7 Interpolaton of the Irregular Curve Network of Shp Hull Form Usng Subdvson Surfaces Kyu-Yeul Lee, Doo-Yeoun Cho and Tae-Wan Km Seoul Natonal Unversty, kylee@snu.ac.kr,whendus@snu.ac.kr,taewan}@snu.ac.kr ABSTRACT One of common problems n free-form surface modelng s how to smoothly nterpolate a gven curve network. In shp desgn, shp hullform s usually desgned wth a curve network, and hullform surfaces are generated by fllng n (or nterpolatng) the curve network. Tensor-product surfaces such as NURBS and Bézer patches are typcal representatons to ths nterpolatng problem. However, tensor-product surface can represent the surfaces of rregular topologcal type by only parttonng the model nto a collecton of ndvdual rectangular patches or usng degenerated patches. Then, adjacent patches should be explctly sttched together and modfed to meet complcated geometrc contnuty constrants. Subdvson surface offers an alternatve. They are capable of modelng everywhere-smooth surfaces of arbtrary topologcal type usng a smple recursve subdvson rule as a whole surface. In ths study, we suggest a method of automatcally generatng hullform surface by nterpolatng the gven rregular curve network usng Catmull-Clark subdvson surfaces. Suggested method has an advantage over prevous works n that t does not requre any addtonal data except the gven curve network. There are three steps n our method; generatng ntal meshes automatcally, modfyng the ntal meshes for curve nterpolaton, and recursvely subdvdng the modfed meshes for smooth hullform subdvson surfaces. In ths paper, we concentrate on descrbng the frst step, automatc generaton of ntal meshes. In addton, we also show several examples of resultng ntal meshes from curve network data of actual shp hullform to justfy the proposed method. Keywords: Subdvson Surface, Curve Network, Interpolaton, Shp Hullform Surface.. INTRODUCTION Curve network s often used to desgn complcated freeform surfaces. Usng the curve network s easer than drectly manpulatng control ponts of surfaces, and t also enables more ntutve modelng. Especally, n shp desgn, shp hullform s usually desgned wth a curve network that conssts of several cross sectonal curves and characterstc curves. Because of the complcated shape of shp hullform, a curve network around the foreand after-body ncludes some rregular (.e. three, fve or hgher sdes) topologcal regons (Fg. ). Hull-surface then can be generated by fllng n (or nterpolatng) the curve network wth approprate surface patches. Tensorproduct surfaces such as NURBS and Bézer patches are typcal representatons to ths fllng problem. However, tensor-product surface can only represent the surfaces of rregular topologcal type by parttonng the model nto a collecton of ndvdual rectangular patches or usng degenerated patches. Then, adjacent patches should be explctly sttched together and modfed to meet complcated geometrc contnuty constrants such as C, C, G, or G condtons. In general, nterpolatng such rregular topologes usng tensor-product surfaces s not a trval problem [, ]. Fg.. Example of a rregular curve network for shp hullform desgn Subdvson surface, frst ntroduced by Doo-Sabn and Catmull-Clark, s an alternatve to tensor-product surfaces. They can easly generate smooth surfaces of arbtrary topologcal type by recursvely subdvdng
