Dscrete Farng of Curves and Surfaces Based on Lnear Curvature Dstrbuton R. Schneder and L. Kobbelt Abstract. In the planar case, one possblty to create a hgh qualty curve that nterpolates a gven set of ponts s to use a clothod splne, whch s a curvature contnuous curve wth lnear curvature segments. In the rst part of the paper we develop an ecent farng algorthm that calculates the dscrete analogon of a closed clothod splne. In the second part we show how ths dscrete lnear curvature concept can be extended to create a farng scheme for the constructon of a trangle mesh that nterpolates the vertces of a gven closed polyhedron of arbtrary topology. x1. Introducton In many elds of computer-aded geometrc desgn (CAGD) one s nterested n constructng curves and surfaces that satsfy aesthetc requrements. A common method to create far objects s to mnmze farness metrcs, but snce hgh qualty farness functonals are usually based on geometrc nvarants, the mnmzaton algorthms can become computatonally very expensve [1]. A popular technque to smplfy ths approach s to gve up the parameter ndependence by approxmatng the geometrc nvarants wth hgher order dervatves. For some mportant farness functonals ths results n algorthms that enable the constructon of a soluton by solvng a lnear equaton system, but such curves and surfaces are n most cases not as far as those dependng on geometrc nvarants only. An nterestng approach to smplfy the constructon process wthout gvng up the geometrc ntrnscs s to use varatonal calculus to derve derental equatons characterzng the soluton of a mnmzaton problem. Mehlum [8] used ths dea to approxmate a mnmal energy curve (MEC), whch mnmzes the functonal R (s) 2 ds, wth pecewse arc segments. In [1] Brunnet and Kefer exploted the property that a segment of a MEC between two nterpolaton ponts satses the derental equaton + 1 2 3 = to speed up the constructon process usng looup tables. Curve and Surface Desgn: Sant-Malo 1999 371 Perre-Jean Laurent, Paul Sablonnere, and Larry L. Schumaer (eds.), pp. 371{38. Copyrght oc 2 by Vanderblt Unversty Press, Nashvlle, TN. ISBN -8265-1356-5. All rghts of reproducton n any form reserved.
372 R. Schneder and L. Kobbelt The usage of such derental equatons based on geometrc nvarants can be seen as a reasonable approach to the farng problem n ts own rght, and t can be appled to curves as well as surfaces. For planar curves, one of the smplest derental equatons s =. Assumng arc length parameterzaton, ths equaton s only satsed by lnes, crcles and clothods. A curvature contnuous curve that conssts of parts of such elements s called a clothod splne. Most algorthms for the constructon of such curves are based on technques that construct the correspondng lne, crcle and clothod segments of the splne [9]. Ths s possble for planar curves, but the dea does not extend to surfaces. In ths paper we wll rst present a fast algorthm to construct an nterpolatng closed dscrete clothod splne (DCS) purely based on ts characterstc derental equaton. Our algorthm uses dscrete data, because ths has been proven to be especally well suted for the ecent constructon of nonlnear splnes [7]. The ecency of our constructon process s largely based on an algorthm called the ndrect approach. Ths algorthm decouples the curvature nformaton from the actual geometry by explotng the fact that the curvature dstrbuton tself s a dscrete lnear splne. Besdes the speed of the planar algorthm, t has another very mportant property: t drectly extends to the constructon of surfaces! In Secton 2 we gve a short revew of related wor. Secton 3 wll rst gve an exact denton of a DCS and then address ts ecent computaton. Secton 4 shows how the planar algorthm extends to surfaces. We construct far surfaces nterpolatng the vertces of a gven closed polyhedron of arbtrary topology by assumng a pecewse lnear mean curvature dstrbuton wth respect to a