An approach for designing a surface pencil through a given geodesic curve

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1 An approach for deigning a urface pencil hrough a given geodeic curve Gülnur SAFFAK ATALAY, Fama GÜLER, Ergin BAYRAM *, Emin KASAP Ondokuz Mayı Univeriy, Faculy of Ar and Science, Mahemaic Deparmen [email protected], [email protected],[email protected], [email protected] ABSTRACT Surface and curve play an imporan role in geomeric deign. In recen year, problem of finding a urface paing hrough a given curve have araced much inere. In he preen paper, we propoe a new mehod o conruc a urface inerpolaing a given curve a he geodeic curve of i. Alo, we analyze he condiion when he reuling urface i a ruled urface. In addiion, developabliy along he common geodeic of he member of urface family are dicued. Finally, we illurae hi mehod by preening ome example. 1. Inroducion A roaion minimizing adaped frame (RMF) {T,U,V} of a pace curve conain he curve angen T and he normal plane vecor U, V which how no inananeou roaion abou T. Becaue of heir minimum wi RMF are very inereing in compuer graphic, including free-form deformaion wih curve conrain [1-6], weep urface modeling [7-10], modeling of generalized cylinder and ree branche [11-15], viualizaion of reamline and ube [15-17], imulaion of rope and ring [18], and moion deign and conrol [19]. There are infiniely many adaped frame on a given pace curve [20]. One can produce oher adaped frame from an exiing one by conrolling he orienaion of he frame vecor U and V in he normal plane of he curve. In differenial geomery he mo familiar adaped frame i Frene frame {T, N, B}, where T i he curve angen, N i he principal normal vecor and B T N i he binormal vecor (ee [21] for deail). Beide of i fame, he Frene frame i no a RMF and i i unuiable fopecifying he orienaion of a rigid body along a given curve in applicaion uch a moion planning, animaion, geomeric deign, and roboic, ince i incur unneceary roaion of he body [22]. Furhermore, Frene frame i undefined if he curvaure vanihe. One of mo ignifican curve on a urface i geodeic curve. Geodeic are imporan in he relaiviic decripion of graviy. Einein principle of equivalence ell u ha geodeic repreen he pah of freely falling paricle in a given pace. (Freely falling in hi conex mean moving only under he influence of graviy, wih no oher force involved). The geodeic principle ae ha he free rajecorie are he geodeic of pace. I play a very imporan role in a geomeric-relaiviy heory, ince i mean ha he fundamenal equaion of dynamic i compleely deermined by he geomery of pace, and herefore ha no o be e a an independen equaion. Moreover, in uch a heory he acion idenifie (up o a conan) wih he fundamenal lengh invarian, o ha he aionary acion principle and he geodeic principle become idenical. The concep of geodeic alo find i place in variou indurial applicaion, uch a en manufacuring, cuing and paining pah, fibergla ape winding in pipe manufacuring, exile manufacuring [23 28]. In archiecure, ome pecial curve have nice properie in erm of rucural funcionaliy and manufacuring co. One example i planar curve in verical plane, whichcan be ued a uppor elemen. Anoher example i geodeic curve, [29]. Deng, B., decribed mehod o creae paern of pecial curve on urface, which find applicaion in deign and realizaion

