Computing quadric surface intersections based on an analysis of plane cubic curves

Graphical Models

Volumn 64, Issue 6, 2002, Pages 335-367

  Wang, Wenping ^a   Joe, Barry ^b   Goldman, Ronald ^c  

^a UNIVERSITY OF HONG KONG   (Hong Kong)

^b ZCS Inc   (Canada)

^c Rice University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGEBRA; ALGORITHMS; COMPUTATIONAL GEOMETRY; COMPUTER SIMULATION; MAPPING; PARAMETER ESTIMATION; QUADRATIC PROGRAMMING; TOPOLOGY;

QUADRIC SURFACE INTERSECTION CURVES (QSIC);

COMPUTER GRAPHICS;

CLASSIFICATION; CURVATURE; MODELING; TOPOLOGY;

EID: 0038576282     PISSN: 15240703     EISSN: None     Source Type: Journal    
DOI: 10.1016/S1077-3169(02)00018-7     Document Type: Article

References (42)

1. S.S. Abhyankar, C. Bajaj, Automatic parameterization of rational curves and surfaces I: conics and conicoids, Comput. Aided Des. 19 (1) (1987) 11-14.
2. S.S. Abhyankar, C. Bajaj, Automatic parameterization of rational curves and surfaces II: cubics and cubicoids, Comput. Aided Des. 19 (9) (1987) 499-502.
3. S.S. Abhyankar, C. Bajaj, Automatic parameterization of rational curves and surfaces III: algebraic space curves, ACM Trans. Graph. 8 (4) (1989) 325-334.
4. H.F. Baker, Principles of Geometry, vol. HI, Cambridge Press, Cambridge, 1923.
5. R. Bix, Conics and Cubics - A Concrete Introduction to Algebraic Curves, Springer, New York, 1998.
6. J.W. Boyse, J.E. Gilchrist, GMSolid: interactive modeling for design and analysis of solids, IEEE Comput. Graph. Appl. 2 (2) (1982) 27-40.
7. T.J. Bromwich, Quadratic forms and their classification by means of invariant-factors, Cambridge Tracts Math. Math. Phys. (3) (1906).
8. C.M. Brown, PADL-2: a technical summary, IEEE Computer Graph. Appl. 2 (2) (1982) 69-84.
9. E. Calabi, Linear systems of real quadratic forms, Bull. Am. Math. Soc. 47 (1964) 844-846.
10. E.W. Chionh, R.N. Goldman, J.R. Miller, Using multivariate resultants to find the intersection of three quadric surfaces, ACM Trans. Graph. 10 (4) (1991) 378-400.
11. L.E. Dickson, Elementary Theory of Equations, Wiley, New York, 1914.
12. R.T. Farouki, C.A. Neff, M.A. O'Connor, Automatic parsing of degenerate quadric-surface intersections, ACM Trans. Graph. 8 (3) (1989) 174-203.
13. T. Garrity, J. Warren, On computing the intersection of a pair of algebraic surfaces, Comput. Aided Geom. Des. 6 (2) (1989) 132-153.
14. R.N. Goldman, Quadrics of revolution, IEEE CG & A 3 (2) (1983) 68-76.
15. R.N. Goldman, J.R. Miller, Combining algebraic rigor with geometric robustness for the detection and calculation of conic sections in the intersection of two natural quadric surfaces, in: Proceedings of ACM Symposium on Solid Modeling Foundations and CAD/CAM Applications, June 5-7, 1991, Austin, Texas, USA, 1991, pp. 221-231.
16. R.N. Goldman, J.D. Warren, A sufficient condition for the intersection of two quadrics of revolution to degenerate into a pair of conic sections (preprint).
17. C.M. Hoffmann, J.E. Hopcroft, Geometric ambiguities in boundary representations, Comput. Aided Des. 19 (3) (1987) 141-147.
18. J.Z. Levin, A parametric algorithm for drawing pictures of solid objects composed of quadrics, Commun. ACM 19 (10) (1976) 555-563.
19. J.Z. Levin, Mathematical models for determining the intersections of quadric surfaces, Comput. Graph. Image Process. 1 (1979) 73-87.
20. Y.M. Li, R.J. Cripps, Identification of inflection points and cusps on rational curves, Comput. Aided Geom. Des. 14 (1997) 491-497.
21. J.R. Miller, Geometric approaches to nonplanar quadric surface intersection curves, ACM Trans. Graph. 6 (1987) 274-307.
22. J.R. Miller, Analysis of quadric-surface-based solid models, IEEE CG & A 8 (1) (1988) 28-42.
23. J.R. Miller, R.N. Goldman, Geometric algorithms for detecting and calculating all conics sections in the intersection of any two natural quadric surfaces, Graph. Models Image Process. 57 (1) (1995) 55-66.
24. S. Ocken, J.T. Schwartz, M. Sharir, Precise implementation of CAD primitives using rational parametrizations of standard surfaces, in: Schwartz, Hopcroft, Sharir (Eds.), Planning, Geometry, and Complexity of Robot Motion, Ablex Publishing Corporation, 1987, pp. 245-266.
25. R.R. Patterson, Parameterizing and graphing non-singular cubic curves, Comp. Aided Des. 20 (10) (1988) 615-623.
26. R.R. Patterson, Parametric cubics as algebraic curves, Comput. Aided Geom. Des. 5 (1988) 139-159.
27. L. Piegl, Geometric method of intersecting natural quadrics represented in trimmed surface form, Comp. Aided Des. 21 (4) (1989) 201-212.
28. M.A. Sabin, A method for displaying the intersection curve of two quadric surfaces, Comput. J. 19 (1976) 336-338.
29. G. Salmon, Higher Plane Curves, G.E. Stechert & Co, New York, 1934.
30. T. Sakkalis, R. Faruoki, Singular points of algebraic curves, J. Symb. Comput. 9 (1990) 405-421.
31. R.F. Sarraga, Algebraic methods for intersections of quadric surfaces in GMSOLID, Comput. Vision Graph. Image Process. 22 (2) (1983) 222-238.
32. G.J. Semple, G.K. Kneebone, Algebraic Projective Geometry, Oxford University Press, London, 1952.
33. G.J. Semple, G.K. Kneebone, Algebraic Curves, Oxford University Press, London, 1959.
34. C.H. Shene, J.K. Johnstone, On the lower degree intersections of two natural quadrics, ACM Trans. Graph. 13 (4) (1994) 400-424.
35. D.M.Y. Sommerville, Analytical Geometry of Three Dimensions, Cambridge University Press, Cambridge, 1947.
36. F. Uhlig, A canonical form for a pair of real symmetric matrices that generate a nonsingular pencil, Linear Algebra Appl. 14 (1976) 189-209.
37. J.V. Uspensky, Theory of Equations, McGraw-Hill, New York, 1948.
38. R.J. Walker, Algebraic Curves, Princeton University Press, Princeton, NJ, 1950.
39. W. Wang, B. Joe, R.N. Goldman, Rational quadratic parameterizations of quadrics, Int. J. Comput. Geom. Appl. 7 (6) (1997) 599-619.
40. I. Wilf, Y. Manor, Quadric surface intersection: shape and structure, Comput. Aided Des. 25 (10) (1993) 633-643.
41. I. Wilf, Y. Manor, Splitting quadric faces, Comput. Aided Des. 22 (1990) 507-514.
42. P.Y. Woon, H. Freeman, A computer procedure for generating visible-line projection of solids bounded by quadric surfaces, Inform. Process. 2 (1971) 1120-1125.