natural parameterzaton. x2. Related Wor A very nterestng algorthm addressng the problem of constructng far curves and surfaces wth nterpolated constrants was presented by Moreton and Sequn. In [1] they mnmzed farng functonals, whose farness measure punshes the varaton of the curvature. The qualty of ther mnmal varatonal splnes s extraordnary good, but due to ther extremely demandng constructon process, the computaton tme needed s enormous. Ther curves and surfaces conssted of polynomal patches. In contrast to ths approach, most farng and nonlnear splnes algorthms are based on dscrete data. Malcolm [7] extended the dscrete lnear splne concept to ecently calculate a dscrete MEC for functonal data. In [13] Welch and Wtn presented a nonlnear farng algorthm for meshes of arbtrary connectvty, based on the stran energy of a thn elastc plate. Recently, Desbrun et al. [3] used the mean curvature ow to derve a dscrete farng algorthm for smoothng of arbtrary connected meshes. In most farng algorthms the orgnal farng functonals are approxmated by smpler parameter dependent functonals. In recent years ths dea was combned wth the concept of subdvson surfaces to create varatonal
Dscrete Farng of Curves and Surfaces 373 subdvson splnes [12,6,4]. Especally [4] should be emphaszed here, snce our local parameterzaton needed n Secton 4 s largely based on deas presented there. Instead of mnmzng a functonal, Taubn [11] proposed a sgnal processng approach to create a fast dscrete farng algorthm for arbtrary connectvty meshes. Closely related but based on a derent approach s the dea presented by Kobbelt et al. [5], where a unform dscretzaton of the Laplace operator s used for nteractve mesh modelng. In [3] Desbrun et al. showed that a more sophstcated dscretzaton of the Laplace operator can lead to mproved results. The drect teraton approach presented later can be seen as a nonlnear generalzaton of such schemes. x3. Dscrete Clothod Splnes In ths secton we show how to nterpolate a closed planar polygon usng a dscretzaton of a clothod splne. Although the algorthms extends to open polygons wth G 1 boundary condtons n a straghtforward manner, we only consder closed curves, because that case drectly extends to surfaces. 3.1 Notaton and dentons In the followng we denote the vertces of the polygon that has to be nterpolated wth P = fp 1 ; : : : ; P n g. Let = f 1 ; : : : ; m g be the vertces of a rened polygon wth P. The dscrete curvature at a pont s de- ned to be the recprocal value of the crcle radus nterpolatng the ponts 1 ; ; +1 or f the ponts le on a straght lne, whch leads to the well nown formula det( 1 ; +1 ) = 2 jj 1 jj jj +1 jj jj +1 1 jj : (1) Denton 1. A polygon wth P s called a dscrete clothod splne (DCS), f the followng condtons are satsed: 1) The nteror of each segment s arc length parameterzed jj 1 jj = jj +1 jj, whenever 62 P, 2) The dscrete curvature s pecewse lnear: 2 = 1 2 + +1 =, whenever 62 P. We wll construct a DCS usng an teraton procedure! +1 untl the above condtons are sucently satsed. The teraton starts wth an ntal polygon that nterpolates the P (Fg. 1). 3.2 Drect approach In ths approach the locaton of a vertex +1 only depends on the local neghborhood 2 ; : : : ; of ts predecessor +2. To satsfy condton 1 n Denton 1 locally, +1 has to le on the perpendcular bsector between
