Implicit and Parametric Curves and Surfaces for Computer Aided Geometric Design,” 1983
7. G.Salmon, Modern Higher Algebra , Hodges, Smith and Co. 1885 (reprinted by Chelsea Publishing Co., New York)
8. T. W.Sederberg, D. C.Anderson, and R. N.Goldman, Computer Vision, Graphics, and Image Processing , 1984 Vol. 28, pp. 72-84
9. D. M. Y.Sommerville, Analytical Geometry of Three Dimensions , Cambridge University Press 1951
10. D. F.Rogers and J. A.Adams, Mathematical Elements for Computer Graphics , McGraw-Hill 1976
11. L. G.Roberts, The Computer Display Review , C. W. Adams Associates 1966
12. E. A.Maxwell, General Homogeneous Coordinates in Space of Three Dimensions , Cambridge University Press 1951
13. J. G.Semple and G. T.Kneebone, Algebraic Projective Geometry , Oxford University Press 1952
14. S. A.Coons, “Surfaces for Computer Aided Design of Space Forms,” 1967 Project MAC, M.I.T.
15. J. M.Lane and R. F.Riesenfeld, “Bounds on a Polynomial,” Nordisk tidskrift for informations-behandlung Vol. 21, No. 1, pp. 112-117 1981 (B.I.T.)
16. W. J.Gordon and R. F.Riesenfeld, “Bernstein-Bézier Methods for the Computer Aided Design of Free-Form Curves and Surfaces,” J. ACM Vol. 21, No. 2, pp. 293-310 1974
17. P.Bézier, R. E.Barnhill, and R. F.Riesenfeld, Computer-Aided Geometric Design , Academic Press 1974
18. J.Ferguson, “Multivariable Curve Interpolation,” J. ACM Vol. 11, No. 2, pp. 221-228 1964
19. R. E.Barnhill, R. E.Barnhill, and R. F.Riesenfeld, Computer-Aided Geometric Design , Academic Press 1974
20. W.Bohm, G.Farin, and J.Kahmann, Computer Aided Geometric Design , 1984 Vol. 1, pp. 1-60
21. F. S.Acton, Numerical Methods That (Usually) Work , Harper and Row 1970
22. G. M.Phillips and P. J.Taylor, Theory and Applications of Numerical Analysis , Academic Press 1973
23. A.Ralston and P.Rabinowitz, A First Course in Numerical Analysis , McGraw-Hill 1978
24. J. H.Ahlberg, E. N.Nilson, and J. L.Walsh, The Theory of Splines and Their Applications , Academic Press 1967
25. W. J.Gordon, “Free-Form Surface Interpolation through Curve Networks,” , General Motors Research Labs 1969 research publication GMR-921
26. J. C.Holladay, Mathematical Tables and Other Aids to Computation , 1957 Vol. 11, pp. 233-243
27. I. J.Schoenberg, Quarterly of Applied Mathematics , 1946 Vol. 4, pp. 45-99 112-141
28. C.de Boor, A Practical Guide to Splines , Springer-Verlag 1978
29. B. A.Barsky, “A Description and Evaluation of Various 3-D Models,” IEEE Computer Graphics and Applications Vol. 4, No. 1, pp. 38-52 Jan. 1984
30. R. H.Bartels, J. C.Beatty, and B. A.Barsky, “An Introduction to the Use of Splines in Computer Graphics,” , University of California at Berkeley 1983
31. B. A.Barsky, Computers in Industry , 1982 Vol. 3, pp. 17-29
32. W. J.Gordon and R. F.Riesenfeld, R. E.Barnhill, and R. F.Riesenfeld, Computer-Aided Geometric Design , Academic Press 1974
33. M. G.Cox, J. Institute for Mathematics & Applications , 1972 Vol. 10, pp. 134-149
34. C.de Boor, J. Approximation Theory , 1972 Vol. 6, pp. 50-62
35. W.Tiller, “Rational B-Splines for Curve and Surface Representation,” IEEE Computer Graphics and Applications Vol. 3, No. 6, pp. 61-69 Sept. 1983
36. R. E.Barnhill, J. R.Rice, Mathematical Software III , Academic Press 1977
37. W. J.Gordon, “An Operator Calculus for Surface and Volume Modeling,” IEEE Computer Graphics and Applications Vol. 3, No. 7, pp. 18-22 Oct. 1983
38. W. J.Gordon, “Spline-Blended Surface Interpolation through Curve Networks,” J. Mathematics and Mechanics Vol. 18, No. 10, pp. 931-952 1969
39. W. J.Gordon, “Blending-Function Methods of Bivariate and Multivariate Interpolation and Approximation,” SIAM J. Numerical Analysis Vol. 8, No. 1, pp. 158-177 1971
