Abstract
This paper addresses the problem of approximating a surface-to-surface intersection curve. Accurate computation of an intersection curve is not practical due to the degree explosion problem, when function decomposition is used, and fundamentally is not possible because of some computational reasons. Therefore, in practical applications, an approximate intersection curve with a low degree is extensively used. However, the approximation of an intersection curve needs to consider the topological and numerical aspects together to produce the approximate curve to be as close to the exact one as possible, since approximation inevitably involves both numerical and topological errors. In this paper, algorithms to compute an approximate intersection curve, which are topologically consistent and numerically accurate with the exact intersection curve, are presented. A set of sufficient conditions for an approximate curve to be topologically consistent with the exact one are provided, and the use of a validated ordinary differential equation solver is discussed. The approximate curve is then refined to reduce the error against the exact curve through optimization. The proposed method is demonstrated with examples.











Similar content being viewed by others
References
Patrikalakis NM, Maekawa T (2002) Shape interrogation for computer aided design and manufacturing. Springer, Heidelberg
Farouki RT (1999) Closing the gap between CAD model and downstream application. SIAM News 32(5):303–319
Song X, Sederberg TW, Zheng J, Farouki RT, Hass J (2004) Linear perturbation methods for topologically consistent representations of free-form surface intersections. Comput Aided Geometric Des 21(3):303–319
Grandine TA, Klein FW (1997) A new approach to the surface intersection problem. Comput Aided Geometric Des 14(2):111–134
Hu Y, Sun T (1997) Moving a B-spline surface to a curve—a trimmed surface matching algorithm. Comput Aided Des 29(6):449–455
Krishnan S, Manocha D (1997) Efficient surface intersection algorithm based on lower-dimensional formulation. ACM Trans Graph 16(1):74–106
Maekawa T, Patrikalakis NM, Sakkalis T, Yu G (1998) Analysis and applications of pipe surfaces. Comput Aided Geometric Des 15(5):437–458
Sakkalis T, Charitos C (1999) Approximating curves via alpha shapes. Graph Model Image Process Arch 61(3):165–176
Sakkalis T, Peters TJ, Bisceglio J (2004) Isotopic approximations and interval solids. Comput Aided Des 36(11):1089–1100
DeRose T, Goldman R, Hagen H, Mann S (1993) Functional composition algorithms via blossoming. ACM Trans Graph 12(2):113–135
Renner G, Weiβ V (2004) Exact and approximate computation of B-spline curves on surfaces. Comput Aided Des 36(4):351–362
Hu CY, Maekawa T, Patrikalakis NM, Ye X (1997) Robust interval algorithm for surface intersections. Comput Aided Des 29(9):617–627
Mukundan H, Ko KH, Maekawa T, Sakkalis T, Patrikalakis NM (2004) Tracing surface intersections with a validated ODE system solver. In: Elber G, Patrikalakis NM, Brunet P (eds) Proceedings of the ninth EG/ACM symposium on solid modeling and applications. EG/ACM, Eurographics Press, Genoa, pp 249–254
Corliss GF, Rihm R (1996) Validating an a priori enclosure using high-order Taylor series. In: Alefeld G, Frommer A, Lang B (eds) Scientific computing and validated numerics: proceedings of the international symposium on scientific computing, computer arithmetic and validated numerics—SCAN’95. Akademie Verlag, Berlin, pp 228–238
Nedialkov NS, Jackson KR, Corliss GF (1999) Validated solutions of initial value problems for ordinary differential equations. Appl Math Comput 105(1):21–68
Patrikalakis NM, Maekawa T, Ko KH, Mukundan H (2004) Surface to surface intersection. Comput Aided Des Appl 1(1–4):449–457
Nedialkov NS (1999) Computing the rigorous bounds on the solution of an initial value problem for an ordinary differential equation. Ph.D. thesis, University of Toronto, Toronto
Löhner RJ (1992) Computation of guaranteed enclosures for the solutions of ordinary initial and boundary value problems. In: Cash J, Gladwell L (eds) Computational ordinary differential equations. Clarendon Press, Oxford, pp 425–435
Sherbrooke EC, Patrikalakis NM (1993) Computation of the solutions of nonlinear polynomial systems. Comput Aided Geometric Des 10(5):379–405
Sakkalis T (1991) The topological configuration of a real algebraic curve. Bull Aust Math Soc 43:37–50
Farouki RT (1987) Computational issues in solid boundary evaluation. Technical Report RC 12454, IBM
Gonzalez-Vega L, Necular I (2002) Efficient topology determination of implicitly defined algebraic plane curves. Comput Aided Geometric Des 19(9):719–743
Sederberg TW, Meyers RJ (1988) Loop detection in surface patch intersections. Comput Aided Geometric Des 5(2):161–171
Sakkalis T, Shen G, Patrikalakis NM (2001) Topological and geometric properties of interval solid models. Graph Models 63(3):163–175
Myles A, Peters J (2005) Threading splines through 3D channels. Comput Aided Des 37:139–148
Zhu C, Byrd RH, Lu P, Nocedal J (1994) L-BFGS-B: FORTRAN subroutines for large scale bound constrained optimization. Technical Report NAM-11, EECS Department, Northwestern University
Acknowledgments
This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute for Information Technology Advancement) (IITA-2009-C1090-0902-0008).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Ko, K.H., Ahn, H.S. Approximation of 3D surface-to-surface intersection curves. Engineering with Computers 26, 49–60 (2010). https://doi.org/10.1007/s00366-009-0133-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00366-009-0133-6