374 R. Schneder and L. Kobbelt -1-2 +1 +1 +2-1 +1 +1 = P 1 1 2 3 m P m-1 2 P n...... Fg. 1. The left sde shows the update steps for n the drect and the ndrect teraton. The rght sde shows the ntal polygon. We smply subsampled the polygon P. and 1 +1 (Fg. 1), reducng the 2 varate problem to a unvarate one. Further we requre +1 to satsfy 2 +1 =. Unfortunately ths equaton s nonlnear n the coordnates of the vertces +1, but snce the update! +1 n the -th teraton step wll be small, we can lnearze the equaton by usng the coordnates of nstead of +1 n the denomnator of equaton (1). Ths allows us to update every solvng a 2 2 lnear system for +1 durng one teraton step. 3.3 Indrect approach If we decouple the curvature values +1 from the actual polygonal geometry and perform the drect teraton scheme only on the curvature values by solvng 2 +1 =, the curvature plot would converge to a pecewse lnear functon. The dea of the ndrect approach s to use ths property to create a new teraton scheme. Ths s done by dvdng each teraton step nto two sub-steps, n the rst sub-step we estmate a contnuous pecewse lnear curvature dstrbuton, and n the second sub-step we use those as boundary condtons to update the ponts. We get the curvature dstrbuton by estmatng the curvature of the polygon at every pont P j, and nterpolate ths curvature values lnearly across the nteror of the segments, assgnng a curvature estmaton ~ +1 to every vertex. In the second step ths curvature nformaton s used to update the ponts, where the poston of the new pont +1 s determned by ~ +1 and the neghborhood 1 ; ; +1 of ts predecessor (Fg. 1). Agan one degree of freedom s reduced by restrctng +1 to le on the perpendcular bsector between the ponts and 1, +1 and further we requre +1 to satsfy +1 = ~ +1. Usng an analogous lnearzaton technque as n the prevous case, we can agan update every solvng a 2 2 lnear system. It s very mportant to alternate both sub-steps durng each teraton. Snce the ~ +1 are only estmates of the nal DCS, an teraton process that merely terates the second sub-step mght not converge. Compared to the drect approach, the ndrect teraton scheme has some nterestng advantages. The nterpolaton ponts mply forces everywhere durng one teraton step; there s no slow propagaton as n the drect approach. The drect approach acts due to ts local formulaton as a lowpass lter, here because of ts global strategy the ndrect teraton s much better suted to reduce the low frequency error. One teraton step s cheaper than one teraton step of the drect approach. Ths fact wll become much more mportant
Dscrete Farng of Curves and Surfaces 375 1 a) 1 b) 1 c) 5 5 5-5 -5-5 -1-5 5 1-1 -5 5 1-1 -5 5 1 1.5 1 d).5 -.5-1 -1.5 1 2 3 4 5 6 7 1.5 1 e).5 -.5-1 -1.5 1 2 3 4 5 6 7 Fg. 2. a) A polygon P wth 8 vertces. b) The DCS nterpolatng the vertces of P. The structure of the polygon s equvalent to a 5 tmes subdvded P. The calculaton tme was < :1 seconds. c) Perodc C 2 cubc splne nterpolaton of P. d) Curvature plot of the DCS. e) Curvature plot of the perodc cubc splne. f we extend the algorthms to the surface case. Fnally, snce the estmated curvatures are constructed by lnear nterpolaton they are well bounded, a fact that stablzes the teraton procedure. 3.4 Detals and remars Both schemes were mplemented usng a generc multgrd scheme, a technque that has proven to be valuable when herarchcal structures are avalable [4]. We rst constructed a soluton on a coarse level, and used a prolongaton of ths coarse soluton as startng pont for an teraton on a ner level. Applyng ths strategy across several levels of the herarchy, the convergence of both approaches are ncreased dramatcally. On the coarsest herarchy level the ntal polygon was constructed by lnearly samplng the polygon P (Fg. 1). The curvature values at the nterpolaton ponts 2 P needed n the estmaton step were calculated by smply applyng formula (1) on. Wthout the multgrd approach, such a smple strategy would not be reasonable, because for ne polygons the occurrng curvature values could become too large. Our nal teraton scheme for the DCS s a combnaton of the two prmary multgrd teratons. Such an approach has the advantage that the estmated curvatures are usually better, snce the hgh frequency error near the nterpolaton ponts are smoothed out by the drect step. Ths multgrd hybrd approach turned out to be a very relable solver. All examples throughout the paper were mplemented n Java 1.2 for Wndows runnng on a PII wth 4MHz. In Fgure 2 we see that our algorthm stll produces a far soluton n cases where classcal perodc cubc splne nterpolaton usng chord length parameterzaton fals completely. In ths example we can also see why t s not only comfortable to be able to start wth such a smple ntal polygon, but also can be very mportant n some