40. M. P.do Carmo, Differential Geometry of Curves and Surfaces , Prentice-Hall 1976
41. B. A.Barsky and A. D.DeRose, “Geometric Continuity of Parametric Curves,” , University of California at Berkeley 1984
42. B. A.Barsky, “The Beta-spline: A Local Representation Based on Shape Parameters and Fundamental Geometric Measures,” 1981
43. B. A.Barsky and J. C.Beatty, “Local Control of Bias and Tension in Beta-splines,” ACM Trans. Graphics Vol. 2, No. 2, pp. 193-218 1983
44. J. K.Hinds and L. P.Kuan, “Surfaces Defined by Curve Transformations,” Proc. 15th Ann. Numerical Control Society Ann. Meeting and Technical Conf. 1978pp. 325-340
45. J. K.Hinds and L. P.Kuan, “Sculptured Surface Technology as a Unified Approach to Geometric Definitions,” , Computer and Automated Systems Assn. of SME 1979 tech. paper MS79-146
46. G. E.Collins, J. R.Rice, Mathematical Software III , Academic Press 1977
47. R.Klass, “An Offset Spline Approximation for Plane Cubic Splines,” Computer Aided Design Vol. 15, No. 5, pp. 297-299 1983
48. W.Tiller and E. G.Hanson, “Offsets of Two-Dimensional Profiles,” IEEE Computer Graphics and Applications Vol. 4, No. 9, pp. 36-46 Sept. 1984
49. A. A. G.Requicha and H. B.Voelcker, “Solid Modeling: A Historical Summary and Contemporary Assessment,” IEEE Computer Graphics and Applications Vol. 2, No. 2, pp. 9-24 Mar. 1982
50. A. A. G.Requicha and H. B.Voelcker, “Solid Modeling: Current Status and Research Directions,” IEEE Computer Graphics and Applications Vol. 3, No. 7, pp. 25-37 Oct. 1983
51. R. B.Tilove, “Extending Solid Modeling Systems for Mechanism Design and Kinematic Simulation,” IEEE Computer Graphics and Applications Vol. 3, No. 3, pp. 9-19 May/June 1983
52. M. J.Pratt, “Solid Modeling and the Interface Between Design and Manufacture,” IEEE Computer Graphics and Applications Vol. 4, No. 7, pp. 52-59 July 1984
53. R. B.Tilove and A. A. G.Requicha, “Closure of Boolean Operations on Geometric Entities,” Computer Aided Design Vol. 12, No. 5, pp. 219-220 1980
54. J. T.Kajiya, “Ray Tracing Parametric Patches,” ACM Computer Graphics (Proc. SIGGRAPH 82) July 1982 Vol. 16, No. 3, pp. 245-254
55. J. Z.Levin, Computer Graphics and Image Processing , 1979 Vol. 11, pp. 73-87
56. A. P.Morgan, “A Method for Computing All Solutions to Systems of Polynomial Equations,” ACM Trans. Mathematical Software Vol. 9, No. 1, pp. 1-17 1983
57. R. F.Sarraga, Computer Vision, Graphics and Image Processing , 1983 Vol. 22, pp. 222-238
58. E.Cohen, T.Lyche, and R.Riesenfeld, Computer Graphics and Image Processing , 1980 Vol. 14, pp. 87-111
59. T. W.Sederberg and D. C.Anderson, “Steiner Surface Patches,” IEEE Computer Graphics and Applications Vol. 5, No. 5, pp. 23-36 May 1985
60. D. H.Hoitsma and M.Rochepp. 15-18 1983 Computer-Aided Geometry Modeling NASA C.P. 2272
61. R. B.Lee and D. A.Fredericks, “Intersection of Parametric Surfaces and a Plane,” IEEE Computer Graphics and Applications Vol. 4, No. 8, pp. 48-51 Aug. 1984
62. K. J.Weiler, “Edge-Based Data Structures for Solid Modeling in Curved Surface Environments,” IEEE Computer Graphics and Applications Vol. 5, No. 1, pp. 21-40 Jan. 1985
63. T. J.Willmore, An Introduction to Differential Geometry , Oxford University Press 1959
64. R. E.Barnhill, “A Survey of the Representation and Design of Surfaces,” IEEE Computer Graphics and Applications Vol. 3, No. 7, pp. 9-16 Oct. 1983
65. C. A.Coulson and A.Jeffrey, Waves: A Mathematical Approach to the Common Types of Wave Motion , Longman 1977
66. G. N.Watson, A Treatise on the Theory of Bessel Functions , Cambridge University Press 1952
67. M.Abramowitz and I.Stegun, Handbook of Mathematical Functions , National Bureau of Standards 1964
68. R. E.Berg and D. G.Stork, The Physics of Sound , Prentice-Hall 1982