Geodesic, Flow Front and Voronoi Diagram

Transcription

1 11 Geodesic, Flow Fron and Voronoi Diagram C. K. Au Nannag Technological Uniersi, ABSTRACT Geodesics and flow frons are orhogonal o each oher. These wo ses consiue he space ime funcion of a source and pla an imporan role in dealing wih wo pes of problem: iniial alue problem and final alue problem. This paper reeals he compuaion of he flow frons from a poin source in a wo dimensional domain wih a gien eloci field. A space ime funcion is esablished for he poin source. The space ime funcions of he higher dimension sources such as line source, cure source and polgonal source can be deried geomericall based on ha of a poin source. Co-operaion and compeiion beween wo sources in a wo dimensional domain and heir relaionship wih Voronoi diagram is presened. 1. INTRODUCTION Orhogonal o he direcion of he flow is he so-called flow fron hese are erms come from fluids dnamics. Indeed, he ranspor of maer or energ is fundamenal o hermal and fluid sudies. Bu flow and fron are no unique o hese areas. An orhogonal se such as he normal and he angen also arises in opics. There, he pair is called ra and wae fron, respeciel. Acuall, he roos of he orhogonal pair dae back seeral hundred ears. Perhaps aking a hin from Ferma, Bernoulli blended mechanics wih opics in he celebraed formulaion of he Brachisochrone (or leas ime) problem. As science maures, he orhogonal se of he shores pah and he Hamilon- Jacobi characerisic surface, arising from heoreical mechanics, are now firml embraced b opimizaion, as well as b conrol heor. As geomer is relaed o mechanics, i should no be surprising ha here are counerpars in differenial geomer; he are called: geodesic and isosurface. Table 1 summarizes he seemingl disparae fields and he erms; hose in ialics are adoped in his paper. Thermal/fluids Opics mechanics/opimizaion/conrol geomer flow Ra shores pah geodesic flow fron wae fron (or eikonal) Hamilon-Jacobi surface Iso-surface (or iso-cure) Tab. 1: Orhogonal Ses. The mahemaical deails and properies of he geodesics can be found in he reference 1. Mos of he researches in geodesics concenrae on arious mehods o compue he cure for a gien surface which is eiher analical, parameric 3 or discree 4-8. Oher generaes a surface pencil from a gien geodesic 9. The geomeric aspecs of a geodesic are eplored in hese works. This aricle addresses he compuaion of a geodesic from he iew of is phsical meaning. I begins b saing he problem ha he calculus of ariaions soles: find a pah beween wo poins, which minimizes he ime funcion, in a domain wih a gien eloci field. Using wo eloci fields, = k and = k (where k is a consan). Based on he geodesics, he flow frons from a poin source can be deried. Two phenomena, cooperaion and compeiion of haing more han one source in a domain are discussed. The filling paern predicion in plasic injecion moulded par is discussed as an applicaion eample.. GEODESICS, POINT SOURCE AND FLOW FRONT A geodesic is he minimal pah beween wo poins. The spaial coordinaes will be denoed b (, Gien a pah paramerized in u, wih componens (u) and (u), is line elemen useful, he conersion facor beween he parameer u and he arc lengh s is ( ds ) = ( d) + ( d is inarian. As i will be u ds= ( ) + ( ) du where u d u = du Compuer-Aided Design & Applicaions, Vol. 4, Nos. 1-4, 007, pp 11-18

2 1 d ds and u =. The ime o rael along a pah γ wih eloci is gien as = du. The calculus of ariaion allows γθ one obain a differenial equaion for describing he pah ha minimizes he ime of rael beween wo poins which is a geodesic in he form of η d d = η (1.1) ds ds η d d = η ds ds 1 where η =. Since he lef hand side of equaion (1.1) and (1.) inoles he Euclidean coordinaes and is righ hand side is he arc lengh, i ma be referred o as he spaial form of a geodesic. The geomer of a geodesic beween wo poins depends upon he eloci field wihin a domain. A poin source j in a domain D is characerized b wo parameers: emission eloci (1.) j and he insan i sars emission wih respec o a reference ime = 0. Based on he principle of leas acion, he pah γ θ from source j a he insan o a poin q a ime in a domain D wih eloci field is a geodesic goerning b he equaion = + ds. γθ The flow fron from an emied poin source j is he fron line of he emission. I is a opological circle in he domain D. An adancing flow fron is a funcion of ime. The flow fron possesses he geomerical proper ha i mus be orhogonal o he geodesic from he source, hence, he equaion of he flow fron a ime is gien b ds j + = 0, θ [ 0,π ] γ θ () For insance, in a domain D wih a consan eloci field, he geodesic from a poin, ) akes on he form of a ( 0 0 sinθ sraigh line. Le he locaion of a source j be (0,0). The geodesic from j o q is = where is he eloci. cosθ + Therefore, he flow fron is epressed as = 0 (wih = 0 ), which is a famil of concenric circles wih ariable. Figure 1 shows he geodesics and he flow frons from a source j wih consan emission eloci. flow frons j geodesics Fig. 1: Flow frons of a poin source j wih consan eloci. Compuer-Aided Design & Applicaions, Vol. 4, Nos. 1-4, 007, pp 11-18