376 R. Schneder and L. Kobbelt Fg. 3. Wreframe of a mesh P and the solutons of a 2 tmes resp. 3 tmes subdvded mesh. Iteraton tmes: mddle :2s, rght :4s. cases. Our algorthm searches the soluton next to the startng polygon, so tang to be a a lnearly sampled P wll prefer solutons wthout loops. A technque that would use the perodc cubc splne to ntalze n ths example would ether fal because of the enourmous curvatures that turn up, or produce a DCS wth loops. It s well nown that clothods can be parameterzed usng Fresnel Integrals [9]. If such a contnuous soluton s preferred, the dscrete soluton could be used to derve a closed form representaton. x4. Closed Surfaces of Arbtrary Topology In ths secton we want to show how to extend the concept of planar clothod splnes to closed surfaces of arbtrary topology. Gven a closed polyhedron P of arbtrary topology, we construct a dscrete far surface that nterpolates the vertces of P. To be able to extend the planar algorthms, we rst have to determne what curvature measure for surfaces has to be used. Accordng to Bonnet's unqueness problem, the mean curvature seems to be most approprate. 4.1 Notaton and dentons In the followng, we denote the vertces of the polyhedron that has to be nterpolated by P = fp 1 ; : : : ; P n g. Let = f 1 ; : : : ; m g be the vertces of a rened polyhedron wth P, where we requre the topologcal mesh structure of to be equvalent to a unformly subdvded P (Fg. 3). Because of ths structure, we can partton the vertces of nto three classes. Vertces 2 P are called nterpolaton vertces, that can be assgned to an edge of P are called edge vertces, and the remanng ponts are called nner vertces. Edge and nner vertces always have valence 6; for those ponts let ;l ; l = 1::6 be ther adjacent vertces. For edge vertces we mae the conventon that the adjacent vertces are arranged such that ;1 and ;4 are edge or nterpolaton vertces. Let H resp. H ;l be the dscrete mean curvature at resp. ;l and let n be the dscrete unt normal vector at. Fnally, let us dene the followng operators: P 1=6 6 g p ( ) = l=1 ;l; f s an nner vertex, 1=2 ( ;1 + ;4 ); f s an edge vertex,
Dscrete Farng of Curves and Surfaces 377 c a b c b a a c c b c a a b b d e Fg. 4. Left) Parameter doman for an nner vertex. Mddle) Parameter doman for an edge vertex. Rght) The parameter doman for an nterpolaton vertex of valence 5 s a subset of ths doman. P 1=6 6 g h (H ) = l=1 H ;l; f s an nner vertex, 1=2 (H ;1 + H ;4 ); f s an edge vertex. We are now n the poston to extend Denton 1 to the surface case. Denton 2. The polyhedron wll be called a soluton to our dscrete farng problem, f the followng condtons are satsed: 1) The vertces are regularly dstrbuted. For all edge and nner vertces, there should be a t 2 IR such that = g p ( ) + t ~n, 2) The curvature s lnearly dstrbuted over each face: g h (H ) = H. Condton 2 s straghtforward, t states that the mean curvature s changng lnearly wth respect to the barycentrc parameterzaton of the nner and edge vertces. Condton 1 s more dcult to understand. It generalzes the DCS property that vertces n the nteror of a segment were on the perpendcular bsector between ts neghbors. As n the planar case, we construct the surfaces usng an teraton! +1, startng wth an ntal polyhedron. 