A dimensionality paradigm for surface interrogations. (English) Zbl 0712.65010

The author notes that in dealing with problems of graphing surfaces it is much better to write down the problem in the form of a system of equations to be solved by Newton’s method instead of trying to reduce the problem algebraically. Applications are given to offset surfaces (in mathematics: parallel surfaces), Voronoi surfaces (equidistant surfaces to two given surfaces) and blending surfaces (Monge “surfaces moulure”) used to smoothly connect two surfaces \(\{\) these seem somewhat easier to handle than cyclides\(\}\).
Reviewer: H.Guggenheimer


65D17 Computer-aided design (modeling of curves and surfaces)
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
Full Text: DOI


[1] Abhyankar, S.; Bajaj, C., Automatic rational parameterization of curves and surfaces IV: Algebraic space curves, ACM Trans. Graphics (1987), to appear · Zbl 0655.65017
[2] Bajaj, C.; Hoffmann, C.; Hopcroft, J.; Lynch, R., Tracing surface intersections, Computer Aided Geometric Design, 5, 285-307 (1988) · Zbl 0659.65012
[3] Blutel, E., Recherches sur les surfaces qui sont même temps lieux de coniques et enveloppes des cônes du second degré, Ann. Sci. École Norm. Super., 7, 155-216 (1890)
[4] Boehm, W.; Farin, G.; Kahmann, J., A survey of curve and surface methods in CAGD, Computer Aided Geometric Design, 1, 1-60 (1984) · Zbl 0604.65005
[5] Buchberger, B., Gröbner bases: An algorithmic method in polynomial ideal theory, (Bose, N. K., Multidimensional Systems Theory (1985), Reidel: Reidel Dordrecht), 184-232 · Zbl 0587.13009
[6] Buchberger, B.; Collins, G.; Kutzler, B., Algebraic methods for geometric reasoning, Annl. Reviews in Comp. Science, Vol. 3 (1988)
[7] Degen, W., Surfaces with a conjugate net of conics in projective space, Tensor, 39, 167-172 (1982) · Zbl 0514.53007
[8] Chandru, V.; Dutta, D.; Hoffmann, C., On the geometry of Dupin cyclides, The Visual Computer, 5, 227-290 (1989)
[9] Chandru, V.; Dutta, D.; Hoffmann, C., Variable radius blending with cyclides, (Preiss, K.; Turner, J.; Wozny, M., Geometric Modeling for Product Engineering (1990), North-Holland: North-Holland Amsterdam), 39-58
[10] Farouki, R., Trimmed surface algorithms for the evaluation and interrogation of solid boundary representation, IBM J. Res. Develop., 31, 314-334 (1986)
[11] Farouki, R., The approximation of non-degenerate offset surfaces, Computer Aided Geometric Design, 3, 15-43 (1986) · Zbl 0621.65003
[12] Farouki, R.; Neff, C., Some analytic and algebraic properties of plane offset curves, Res. Rept. RC-14364 (1989), IBM: IBM Yorktown Heights
[13] Farouki, R.; Rajan, V., On the numerical condition of Bernstein polynomials, Res. Rept. RC-12626 (1987), IBM: IBM Yorktown Heights · Zbl 0636.65012
[14] Garrity, T.; Warren, J., On computing the intersection of a pair of algebraic surfaces, Computer Aided Geometric Design, 6, 137-153 (1989) · Zbl 0699.65012
[15] Geisow, A., Surface interrogation, Ph.D. Diss. (1986), School of Computing and Accountancy, Univ. of East Anglia
[16] Golub, G.; Van Loan, C., Matrix Computations (1983), Johns Hopkins Univ. Press: Johns Hopkins Univ. Press Baltimore, MD · Zbl 0559.65011
[17] Hoffmann, C., Algebraic curves, (Rice, J., Mathematical Aspects of Scientific Software (1987), Springer: Springer Berlin), 101-122, IMA Vols. in Math. and Appl. · Zbl 0663.14019
[18] Hoffmann, C., Geometric and Solid Modeling (1989), Morgan Kaufmann: Morgan Kaufmann San Mateo, CA, Chapter 7
[19] Hoffmann, C., Algebraic and numerical techniques for offsets and blends, (Dahmen, W.; Gasca, M.; Micchelli, C., Computations of Curves and Surfaces (1989), Kluwer Academic Publ: Kluwer Academic Publ Dordrecht), 499-528 · Zbl 0705.68102
[20] Pegna, J., Variable sweep geometric modeling, Ph.D. Diss. (1988), Mech. Engr., Stanford Univ
[21] Prakash, P.; Patrikalakis, N., Algebraic and rational polynomial surface intersections, Computer Vision, Graphics, and Image Processing (1988), to appear
[22] Pratt, M.; Geisow, A., Surface/surface intersection problems, (Gregory, J., The Mathematics of Surfaces (1986), Oxford Univ. Press: Oxford Univ. Press Oxford), 117-142
[23] Morgan, A., An algorithm for solving the line-tube classification problem, Rept. GMR-3858 (1981), Dept. of Math., General Motors Res. Labs: Dept. of Math., General Motors Res. Labs Warren, MI
[24] Rossignac, J.; Requicha, A., Constant radius blending in solid modeling, Comp. Mech. Engr., 3, 65-73 (1984)
[25] Sederberg, T., Implicit and parametric curves and surfaces for computer aided geometric design, Ph.D. Diss. (1983), Mech. Engr., Purdue Univ
[26] Sederberg, T., An algorithm for algebraic curve intersection, TR ECGL 88-3 (1988), Civil Engr., Brigham Young Univ · Zbl 0688.65012
[27] Sederberg, T.; Parry, S., A comparison of curve intersection algorithms, Computer Aided Design, 18, 58-63 (1986)
[28] Spivak, M., A Comprehensive Introduction to Differential Geometry, Vol. III, 255-263 (1975), Wilmingto, DE · Zbl 0306.53001
[29] Wolter, F.-E., Cut loci in bordered and unbordered Riemannian manifolds, Ph.D. Diss., Math. (1985), Tech. Univ: Tech. Univ Berlin, West Germany · Zbl 0674.53049
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