Marching Along A Regular Surface/surface Intersection With Circular Steps

This paper presents a simple and elegant algorithm to estimate adaptively the stepping direction and size for tracing a branch of the intersection curve between two regular surfaces without any nonlinear equation system solver. The step is neither along the tangent vector at the current point nor along a parabola in a vicinity of the current point; it is along a circle at the current point. Although no curvature analysis or power series expansions about each point of the intersection curve were used in its construction, we demonstrate that our circle tends to the exact osculating circle, when the distance between two subsequent sampling points tends to zero. Through numerical examples, we also show that the performance of our algorithm by handling singular points, bifurcations, and points on the closely spaced branches, is equivalent to the ones based on embedding schemes.164249268Abdel-Malek, K., Yeh, H.J., Determining intersection curves between surfaces of two solids (1996) Computer-Aided Design, 28, pp. 539-549Asteasu, C., Intersection of arbitrary surfaces (1988) Computer-Aided Design, 20, pp. 533-538Blachman, N., (1992) Matematica: A Practical Approach, , Prentice-Hall Inc., New JerseyBajaj, C.L., Hoffmann, C.M., Hopcroft, J.E., Lynch, R.E., Tracing surface intersections (1988) Computer Aided Geometric Design, 5, pp. 285-307De Andrade, L.N., (1998) Tracing Along Regular Surface Intersection with Circular Steps, , PhD thesis, FEEC - Unicamp, Campinas (in Portuguese)Kriezis, G.A., Patrikalakis, N.M., Rational polynomial surface intersections (1991) Advances in Design Automation - 1991, 2, pp. 43-52. , Gariele, G.A. ed., Miami, FloridaGrandine, T.A., Klein F.W. IV, A new approach to the surface intersection problem (1997) Computer Aided Geometric Design, 14, pp. 111-134Hoschek, J., Lasser, D., (1993) Fundamentals of Computer Aided Geometric Design, , A K Peters, Ltd., Wellesley, MA, USA. Translation of: Grundlagen der geometrischen Datenverarbeitung, by Schumaker, L.LHoughton, E.G., Emnett, R.F., Factor, J.D., Sabharwal, C.L., Implementation of a divid-and-conquer method for intersection of parametric surfaces (1985) Computer Aided Geometric Design, 2, pp. 173-183Coxeter, H.S.M., (1989) Introduction to Geometry, , Wiley, Inc., 2nd edKoparkar, P., Surface intersection by switching from recursive subdivision to iterative refinement (1991) The Visual Computer, 8, pp. 47-63Mortenson, M.E., (1985) Geometric Modeling, , Wiley, USA, 1st edCarmo, M.P., (1976) Differential Geometry of Curves and Surfaces, , Prentice-Hall Inc., New Jersey, 1st edMüllenheim, G., On determining start points for a surface/surface intersection algorithm (1991) Computer Aided Geometric Design, 8, pp. 401-408Preusser, A., Computing area filling contours for surfaces defined by piecewise polynomials (1986) Computer Aided Geometric Design, 3, pp. 267-279Barnhill, R.E., Farin, G., Jordan, M., Piper, B.R., Surface/surface intersection (1987) Computer Aided Geometric Design, 4, pp. 3-16Barnhill, R.E., Kersey, S.N., A marching method for parametric surface/surface intersection (1990) Computer Aided Geometric Design, 7, pp. 257-280Sederberg, T.W., Meyers, R.J., Loop detection in surface patch intersections (1988) Computer Aided Geometric Design, 5, pp. 161-171Stoyanov, Tz.E., Marching along surface/surface intersection curves with an adaptative step length (1992) Computer Aided Geometric Design, 9, pp. 485-48