4.2 Dscretzaton of the geometrc nvarants Our dscretzaton technque s based on the well-nown dea of constructng a local quadratc least square approxmaton of the dscrete data and estmatng the needed geometrc nvarants from the rst and second fundamental forms. For the mean curvature H, ths means [2] H = 1 2 eg 2fF + ge EG F 2 ; (2) where E; F; G are the coecents of the rst fundamental form, and e; f; g are those of the second fundamental form of the least square approxmaton. Instead of recalculatng a local parameterzaton durng each step n the teraton, we explot the fact that our meshes are well structured to determne a local parameterzaton n advance, thus rasng the speed of our algorthms consderably. The decson, whch local parameterzaton should be assgned to a vertex was nuenced by three mayor constrants: smplcty, regularty and unqueness of the quadratc approxmaton. These constrants lead us to the followng local parameterzaton classes (Fg. 4). For nner vertces, the
378 R. Schneder and L. Kobbelt parameter doman s a regular hexagon, for edge vertces t s composed of two regular hexagon halves. Calculatng the matrces needed for a least square approxmaton n both cases, t s easy to see that ths parameterzatons always lead to a unque least square approxmaton. Only at the nterpolaton vertces t s not always sucent to use the 1-neghborhood. Ths s obvous f the valence v of the vertex s 3 or 4, but even f the valence s hgher, the least square approxmaton can fal n the 1-neghborhood. Usng parts of the 2- neghborhood consstng of regular sectors solves that problem. For example, the least square approxmaton s already unque by usng the 1-neghborhood and one complete sector of the 2-dsc (mared wth a,b,c n Fg. 4). Snce we are only nterested n ntrnsc values, we can use the fact that an ane mappng of the parameter doman only changes the parameterzaton of our quadratc approxmaton, but not geometrc nvarants. Ths fact allowed us to chose one xed equlateral hexagon to serve as parameter doman for all nner vertces, and guaranteed a smple update step for such ponts. At edge and nterpolaton vertces, we calculated the parameter domans by applyng the blendng technque proposed n [4]. Ths algorthm constructs a local parameterzaton that adapts to the underlyng geometry dened by P, thus approxmatng a local sometrc parameterzaton. For a detaled descrpton of that algorthm we have to refer to that wor. 4.3 Drect approach When updatng the vertex n an teraton step, the new poston s determned by the two equatons +1 = g p ( ) + t ~n and H +1 = g h (H ). Wth analogous arguments as n the curve settng, we lnearzed the mean curvature expresson (2) for H +1 by usng the vertces of to calculate the coecents of the rst fundamental form E; F; G and by replacng the normal n +1 by n when calculatng the coecents of the second fundamental form. Fnally, the vertex +1 s determned by solvng a lnear equaton for t. 4.4 Indrect approach The planar ndrect teraton scheme drectly extends to the surface settng. In the rst sub-step we estmate the mean curvature of the polyhedron at the nterpolaton vertces and use smple lnear nterpolaton to assgn a mean curvature value H ~ +1 to every edge and nner vertex. Ths means the estmated values satsfy H ~ +1 = g h ( H ~ +1 ). In the second sub-step we then use these estmated values to determne +1 usng the formula H +1 = H ~ +1 wth the same lnearzaton method as n the drect approach. 4.5 Detals and remars Snce the vertces are updated along the surface normals, the drect teraton can be nterpreted as some nd of curvature ow, where the speed s determned by a local equaton system. As was ponted out by Desbrun et al. [3], ows dependng only on local surface propertes do not have to be numercally stable for large steps durng the teraton process. To allow larger