8 ntal polyhedral meshes wth C or C contnuty [3, 4]. But, boundary control and curve nterpolaton of subdvson surfaces are mportant problems to be solved. Recently, several researches on nterpolatng specfed curves usng subdvson surfaces are presented. Levn [5, 6] proposed Combned Subdvson Scheme whch modfes the subdvson rules around curves to be nterpolated, whle Nasr [7-] modfes the ntal polyhedral meshes for curve nterpolaton nstead of changng subdvson rules. However, n addton to user-specfed curves that should be nterpolated, both of them requre ntal polyhedral meshes as nput, whch are usually not avalable n ntal shp desgn stage n practce. We gve a method how to buld the ntal polyhedral meshes utlzng Coons patches [6]. We suggest a method that can automatcally generate hullform surfaces from gven rregular curve network usng Catmull-Clark subdvson surfaces wthout any addtonal nputs except gven curve network. Suggested method conssts of three steps. In the frst step, ntal polyhedral meshes are automatcally generated by usng pecewse blnearly blended Coons patches and regular N-sded control meshes [8]. Then, n the second step, the ntal meshes are modfed for ther lmtng subdvson surfaces to nterpolate gven curve network. In the last step, smooth subdvson surfaces are obtaned by recursvely subdvdng the modfed ntal meshes. In ths paper, we concentrate on descrbng the frst step, automatc generaton of ntal meshes (Secton 3). Several examples of ntal meshes generated from actual hullform curve network data are ncluded to justfy our proposed method n Secton 4. Fnally, concludng remarks and future researches are dscussed n Secton 5. whch s frequent n curve network especally for shp hullform desgn. And ntal polyhedral meshes whose lmtng surfaces nterpolate the gven curve network are also requred as nput, whch are not avalable n shp hullform desgn. Nasr [7-5] suggested algorthm whch generates face strps that converge to unform quadratc (or cubc) B- splne curve by Doo-Sabn (or Catmull Clark) subdvson rules. These face strps are named as polygonal complexes. Usng ths algorthm, Nasr proposed two approaches that generate smooth subdvson surfaces nterpolatng gven unform quadratc B-splne curves: complex approach [] and polygonal approach [0]. In complex approach as shown n Fg. (a), polygonal complexes of the gven nterpolated curves should frstly be provded by the user based on Nasr s algorthm. Next, the user has to complete the polyhedral meshes by fllng n some vertces nsde the regons enclosed by the polygonal complexes. So, the complex approach s not sutable for automatc surface generaton. On the other hand, polygonal approach as shown n Fg. (b) s to start wth an ntal polyhedral meshes and tagged control polygons whose vertces and edges are chosen from those of ntal meshes to defne the curves to be nterpolated. The basc dea s to construct polygonal complexes, one for each curve, by modfyng some panels of the ntal meshes or ts one tme unform subdvson dependng on whether the curve to be nterpolated s a boundary one or an nteror one. However, Nasr's method also needs ntal polyhedral meshes as gven data. Fg. shows the procedure of generatng subdvson surfaces nterpolatng gven unform quadratc B-splne curve usng complex approach and polygonal approach, respectvely.. BASIC IDEA Whle subdvson scheme can generate smooth lmtng surfaces of arbtrary topologcal types by recursvely applyng smple subdvson rules, t s dffcult to generate surfaces nterpolatng specfed curves, whch s actually an nverse problem. Related works on ths problem and basc dea of our proposed algorthm are summarzed n ths secton.. Related Works Levn [5] proposed combned subdvson scheme whch modfes subdvson rules for nterpolatng boundary curves. And Levn also extended hs scheme to nterpolate curve network that consst of nner curves and boundary curves [6]. Levn s scheme s smple to generate smooth surfaces wth arbtrary topologcal type that nterpolate networks of curves gven n any parametrc representaton, but t cannot handle extraordnary ponts where more than three curves meet, Fg.. Procedure of complex approach (a) and polygonal approach (b) n Doo-Sabn Subdvson settng. Basc dea of proposed algorthm In ths study, we suggest a method that can automatcally generate hullform surfaces from gven rregular curve