3 13 3. SPACE TIME FUNCTION OF A SOURCE A space ime describes he posiion in domain D wih a eloci field a a specific ime insan. The equaion of flow fron, Ψ(, = 0 (3) wih ds Ψ(, = + gies he relaionship beween a wo dimensional space (in erms of, ) and ime. Hence, j γθ he space ime funcion of a poin source j a 0, ) in a eloci field + + ( k 0 ) = 0 ( 0, where is he ime dela of source j. = k (where k is a consan) is Figure shows space ime funcion of a poin source j a ( 0, 0 ) in a consan eloci field wih a ime dela. The black cures link all he eens happen simulaneousl while he gre lines are he world lines. Projecing hese black and gre cures ono he space ( plane) gies he flow frons and he geodesics in he domain D. - iew 3D iew - iew - iew Fig. : Space ime funcion of a poin source in a consan eloci field. When here ei more han one source in a domain, wo possible phenomena arise: 1. hese agens co-operae o acquire erriories and;. hese agens counerac o compee for erriories. Le = Ψ (, and = Ψ (, be he space ime funcions for wo sources j and k in a domain D. The resulan k space ime funcion is: = min[ Ψ j (,, Ψ (, ], (, D (4) k If hese agens co-operae, hen he resulan flow fron a ime is gien b he Boolean union of he funcion Ψ (, Ψ (,. j k For insance, a cure source is a pical eample of co-operaion. A cure source c in a domain D can be considered as an aggregaion of infinie number of poin sources along he cure. All hese poin sources co-operae o acquire erriories. The flow fron of he cure source a ime is j Ψ j (,, j c, (, D. Hence, he space ime Compuer-Aided Design & Applicaions, Vol. 4, Nos. 1-4, 007, pp 11-18

4 14 funcion of a cure source is he enelope obained b sweeping he space ime funcion of a poin source wih ime delas. Figure 3(a) and 3(b) show he space ime funcions of a line and a cure source wih arious ime dela in a domain wih a consan eloci field respeciel. Similarl, he space ime funcion of a polgonal source in a domain wih consan eloci field is shown in figure 3(c). (a) line source wih ariable ime dela in a consan eloci field (b) cure source wih consan ime dela in a consan eloci field (c) polgonal source wih consan ime dela in a consan eloci field Fig. 3: Various cure sources. If he agens counerac and compee for erriories, a fronierµ eiss o pariion he domain. This fronier is he algebraic inersecion of heir space ime funcions. Therefore, µ (,, ) = (,, ) =Ψ (, = Ψ (, ) (5) { ( ) ( )} j, k j k Voronoi diagram in compuaional geomer demonsraes his phenomenon. 4. VORONOI DIAGRAM A Voronoi diagram pariions he space based on a se of gien sies according o a membership funcion. A disance funcion measuring he Euclidean disance is commonl emploed. Such a disance funcion assumes a consan eloci field on he Euclidean space. Considering each sies as a source, he flow fron propagaes along he geodesic from he sources. A Voronoi diagram can be generaed b projecing he space ime funcion on he space as shown in figure 4. Compuer-Aided Design & Applicaions, Vol. 4, Nos. 1-4, 007, pp 11-18

5 15 Ψ 1 Ψ3 Ψ4 Ψ5 µ 3,4 Sie Characerisics (, j ) 1 (1.5,1.73) (10,0.68) 3 (10,1.73) 4 (10,1.73) 5 (.5,0.577) 6 (0,0.68) Fig. 4: A Voronoi diagram. Hence, i can be seen ha he flow fron is he boundar of he growing space ime funcion while he fronier is he inernal edges. This informaion is obained b modeling he space ime funcion as solid geomer and performing he Boolean operaion union. An algorihm for consrucing a Voronoi diagram wih aries sie characerisics based on solid modeling is lised in figure 5: creae a solid geomer Κ 0 based on 0 and 0 a q 0 while (no done) do begin creae a solid geomer Κ 0 Κ 0 Κ j end eracing he edges in K 0 projecing he edges ono he domain Κ j based on and j a q j Fig. 5: An algorihm for consrucing he Voronoi diagram. Figure 6 shows a Voronoi diagram for sies wih arious dimensionaliies. Sie 1 and are poin sies (source), sie 3 is a line sie (source) while sie 4 is a polgonal sie (source). Their space ime funcions are depiced in figure 3. These sies possess differen sie characerisics. 3 Sies Sie characerisics (, ) j j 1 (3.5, 1.73) 4 (1.5, 0.7) 3 (.50, 1.73) 1 4 (.50, 1.00) Fig. 6: A Voronoi diagram for sies wih arious dimensionaliies. Compuer-Aided Design & Applicaions, Vol. 4, Nos. 1-4, 007, pp 11-18

