Abstract. We describe a parametric hybrid Bezier patch that, in addition. schemes are local in that changes to part of the data only aect portions of

Size: px

Start display at page:

Download "Abstract. We describe a parametric hybrid Bezier patch that, in addition. schemes are local in that changes to part of the data only aect portions of"

Veronica Todd
6 years ago
Views:

1 A Parametri Hyrid Triangular Bezier Path Stephen Mann and Matthew Davidhuk Astrat. We desrie a parametri hyrid Bezier path that, in addition to lending interior ontrol points, lends oundary ontrol points. This oundary lend is neessary to generalize a funtional ross-oundary onstrution that relies on the natural parameterization of the funtional setting. When interpolating irregularly sattered data and when inreasing the tessellation of the data mesh, the new sheme shows improvement over representative parametri data tting shemes. x1. Introdution A large numer of loal parametri triangular surfae shemes have een developed over the past fteen years (see [10] for a survey of suh shemes). These shemes are loal in that hanges to part of the data only aet portions of the surfae near the hanged data. Surprisingly, all of these shemes exhiit similar shape defets. On loser inspetion, it is seen that these shemes all have a large numer of free parameters that are set using simple heuristis. By manually adjusting these parameters, one an improve the shape of the surfaes [7]. One way to improve automatially the shape of the onstruted surfaes is to use variational methods. Several authors have used suh shemes to improve the shape of the onstruted surfaes, ut usually at a high omputational ost due to the gloal nature of the solution (e.g., [11]). Similarly, we an use loal optimization methods to set the free parameters and improve the shape, although the results are not as good as the gloal methods [8]. In this paper, we will investigate a loal, non-optimization method for improving the onstrution of the ross-oundary derivatives in a triangular parametri sheme. This is a generalization of the hyrid sattered data tting sheme of Foley and Opitz [2] (their method is similar to a result independently developed y Goodman and Said [4]). The resulting surfaes show large improvement in shape over other loal, parametri, triangular surfae tting tehniques. More preisely, given a triangle of data (a set of three verties with normals), our sheme onstruts a hyrid, parametri path that Mathematial Methods for Curves and Surfaes II 1 Morten Dhlen, Tom Lyhe, Larry L. Shumaker (eds.), pp. 1{3. Copyright o 1998 y Vanderilt University Press, Nashville, TN. ISBN 1-xxxxx-xxx-x. All rights of reprodution in any form reserved.

2 2 S. Mann and M. Davidhuk p q Fig. 1. Domain ontrol net for the Foley-Opitz hyrid Bezier path. interpolates the positions and normals at the orners. When used to ll a triangular polyhedron, the resulting surfae pathes will meet with tangent plane ontinuity. x2. The Foley-Opitz Sheme Foley and Opitz [2] present a method for interpolation of sattered data aove the plane using a \hyrid" ui Bezier path ased on Nielson's sheme [12]. A hyrid ui path is similar to a ui Bezier path, exept the interior ontrol point is a rational lend of three points. With the Foley-Opitz method, the ui path oundaries are ompletely determined y the triangle verties and normals. The three inner ontrol points are onstruted using a C 1 ross oundary onstrution that gives the hyrid path ui preision. Figure 1 shows the domain ontrol net for two neighoring triangles. p2 is one of the three interior ontrol points assoiated with the left triangle and q2 is one of the three interior ontrol points assoiated with the right triangle. Foley and Opitz ompute p2 as follows. Let r, s, and t e the aryentri oordinates of 003 with respet to 003, 030, and 003. If oth pathes if Figure 1 form a single ui, then from sudividing Bezier uis it an e shown that p r 2 ( ) 2rs( ) 2rt201 2st021 s 2 ( ) t 2 ( ) (2(r + s)t) The point q2 is fored y ontinuity onditions to e q2 rp2 + s120 + t : When applied to data that does not ome from a ui, the Foley-Opitz onstrution of p2 and q2 ensures that the two triangles have a C 1 join along their ommon order. Idential alulations would e made to ensure C 1 ontinuity aross the remaining two edges giving three settings for the interior ontrol points of eah of the two pathes. :