9 network by usng subdvson surfaces based on Nasr s polygonal approach. Whle the polygonal approach requres ntal polyhedral meshes that nclude control polygons of curves to be nterpolated, only curve network s avalable n shp hullform desgn. The basc dea to obtan ntal meshes s to use blnearly blended Coons patches that can be generated only wth 4 boundary curves. If boundary curves can be represented n B-splne or Bézer form, ther blnearly blended Coons patches becomes B-splne or Bézer surfaces. Snce the control meshes of the blnearly blended Coons patches ncludes control polygons of ther boundary curves (Fg. 3), these meshes can be used for ntal polyhedral meshes for polygonal approaches. The procedure and several ssues of constructng ntal polyhedral meshes are descrbed n the followng secton. polyhedral meshes for polygonal approaches (Fg. 4). Ths pecewse blnearly blended Coons patches have a drawback that they, n general, are C 0 contnuous. However, the control meshes of blnearly blended Coons patch are only used for ntal nput polyhedral meshes of polygonal approach and wll be modfed to nterpolate gven all curve network. So, the resultng subdvson surfaces are not affected by ths drawback. 0, 0) ), 3. AUTOMATIC GENERATION OF INITIAL SUBDIVISION MESHES 3. Control meshes of blnearly blended Coons patches Blnearly blended Coons patch s the smplest soluton to generate surfaces that nterpolate gven four boundary curves [6]: 0,, [ u u] + [ 0) ) ] v v 0, 0) 0, ) v [ ],, 0),) v where x ( 0), ), 0,,, are gven boundary curves. If all boundary curves are gven n B-splne or Bézer form,.e. by ther control polygons, and opposte boundary curves are defned over the same knot sequence and of the same degree, blnearly blended Coons patches are also represented n B-splne or Bézer form. And ts control meshes can be obtaned n a smple way: nterpretng the boundary polygons of gven boundary curves as pecewse lnear curves, the resultng Coons patch would then be the control meshes of a B- splne or Bézer surface that nterpolates the boundary curves [7]. Fg. 3 shows the control meshes of blnearly blended Coons patches that nterpolates gven four boundary curves. The control meshes keep the control polygons (bold lnes) of gven boundary curves as boundary polygons. So, f ths blnearly blended Coons patch scheme s appled to each regon bounded by nets of curve, we can get the control meshes of pecewse blnearly Coons patch that s a good canddate of ntal Fg. 3. Control mesh of blnearly blended Coons patch Fg. 4. Generaton of ntal meshes for polygonal approach usng blnearly blended Coons patches 3. Gap problem of the ntal polyhedral meshes generated by Coons patches To get the control meshes of blnearly blended Coons patches, opposte curves must have the common knot sequence (or same number of curve segments). If ther knot sequences are dfferent from each other, knot nserton can be appled. For example, n Fg. 5, all boundary curves are cubc and have two dfferent knot sequences: {0, 0, 0,,, } and {0, 0, 0, 0.5,,, }. Knot nserton operaton at parameter 0.5 makes the knot sequences and the number of control polygons of opposte boundary curves same. Then, we can get the control meshes by usng of blnearly blended Coons scheme. However, there s one problem to obtan the ntal polyhedral meshes for polygonal approach from