6 16 Howeer, hese agens usuall co-operae o acquire erriories and counerac inernall simulaneousl. Hence, he combinaion of he wo phenomena usuall arises. The filling of he injecion moulding is an eample o illusrae his siuaion. The mel flow in he cai is spli due o he changes in flow eloci. These spli flows are co-operaing o earn he errior (o fill up he cai and compeing indiiduall so ha weld lines eis. D 1 gae space ime funcion edge e (a) a bo D unfolded model weld line (b) space ime funcion edge e (c) flow frons folded model (d) weld lines Fig. 7: Filling simulaion of a molding. unfolded model Figure 7(a) shows a moulding wih wo differen wall hicknesses. The hickness of domain D 1 is greaer han ha of D. Mel flows ino he cai from he gae (a poin source) wih consan eloci. For simplici, he flow eloci is assumed o be direcl proporional o he wall hickness. The edge e is common o boh domains D 1 and D and he eloci in D 1 is faser. Hence, e is a line source in domain D. The space ime funcion is shown in figure 7(b) wih he model unfolded. Secioning his funcion and projecing ono he unfolded model ields he flow frons as depiced in figure 7(c). A disconinui in he space ime funcion indicaes a weld line. Figure 7(d) gies he flow frons and weld lines for boh folded and unfolded model. Figure 8(a) gies a flow fron paern generaed b finie elemen mehod wih deail calculaion of he flow eloci (insead of assuming i as a consan) while he flow fron paern in figure 8(b) of he same molding is obained b he kineic source approach. I can be seen ha boh paerns are basicall agreeable, paricularl he locaion of he weld line. Compuer-Aided Design & Applicaions, Vol. 4, Nos. 1-4, 007, pp 11-18

7 17 (a) flow frons generaed b finie elemen mehod (b) flow frons generaed b kineic source approach Fig. 8: Flow frons. 5. CONCLUSION From Minkowski o Einsein 1, space and ime are no separaed. Hence, a minimal pah in ime beween wo poins in he space is he geodesic beween wo eens in he space ime. The space ime funcion plas an imporan role in soling wo pes of problem: iniial alue problem and boundar alue problem. The iniial alue problem refers o he compuaion of he geodesic in space ime for a gien iniial condiions: iniial posiion, eloci and inciden angle while he boundar alue problem compues he geodesic beween wo eens in he space ime. Geodesics, flow frons and space ime funcions are discussed in his aricle. Based on he space ime funcion of a poin source, space ime funcions of he higher dimension source such as cure and polgonal source can be obained. Hence, he flow frons of a cure source and a polgonal source is generaed b slicing he space ime funcion and projecing on he domain. While geodesics beween wo poins in he domain are deried since geodesics and flow frons are orhogonal se. The space ime funcions of muliple sources describe he hree phenomena: co-operaion beween wo agens, compeiion beween wo agens, and he combinaion of hese wo phenomena. An approach for spaial pariioning and consrucing filling paerns for laminar flow simulaion wih some known eloci fields are demonsraed. 6. REFERENCES [1] Do Carmo, M. P.: Differenial Geomer of Cures and Surfaces. Prenice-Hall, Englewood Cliffs, NJ [] Parikalakis, N. M.; Badris, L.: Offses of Cures on Raional B-spline Surfaces, Engineering wih Compuers 5, 1989, [3] Spiak, M.: A Comprehensie Inroducion o Differenial Geomer. nd ed. Houson [4] Kimmel, R.; Sehian J. A.: Compuing Geodesic Pahs on Manifolds, Proceedings of Naional Academ of Sciences, 95(15), 1998, [5] Kimmel, R.; Kirai, N.; Brucksein, A. M.: Muli-alued Disance Maps in Finding Shores Pahs Beween Moing Obsacles, IEEE Trans. Robo Auomaion, 14(3), 1998, [6] Nooni, M.; Klein, R.: Compuing Geodesic Pahs on Triangular Meshes, Proceedings of The 10-h Inernaional Conference in Cenral Europe on Compuer Graphics, Visualizaion and Compuer Vision'00 (WSCG'00). 00. [7] Polhier, K.; Schmies, M.: Sraighes geodesics on polgonal surfaces. In: Hege, H. C.; Polhier, K. ediors. Mahemaical Visualizaion, Berlin: Springer [8] Rai Kumar, G. V. V.; Sriniasan P.; Dearaja Holla, V.; Shasr, K. G.: Geodesic cure compuaions on surfaces, Compuer Aided Geomeric Design, 0(), 003, [9] Wang, G.; Tang, K.; Tai, C.: Parameric represenaion of a surface pencil wih a common spaial geodesic, Compuer-Aided Design, 36(5), 004, [10] Sahl, S.: The Poincare half plane, Jones and Barle Publishers, Compuer-Aided Design & Applicaions, Vol. 4, Nos. 1-4, 007, pp 11-18

8 18 [11] Okabe, A.; Boos B.; Sugihara, K.: Spaial Tessellaions: Conceps and Applicaions of Voronoi Diagrams, John Wile & Sons, New York, 199. [1] hp://phsics.sr.edu/courses/modules/lightcone/minkowski.hml Compuer-Aided Design & Applicaions, Vol. 4, Nos. 1-4, 007, pp 11-18