Dscrete Farng of Curves and Surfaces 379 Fg. 5. Left) Tetra Thng mesh. Rght) Soluton of the multgrd ndrect teraton after the Tetra Thng has been subdvded 5 tmes. The reecton lnes ndcate that the surface s quas G 2 contnuous. Iteraton tme: 4 seconds. update steps, Desbrun et al. used the bacward Euler method to develop an algorthm called mplct ntegraton for farng of arbtrary meshes based on Laplacan and on mean curvature ow. The dea behnd ths approach s to use global surface nformaton nstead of consderng the local neghborhood only. The ndrect approach follows that dea and explots global surface propertes ecently. In the surface case the ndrect approach shows ts true power. The estmaton sub-step s mostly smple lnear nterpolaton and usng the estmated curvature, we only need the 1-neghborhood of a vertex durng ts update process. The update process for nteror vertces s especally smple because of the equlateral hexagon as parameter doman. For edge and nterpolaton vertces, we calculated the least square matrces before the rst teraton step and cached the result. Our surface examples were created usng a mutgrd teraton scheme based on the ndrect approach. As mentoned earler, the constructon of our local parameterzaton s based on the technque presented n [4], where dvded derence operators to create dscrete thn plate splnes had to be derved. A comparson of our results showed that the deas presented n ths paper allow the constructon of consderably mproved meshes wthout ncreasng the computaton tme. The mproved qualty s due to the fact that our dscrete surfaces are based on geometrc nvarants nstead of second order partal dervatves, and thus the error made by guessng an sometrc local parameterzaton n advance has less nuence. Due to the fact that we estmated the local parameterzaton once, we do not get an exact G 2 contnuty at the edges and nterpolatng ponts, but as can be seen n Fgure 5, our surfaces are good approxmatons to G 2 surfaces. To acheve true G 2 contnuty, one would have to ncrease the eort for the handlng of edge and nterpolatng vertces. Future wor wll nvestgate f such eorts are worthwhle.
38 R. Schneder and L. Kobbelt x5. Concluson The qualty of the resultng curves and surfaces s superor to approaches based on quadratc functonals as the thn plate energy due to the more sophstcated approxmaton of geometrc nvarants. Although our objects are nonlnear splnes, they can be calculated fast enough to be applcable n nteractve desgn. The presented deas are optmally suted for the constructon of curves and surfaces wth pecewse lnear curvature dstrbuton, but could also be extended to hgher order dscrete farng approaches. References 1. Brunnett G., and J. Kefer, Interpolaton wth mnmal-energy splnes, Computer-Aded Desgn 26(2) (1994), 137{144. 2. do Carmo, M. P., Derental Geometry of Curves and Surfaces Prentce- Hall, Inc Englewood Cls, New Jersey, 1993. 3. Desbrun, M., M. Meyer, P. Schroder, and A. H. Barr, Implct farng of rregular meshes usng duson and curvature ow, SIGGRAPH 99 Conference Proceedngs, 317-324. 4. Kobbelt, L., Dscrete farng, Proceedngs of the Seventh IMA Conference on the Mathematcs of Surfaces '97, 11{131. 5. Kobbelt, L., S. Campagna, J. Vorsatz, and H. P. Sedel, Interactve multresoluton modelng on arbtrary meshes, SIGGRAPH 97 Conference Proceedngs, 15{114. 6. Kobbelt, L., A varatonal approach to subdvson, Comput. Aded Geom. Desgn 13 (1996), 743{761. 7. Malcolm M. A., On the computaton of nonlnear splne functons, SIAM J. Numer. Anal. 15 (1977), 254{282. 8. Mehlum, E., Nonlnear splnes, Computer Aded Geometrc Desgn, Academc Press, London, 173{27, 1974. 9. Mee, D. S., and D. J. Walton, A guded clothod splne, Comput. Aded Geom. Desgn 8 (1991), 163{174. 1. Moreton H. P., and C. H. Sequn, Functonal optmzaton for far surface desgn, SIGGRAPH 92 Conference Proceedngs, 167{176. 11. Taubn G., A sgnal processng approach to far surface desgn, SIG- GRAPH 95 Conference Proceedngs, 351{358. 12. Wemer, H., and J. Warren, Subdvson Schemes for Thn Plate Splnes, Computer Graphcs Forum 17 (1998), 33{314. 13. Welch, W., and A. Wtn, Free-Form shape desgn usng trangulated surfaces, SIGGRAPH 94 Conference Proceedngs, 247{256. Lef Kobbelt and Robert Schneder Max-Planc-Insttut fur Informat 66123 Saarbrucen, Germany fobbelt, schnederg@mp-sb.mpg.de