0 pecewse blnearly blended Coons patches. Fg. 6 shows ths stuaton. The control mesh of the left regon can be generated by blnearly blended Coons scheme wth four boundary curves that have the common knot sequence, {0, 0, 0,,, }. But, the rght regon needs knot nserton operaton for all boundary curves to have the common knot sequence, {0, 0, 0, 0.5,,, }. N 3 N 5 M M 3 M 4 0 0.5 0 0.5 0 0 0 0.5 0 0.5 0 0.5 Fg. 5. Coons patch nterpolatng boundary curves wth dfferent knot sequence 0 0.5 N 6 Fg. 7. Regular N-sded mesh topology (N s the number of sdes, M s the level of mesh) Regular N-sded control mesh topology and constructon algorthm are orgnally suggested to fll N-sded holes usng Doo-Sabn subdvson surfaces [8]. We extended ths algorthm for Catmull-Clark subdvson surfaces. Constructon algorthm for regular N-sded control mesh s as follows: Gap Fg. 6. Gap problem n generaton of ntal meshes by usng of blnearly blended Coons patches Consequently, the gaps between the left and rght control meshes may occur. Havng ths gap, the control mesh can not be used as ntal nput for the polygonal approach. One soluton to ths problem s mesh sttchng. However, sttchng may create ugly lookng surface and remove 4-sded faces, along the boundary curves, that are necessary to generate Catmull-Clark polygonal complexes for curve nterpolaton n the mesh modfcaton step. So, we solve ths gap problem by approxmatng all gven curve network wth constant number of segments. Ths approxmaton guarantees that ntal mesh can be generated wthout any gap. 3.3 Intal meshes for rregular topologcal regons Intal meshes for rectangular regons can be obtaned by blnearly blended Coons scheme mentoned above. To complete the ntal polyhedral meshes for polygonal approach, we need another algorthm to automatcally generate ntal meshes for non-rectangular regons (.e., trangular regon, pentagonal regon, and so on). For these rregular N-sded regons (N 4), we use regular N- sded control mesh topology (Fg. 7). Step : Generaton of level M() mesh (fg. 8) All boundary curves are approxmated to unform cubc B-splne curves wth (M-) segments. Boundary ponts(black crcle) of the level M mesh are set to control ponts of approxmated boundary curves. N corner twst vectors are estmated by usng of Adn s twst algorthm descrbed n secton 3.3.. N nner-corner ponts(whte crcle) of the level M mesh are determned usng estmated corner twst vectors. Step : Generaton of level M+ mesh (fg. 9) All boundary curves are approxmated to unform cubc B-splne curves wth (M-) segments. Generate ntal level M+ meshes by applyng one step subdvson to the level M mesh. Boundary ponts of the level M+ mesh are set to control ponts of approxmated boundary curves. Inner-boundary ponts of the level M+ are determned usng estmated surface normal and cross boundary dervatves (secton 3.3.). All the others ponts od the level M+ are unchanged. If (M-) s less than user-specfed constant number of segments for curve approxmaton descrbed n secton 3., repeat Step.
Around a corner of regular N-sded control meshes, the adjacent rectangular polyhedron can be consdered as Bézer control mesh. So, f corner twst vectors are estmated, nner-corner ponts of regular N-sded control mesh can be determned. Orgnal cubc non-unform curve network Approxmated curve network (Cubc unform B-splne curves wth segments) Estmaton of corner twst vectors For corner twst vector estmaton, we use well-known Adn s twst vector estmaton method. Adn s method deals wth only 4-sded regons, and requres 4 surface corner ponts and 4 tangent vectors of boundary curves (Fg ). Fg. 8. Constructon of regular 5-sded control mesh of level 0, [ ] v + 0), u v ) 0,) [ ],0),) ) u 0,) twst vector v ) ) xv(,0) xv () + xu (0,) xu () u ( ),0) 0,) +,) ),0) Fg.. Geometrc meanng of Adn s twst vectors,) Orgnal cubc non-unform curve network subdvson Approxmated curve network (Cubc unform B-splne curves wth segments) Estmaton of corner twst vectors Fg. 9. Constructon of regular 5-sded control mesh of level 3 3.3. Estmaton of corner twst vectors In control mesh of bcubc Bézer surface, geometrc