3 Parametri Hyrid Triangular Path { DRAFT 3 Fig. 2. Isophotes of interpolants to the Franke data set. Fig. 3. One plane per path pair. The three interior points (p0 p0, p1, p2) are lended with Nielson's rational lend funtions, giving a i (t0; t1; t2) t j t k t i t j + t i t k + t j t k ; i 6 j 6 k: (1) 111(t0; t1; t2) a0(t0; t1; t2)p0 + a1(t0; t1; t2)p1 + a2(t0; t1; t2)p2 After lending, we are left with a 10 point ui Bezier path, whih is evaluated at (t0; t1; t2) in the standard way. Figure 2 shows isophote plots [5] of the Franke data set [3] (on the left), and isophote plots of this data interpolated with Clough-Toher (middle) and Foley-Opitz (right) pathes. The C 1 disontinuities in the isophote lines our along path oundaries, and are visile as shape artifats in shaded images [9]. The Foley-Opitz interpolant is generally smoother than the Clough-Toher interpolant. x3. A Hyrid, Parametri, Cui Sheme Our goal is to reate a parametri version of the Foley-Opitz sheme to ring its good surfae quality to the parametri setting. There are two prolems we must solve to reate this parametri version of the Foley-Opitz method: Finding loal parameterizations for the Foley-Opitz ross-oundary method, and nding lend funtions for the resulting points that have the appropriate properties. The Foley-Opitz ross-oundary onstrution relies on a natural parameterization etween path pairs sine the aryentri oordinates of neighoring pathes with respet to eah other are key in determining the tangent plane elds. In the parametri setting, there is no predened assoiation etween path domains. Therefore, some assoiation etween neighoring path domains must e made in order to use Foley's tangent plane eld onstrution. p2:

4 4 S. Mann and M. Davidhuk V ,2 201,1 210,1 201,2 120,2 111,2 111,1 102,1 111,0 120,0 102,0 V ,2 012,1 021,0 012,0 003 V 2 Fig. 4. Domain ontrol net for parametri hyrid path. 120,0 111,0 102,0 V ,0 003 V 2 012,0 Fig. 5. Tangent plane eld ontrol points along the edge. Our approah is to hoose a plane for eah path pair (Figure 3), projet the orner points of oth pathes onto the plane, and then perform Foley's C 1 onstrution. Three sets of ontrol points are alulated for eah hyrid path { eah set representing a C 1 onstrution along one triangle edge. The three sets of ontrol points will share the same triangle orner verties ut in general dier in the rest of the oundary and interior ontrol points. We must lend oth oundary and interior ontrol points to produe the nal interpolant. Figure 4 shows the domain ontrol net for a parametri version of Foley's sheme. The struture is similar to Foley's { the ontrol points are organized like the ontrol points of a ui triangular Bezier path exept a group of ontrol points orrespond to a single, regular ui ontrol point. When onstruting the tangent plane eld along a partiular oundary, only two parametri hyrid pathes are involved and onsequently only two Bezier pathes are needed for the onstrution. Eah Bezier path ontriutes seven ontrol points to a parametri hyrid path, as in Figure 5. These ontrol points determine the tangent plane eld along that oundary. To onstrut the seven points, a plane is hosen as a parameterization for the two Bezier pathes allowing the onstrution as in Foley's funtional sheme. One the plane is hosen, the Bezier path ontrol points are ompletely determined y the triangle verties and the assoiated normals { Hermite interpolation over a plane ompletely determines the ui oundary urves and Foley's

