Journal of Mechanical Design, ASME Transactions, Vol. 118 No. 4, pages 499-508, 1996. Computation of Shortest Paths on Free-Form Parametric Surfaces Takashi Maekawa Massachusetts Institute of Technology Department of Ocean Engineering Design Lab oratory, Cambridge, MA 02139-4307, USA Abstract Computation of shortest paths on free-form surfaces is an imp ortant problem in ship design, rob ot motion plan- ning, computation of medial axis transforms of trimmed surface patches, terrain navigation and NC machining. The ob jectiveof this pap er is to provide an ecient and reliable metho d for computing the shortest path be- tween two points on a free-form parametric surface and the shortest path between a point and a curve on a free-form parametric surface. These problems can b e reduced to solving a two p oint b oundary value problem. Our approach for solving the twopoint b oundary value problem is based on a relaxation metho d relying on nite di erence discretization. Examples illustrate our metho d. Keywords: geo desics, b oundary value problem, nite di erence metho d, relaxation metho d 1 Intro duction The history of geo desic lines b egins with the study by Johann Bernoulli, who solved the problem of the shortest distance b etween twopoints on a convex surface in 1697, according to [15]. He showed that the osculating plane of the geo desic line must always b e p erp endicular to the tangent plane. The equation of the geo desics was rst obtained by Euler (1732) for a surface given by F (x; y ; z )=0. His attention to the problem was due to Johann Bernoulli, probably through the aid of his nephew Daniel, who was at St. Petersburg with Euler [14]. Bliss [2] obtained the geo desic lines on the anchor ring, which has a torus shap e, analytically. Munchmeyer and Haw [10] were the rst to intro duce the geo desic curvetothe CAGD community. They applied the geo desic curve in ship design, namely to nd out the precise layout of the seams and butts in the ship hull. Beck et al [1] computed b oth the initial-value integration and b oundary-value integration of geo desic paths, using the fourth order Runge-Kutta metho d on a bicubic spline surface. However their pap er do es not provide any details of the sho oting metho d for the solution of the b oundary value problem. Patrikalakis and Bardis computed geo desic o sets of curves on rational B-spline surfaces using the initial-value integration [11]. One application of such o sets is automated construction of linkage curves for free-form pro cedural blending surfaces. Sneyd and Peskin [13] investigated the computation of geo desic paths on a generalized cylinder as initial value problem using a second order Runge-Kutta metho d. Their work was motivated by constructing the great vessels of the heart out of geo desic b ers. Very recently, Kimmel et al. [7] presented a new numerical metho d for nding the shortest path on surfaces by calculating the propagation of an equal geo desic-distance contour from a p oint or a source region on the surface. The algorithm works on a rectangular grid using the nite di erence approximation. The shortest path problem is also very active among the rob ot motion planning and terrain navigation communities, 1 however they usually representthe surface as a p olyhedral surface and solve the problem in the eld of linear computational geometry [9]. In this work a new approach for nding the shortest path b etween twopoints on a free-form parametric surface is intro duced. Also a metho d for computing the shortest path b etween a p oint and a curve on a free-form surface is describ ed. The geo desic path between two points is obtained by rst discretizing the governing equations by nite di erence on a mesh with m p oints. Secondly we start with an initial guess and improve the solution iteratively or in other words relax to the true solution. The shortest path b etween a p ointandacurve is obtained in a similar way but with nested iterations, since we do not know the p oint on the curvewhich forms the shortest path b eforehand. This pap er is structured as follows. Section 2 b egins with a brief review of relevant di erential geometry prop erties of geo desics on a parametric surface. Section 3 describ es the nite di erence approach to the two p oint b oundary value problem. Section 4 explains how to obtain a go o d initial approximation for the twopoint b oundary value problem and provides a robust algorithm to nd all the geo desic paths b etween twopoints on a surface. Section 5 discusses an algorithm to obtain a geo desic path from a p oint to a curve on a surface. Finally, section 6 illustrates our metho d with examples and concludes the pap er. 2 Review of Di erential Geometry The b o oks by Struik [15], Kreyszig [8], doCarmo [4] o er rm theoretical basis to the di erential geometry asp ects of geo desics. In this section, we summarize the relevant de nitions employed in this work. A general parametric surface S can b e de ned as a vector-valued mapping from two-dimensional uv parameter domain to a set of three-dimensional xy z co ordinates T r(u; v )=[x(u; v ); y (u; v ); z (u; v )] : (1) Let C b e an arc length parametrized regular curve on surface r(u; v )which passes through p oint P as shown in Figure 1 and denote by r(s)=r(u(s);v(s)): (2) Let t b e a unit tangentvector of C at P , n be a unit normal vector of C at P , N b e a unit surface normal vector N P S u t kg n C kn k Figure 1: De nition of geo desic curvature of S at P and u b e a unit vector p erp endicular to t in the tangent plane de ned by N t. We can decomp ose the curvature vector k of C into N comp onent k , whichis called normal curvature vector, and u comp onent n k , which is called geo desic curvature vector as follows: g k = k + k = N + u (3) n g n g 2 Here and are the normal and geo desic curvatures, resp ectively and de ned as follows: n g = k N (4) n = k u (5) g The minus sign in equation (4) ensures that if is p ositive, the center of curvature lies opp osite to the direction n of the surface normal. Consequently, dt = (N t) (6) g ds The unit tangentvector of the curve C can b e obtained bydi erentiating equation (2) with resp ect to the arc length. dr(u(s);v(s)) du dv t = = r + r (7) u v ds ds ds where the chain rule is used and the subscripts u and v denote the rst partial derivatives with resp ect to u and v parameter resp ectively. Hence 2 2 2 2 dt du du dv dv d u d v = r +2r + r + r + r (8) uu uv vv u v 2 2 ds ds ds ds ds ds ds so that 3 2 2 du du dv dv du +(2r r + r r ) +(r r +2r r ) = [(r r ) u uv v uu u vv v uv g u uu ds ds ds ds ds 3 2 2 d v d u dv du dv (9) ] N +(r r ) N + (r r ) u v v vv 2 2 ds ds ds ds ds 2 2 3 2 2 3 dv du du dv dv du d v dv d u du We can easily notice that the co ecients of , , , are all functions , 2 2 ds ds ds ds ds ds ds ds ds ds of the co ecients of the rst fundamental form E , F and G and their derivatives, E , F , G , E , F , G . It u u u v v v is interesting to note that the normal curvature dep ends on both the rst and second fundamental forms, n i while the geo desic curvature dep ends only on the rst fundamental form. Using the Christo el symb ols jk (i; j; k =1; 2) de ned as follows 2EF EE + FE GE 2FF + FE u v u u u v 2 1 ; = = 11 11 2 2 2(EG F ) 2(EG F ) GE FG EG FE v u u v 1 2 = ; (10) = 12 12 2 2 2(EG F ) 2(EG F ) EG 2FF + FG 2GF GG + FG v v u v u v 2 1 ; = = 22 22 2 2 2(EG F ) 2(EG F ) geo desic curvature can b e reduced to 3 2 2 du du du dv dv 2 1 1 2 2 = [ ) ) 2 +(2 +( g 11 11 12 12 22 ds ds ds ds ds 3 2 2 p du d u dv d v dv 1 2 + ] EG F (11) 22 2 2 ds ds ds ds ds Geo desic paths are sometimes de ned as shortest path b etween p oints on a surface, however this is not always a satisfactory de nition. In this pap er we de ne as follows [15]: De nition 2.1 Geodesics are curves of zerogeodesic curvature. In other words, the osculating planes of a geo desic contain the surface normal.