meanng of corner twst vector s a dfference vector between an nner-corner pont and parallelogram generated by corner pont and two adjacent boundary ponts (Fg. 0). To apply Adn s twst estmaton method to nonrectangular regons, we determned the mdpont of opposte boundary curve as opposte corner pont, and estmated 4 tangent vectors by smple geometrc operatons such as pont-plane projecton, lnear nterpolaton, and so on (Fg., 3). After estmaton of corner twst vectors, we can determne nner-corner ponts of regular N-sded mesh by usng of the geometrc meanng of corner twst vectors. B D A C Adn s Twst Vector at A + -(A B C + D) Fg.. Adn s twst vector estmaton for 3-sded regon Twst( b( b( 3 3 j 0 0, b() 3 3Δ b, Δ b B ( u) B (, 9 (( Δ j 0, 0, 9 ( Δ b Δ b ),0 j b -Δ, b ) -( Δ,0 b -Δ b 0, )) Fg. 0. Geometrc meanng of corner twst vectors Fg. 3. Adn s twst vector estmaton for 5-sded regon
3.3. Determnaton of nner-boundary ponts In N-sded control mesh of the level M( 3), nnerboundary ponts (whte rectangle n Fg. 4) are also determned for reasonable nner shape. At each sde of the N-sded regon, frst, surface normal vectors (N ) along boundary curves are estmated by lnearly nterpolatng both end surface normal vectors (N 0 and N 4 ) at the corner, and the cross boundary dervatve vectors (CBD(0.5)) should be perpendcular to estmated surface normal vectors (N ). Snce regular N- sded control meshes can be consdered as control meshes of bcubc B-splne surface around the boundary curves, the cross boundary dervatve vectors can be derved from nner-boundary ponts. Then we can determne nner-boundary ponts by smple lnear equatons. N 0 hullform s usually desgned wth non-unform cubc B- splne curves, approxmaton of them to unform cubc form s an ndspensable process. For avodng gap problems n ntal meshes, there s another lmtaton of constant number of segments n curve approxmaton process. Fg. 5 shows the curve network of actual shp hullform data, and the ntal meshes generated by proposed algorthm. In general, the lmtng subdvson surfaces from ths ntal meshes does not nterpolate the gven curves network. Ths s why mesh modfcaton process s needed. Mesh modfcaton s to embeddng Catmull- Clark polygonal complexes that converge gven curve network to the resultng ntal meshes. Ths mesh modfcaton algorthm for Catmull-Clark subdvson settng s almost developed. Snce we are preparng ths algorthm as another paper, ths algorthm was excluded n ths paper. CBD( u) 4 0 4 0 3 ( Q P ) N ( u) 3 V N ( u) P P 0 P V 0 V V Q 0 Q Q T N curve network CBD(0.5) V 3 P 3 Q 3 V 4 P 4 N 4 Q 4 0 / Intal meshes CBD( ) V + V+ V3 4 4 4 T N CBD( ) T N ( Q P) + ( Q P ) + ( Q P) where λ λ, V + 0 V 4 4 3 3 Resultng surfaces wthout mesh modfcaton Fg. 4. Determnaton of nner-boundary ponts usng cross boundary dervatve vectors 4. RESULT OF INITIAL MESHES Algorthm proposed n ths paper s bascally based on Nasr s polygonal approaches except ntal mesh generaton usng pecewse blnearly Coons patches and regular N-sded control meshes. Nasr s approach has restrcton that curves of network should be unform cubc B-splne curves for Catmull-Clark settng, and ths also s the same lmtaton n our algorthm. Snce shp Fg. 5. Gven curves network of shp hullform surfaces, resultng ntal meshes and subdvson surfaces wthout mesh modfcaton 6. CONCLUSIONS AND FUTURE WORK In ths paper, we suggested a method that can automatcally generate hullform surfaces from gven rregular curve network by usng subdvson surfaces