5 Parametri Hyrid Triangular Path { DRAFT 5 V V V Fig. 6. Control net after lending parametri hyrid ontrol points. ross oundary onstrution determines the interior ontrol points. After the ontrol points from three Bezier pathes are alulated, they an e used to formulate the parametri hyrid path. However, lending is not as straight forward as in the funtional ase { oundary points are inluded in the lend. Figure 6 illustrates the ontrol net for the parametri path. Eah point ijk is a rational lend of the assoiated ontrol points from the three Bezier pathes shown in Figure 4. The points 300; 030, and 003 are onstants, the remaining oundary ontrol points are rational lends of two Bezier path oundary ontrol points, and the interior point 111 is a lend of three points. The lend formulation must preserve the important properties of the three underlying Bezier pathes, oundary urves and ross oundary derivatives, when evaluating along the edges. The orner ontrol points are onstants so no lending funtion is needed. The lending funtion for the interior ontrol points is the same as used in Foley's funtional onstrution, giving us 111 a0111;0 + a1111;1 + a2111;2 where the a i are dened in (1). The oundary ontrol points are lended to give two properties: 1) When evaluated along a oundary the parametri Foley ontrol points eome the ontrol points of one of the three Bezier pathes 2) The tangent plane eld of the parametri hyrid path along a oundary mathes the tangent plane eld along the same oundary of one of the three Bezier pathes. An asymmetri lend of the following form has oth of these properties h ij (u0; u1; u2) (1 u i )u j 2 (1 u i )u j 2 + (1 uj )u i 2 (2) This lending funtion is used to weight all the non-vertex oundary Bezier ontrol points: ijk;i ijk (u0; u1; u2) h ij (u0; u1; u2) ijk;i ijk;j ijk;i + h ji ijk;k ijk;j ;

6 6 S. Mann and M. Davidhuk for ijk eing any permutation of 012. For example, the V1V2 oundary ontrol points would e 012(u0; u1; u2) h01(u0; u1; u2)012;0 + h10(u0; u1; u2)012;1 021(u0; u1; u2) h02(u0; u1; u2)021;0 + h20(u0; u1; u2)021;2 When evaluated along the V1V2 edge u0 0 we get 012(0; u1; u2) 012;0; 021(0; u1; u2) 021;0 sine Equation (2) gives h0j(0; u1; u2) 1 and h i0(0; u1; u2) 0. The V1V2 oundary urve is the ui Bezier urve given y the ontrol points 030, 021;0, 012;0, and 003, whih gives us C 0 ontinuity. This lend also gives us C 1 ontinuity, as desried in Davidhuk's thesis [1]. The ontrol points f ; 030; 003; ~i;j g are lended to form the ten ontrol points of a standard ui Bezier path, laeled as ~i. The Bezier path dened y the ~i is then evaluated X at u. The onise denition is F B 3 ~ ~i i j ij3 ~ x4. Choie of plane There is freedom in the hoie of the projetion plane, and there are some restritions. The orientation, not the position, of the plane determines the positions of the ontrol points thus giving two rotational degrees of freedom. The plane should e onstruted geometrially from the given information { triangle verties and normals. The two Bezier pathes must not have overlapping domains on the plane so the orientation is restrited to eing \underneath" oth pathes. One failsafe method of hoosing a plane is to take the plane perpendiular to the iseting plane of the two neighoring triangles and that ontains their ommon edge. With this hoie of plane, the projetion of the triangles will lie on opposite sides of the projetion of their ommon edge, so the path domains will never overlap. However, a etter hoie is the plane perpendiular to the average of the normals at the two data points, and either date point. Although this onstrution is not guaranteed to give us a valid plane (i.e., the two projetion of the two triangles along the edge may overlap), it does in general give us etter shaped surfaes [1].

Parametri Hyrid Triangular Path { DRAFT 7 Fig. 7. Triangular Gregory pathes and our sheme t to a torus. Fig. 8. Triangular Gregory pathes and our sheme t to a at data set.

We ompared our method to Triangular Gregory pathes [6]. In Figure 7, we see isophote plots of oth methods t to a 10x10 sampling of the torus. The isophotes for our method are notiealy smoother.

7 Parametri Hyrid Triangular Path { DRAFT 7 Fig. 7. Triangular Gregory pathes and our sheme t to a torus. Fig. 8. Triangular Gregory pathes and our sheme t to a at data set. xresults To test our surfae onstrution method, we used it to t pathes to a at data set (where normals are estimated) and to samplings of a torus (where normals ome from the torus). We ompared our method to Triangular Gregory pathes [6]. In Figure 7, we see isophote plots of oth methods t to a 10x10 sampling of the torus. The isophotes for our method are notiealy smoother. In Figure 8, we see shaded images of oth methods t to a at data set. This data set has 366 verties and 698 faes. Normals to the verties were estimated y a simple averaging of fae normals. Many of the shape artifats that appear on triangular Gregory pathes our not present on the surfae onstruted y our sheme. One drawak to our sheme is that as the tangent planes on either side of an edge eome perpendiular to the edge, the interior points of the oundary urve move towards innity. Thus, normal estimation eomes a ritial step.

8 8 S. Mann and M. Davidhuk Aknowledgments. This researh was funded y the Natural Sienes and Engineering Researh Counil of Canada Referenes 1. Davidhuk, M., A Parametri Tyrid trianglar Bezier Path, dissertation, Master's, University of Waterloo (Waterloo ON), Foley, T. A. and K. Opitz, Hyrid Cui Bezier Triangle Pathes, in Mathematial Methods in Computer Aided Geometri Design II, T. Lyhe and L. Shumaker (eds), Aademi Press, New York, 1992, 275{ Franke, R., A ritial omparison of some methods for interpolation of sattered data, Report NPS , Naval Postgraduate Shool, Goodman, T. N. T. and H. B. Said, A C 1 triangular interpolant suitale for sattered data interpolation, Commun. Appl. Numer. Methods 7 (1991), 479{ Hagen, H., S. Hahmann, T. Shreier, Y. Nakajima, B. Wordenweer, and P. Hollemann-Grundetedt, Surfae interrogation algorithms, Comp. Graphis and Applis. 12 (1992), 53{ Longhi, L., Interpolating pathes etween ui oundaries, Tehnial Report T.R. UCB/CSD 87/313, University of California, Berkeley, Mann, S., Surfae Approximation Using Geometri Hermite Pathes, dissertation, Dotoral, University of Washington (Seattle, WA), Mann, S., Using loal optimization in surfae tting, in Mathematial Methods for Curves and Surfaes, Morten Dhlen, Tom Lyhe, Larry L. Shumaker (eds), Vanderilt University Press, Nashville & London, 1995, 323{ Mann, S., Cui preision Clough-Toher interpolation, sumitted for puliation. 10. Mann, S., C. Loop, M. Lounsery, D. Meyers, J. Painter, T. DeRose, and K. Sloan, A survey of parametri sattered data tting using triangular interpolants, in Curve and Surfae Design, H. Hagen (ed), SIAM Puliations, SIAM, Philadelphia PA, 1992, 145{ Moreton, H. P. and C. H. Sequin, Funtional Optimization for Fair Surfae Design, Computer Graphis (ACM SIGGRAPH) 26 (1992), 167{ Nielson, G. M., A transnite, visually ontinuous, triangular interpolant, in Geometri Modeling: Algorithms and New Trends, G. E. Farin (ed), SIAM Puliations, Philadelphia, 1987, 235{245. Stephen Mann and Matthew Davidhuk Computer Siene Department University of Waterloo, Waterloo, ON, N2L 2R7, CANADA smann@gl.uwaterloo.a, mdavidh@gl.uwaterloo.a

On Optimal Total Cost and Optimal Order Quantity for Fuzzy Inventory Model without Shortage

International Journal of Fuzzy Mathemat and Systems. ISSN 48-9940 Volume 4, Numer (014, pp. 193-01 Researh India Puliations http://www.ripuliation.om On Optimal Total Cost and Optimal Order Quantity for