3 wthout any addtonal nputs except gven curve network. The ntal polyhedral meshes could be automatcally generated from the control meshes of pecewse blnearly blended Coons patches, and regular N-sded control meshes. Snce curve network used for hullform surface s usually desgned based on cubc nonunform B-splne curves, curve approxmaton operaton was needed. In order to generate shp hullform surfaces exactly nterpolatng the gven curve network, researches on algorthm whch makes t possble to drectly nterpolate non-unform cubc B-splne curves wthout curve approxmaton by unform cubc subdvson surfaces can be consdered. 7. ACKNOWLEDGEMENTS We apprecate Prof. Ahmad Nasr and Dr. Thomas Hermann for ther valuable dscusson on the problem of G and n-sded surface nterpolaton. Ths research has been accomplshed by the partal support of the Korean Scence Foundaton Objectve Bass Research. 8. REFERENCES [] Pegl, L. and Tller, W., Fllng n-sded regons wth NURBS patches, The Vsual Computer, Vol. 5, No., pp. 77-89, 999. [] Rhm, J.H. and Lee, K.Y., Remeshng algorthm of curve network of shp hullform and Interpolatng hullform surfaces wth Gregory patches n Proceedngs of the Socety of CAD/CAM Engneers, pp. 37-334, 997. (Wrtten n Korean) [3] Catmull, E. and Clark, J., Recursvely generated B- splne surfaces on arbtrary topologcal meshes, Computer-Aded Desgn, Vol. 0, No. 6, pp. 350-355, 978. [4] Doo, D. and Sabn, M., Behavor of recursve dvson surfaces near extraordnary ponts, Computer-Aded Desgn, Vol. 0, No. 6, pp. 356-360, 978. [5] Levn, A., Combned subdvson schemes for the desgn of surfaces satsfyng boundary condtons, Computer Aded Geometrc Desgn, Vol. 6, No. 5, pp. 345-354, 999. [6] Levn, A. Interpolatng nets of curves by smooth subdvson surfaces n Proc. of SIGGRAPH 999, pp. 57-64, 999. [7] Nasr, A., Polyhedral subdvson methods for free form surfaces, ACM Transactons on Graphcs, Vol. 8, No., pp. 89-96, 987. [8] Nasr, A., Curve nterpolaton n recursvely generated surfaces over arbtrary topology, Computer Aded Geometrc Desgn, Vol. 4, No., pp. 3-30, 997. [9] Nasr, A., Recursve subdvson of polygonal complexes and ts applcatons n CAGD, Computer Aded Geometrc Desgn, Vol. 7, No. 7, pp. -5, 000. [0] Nasr, A., A polygonal approach for nterpolatng meshes of curves by subdvson surfaces n Proc. Geometrc Modelng and Processng, pp. 6-73, 000. [] Nasr, A., Constructng polygonal complexes wth shape handles for curve nterpolaton by subdvson surfaces, Computer-Aded Desgn, Vol. 33, No., pp.753-765, 00. [] Nasr, A., and Abbas, A., Desgnng Catmull-Clark Subdvson Surfaces wth Curve nterpolaton Constrants, Computer & Graphcs, Vol. 6, No. 3, pp. 393-400, 00. [3] Nasr, A., and Abbas, A., Lofted Catmull-Clark Subdvson Surfaces n Proc. of Geometrc Modelng Processng Internatonal 00, pp. 83-93, 00. [4] Abbas, A., and Nasr, A., Interpolatng Multple Intersectng Curves Usng Catmull-Clark Subdvson Surfaces, To appear n Proc. of the Internatonal Conference on Computer Aded Desgn 004, Thaland, May 004. [5] Nasr, A., Interpolatng an unlmted number of curves through an extraordnary pont on subdvson Surfaces, Computer Graphcs Forum, Vol., Vo., pp. 87-97, 003. [6] Coons, S., Surfaces for computer aded desgn of space forms, Techncal report, MIT, 967. Project MAC-TR 4. [7] Farn, G., Commutatvty of Coons and tensor product operators, Rocky Mountan Journal of Mathematcs, Vol., No., pp. 54-546, 99. [8] Hwang, W.C., and Chuang, J.H., N-Sded Hole Fllng and Vertex Blendng Usng Subdvson Surfaces, Journal of Informaton Scence and Engneerng, Vol. 9, No. 5, pp.857-879, 003. [9] Habb, A. and Warren, J., Edge and vertex nserton for a class of C subdvson surfaces, Computer Aded Geometrc Desgn, Vol. 6, No. 4, pp. 3-47, 999. [0] Sederberg, T.W., Zheng, J., Sewel, D., and Sabn, M., Non-unform recursve subdvson surfaces n Proc. of the 5th conference on Computer graphcs and nteractve technques, pp. 387-394, 998. [] Warren, J., and Henrk, W., Subdvson Methods for Geometrc Desgn: A Constructve Approach, Morgan Kaufmann Publshers, 00.
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