2 of freeform archiecure. He preened an evoluion approach o generae a erie of curve which are eiher geodeic or piecewie geodeic, aring from a given ource curve on a urface. In [29], he inveigaed familie of pecial curve (uch a geodeic) on freeform urface, and propoe compuaional ool o creae uch familie. Alo, he inveigaed paern of pecial curve on urface, which find applicaion in deign and realizaion of freeform archiecural hape (for deail, ee [29] ). Mo people have heard he phrae; a raigh line i he hore diance beween wo poin. Bu in differenial geomery, hey ay hi ame hing in a differen language. They ay inead geodeic for he Euclidean meric are raigh line. A geodeic i a curve ha repreen he exremal value of a diance funcion in ome pace. In he Euclidean pace, exremal mean 'minimal',o geodeic are pah of minimal arc lengh. In general relaiviy, geodeic generalize he noion of "raigh line" o curved pace ime. Thi concep i baed on he mahemaical concep of a geodeic. Imporanly, he world line of a paricle free from all exernal force i a paricular ype of geodeic. In oher word, a freely moving paricle alway move along a geodeic. Geodeic are curve along which geodeic curvaure vanihe. Thi i of coure where he geodeic curvaure ha i name from. In recen year, fundamenal reearch ha focued on he revere problem or backward analyi: given a 3D curve, how can we characerize hoe urface ha poe hi curve a a pecial curve, raher han finding and claifying curve on analyical curved urface. The concep of family of urface having a given characeriic curve wa fir inroduced by Wang e.al. [28] in Euclidean 3-pace. Kaap e.al. [30] generalized he work of Wang by inroducing new ype of marching-cale funcion, coefficien of he Frene frame appearing in he parameric repreenaion of urface. Alo, urface wih common geodeic in Minkowki 3-pace have been he ubjec of many udie. In [31] Kaap and Akyıldız defined urface wih a common geodeic in Minkowki 3-pace and gave he ufficien condiion on marching-cale funcion o ha he given curve i a common geodeic on ha urface. Şaffak and Kaap [32] udied family of urface wih a common null geodeic. Lie e al. [33] derived he neceary and ufficien condiion for a given curve o be he line of curvaure on a urface. Bayram e al. [34] udied parameric urface which poe a given curve a a common aympoic. However, hey olved he problem uing Frene frame of he given curve. In hi paper, we obain he neceary and ufficien condiion for a given curve o be boh ioparameric and geodeic on a parameric urface depending on he RMF. Furhermore, we how ha here exi ruled urface poeing a given curve a a common geodeic curve and preen a crieria for hee ruled urface o be developable one. We only udy curve wih an arc lengh parameer becaue uch a udy i eay o follow; if neceary, one can obain imilar reul for arbirarily parameeried regular curve. 2. Background A parameric curve 1 2, L L, ha a conan or -parameer value. In hi paper, repec o arc lengh paramee and we aume ha i a curve on a urface r r For every poin of r, if r 0, he e,, along r, where T r, P P, in 3 ha denoe he derivaive of r wih i a regular curve, i.e. r 0 T N B i called he Frene frame / and N B T N. are he uni angen, principal normal, and binormal vecor of he curve a he poin, repecively. Derivaive formula of he Frene frame i governed by he relaion

3 where r and he curve T 0 0 T d N 0 N d 0 0 B B, repecively [35].,, rr r (2.1) are called he curvaure and orion of Anoher ueful frame along a curve i roaion minimizing frame. They are ueful in animaion, moion planning, wep urface conrucion and relaed applicaion where he Frene frame may prove unuiable or undefined. A frame among he frame on he curve around he angen vecor T.,, T U V i called roaion minimizing if i i he frame of minimum wi,, T U V i an RMF if, U U r V V r r where denoe he andard inner produc in [36]. Oberve ha uch a pair U and V i no unique; here exi a one parameer family of RMF correponding o differen e of iniial value of U and V. According o Bihop [20], a frame i an RMF if and only if each of and i parallel o. Equivalenly, U ' V ' T 3 U ' V 0, V ' U 0, (2.2) i he neceary and ufficien condiion for he frame o be roaion minimizing [37]. There i a relaion beween he Frene frame (if he Frene frame i defined) and RMF, ha i, U and V are he roaion of N and B of he curve in he normal plane. Then, where U co in N V in co B i he angle beween he vecor N and U (ee Fig. 2), [38]., (2.3) Fig. 2 The Frene frame (T(), N(), B()) and he vecor U(), V(). Eqn. (2.3) implie ha,, T U V i an RMF if i aifie he following relaion U ' co T, V ' in T, '. (2.4)

4 Noe ha,, T U V where he Frene frame i undefined. i defined along he curve even if he curvaure vanihe 3. Surface pencil wih an geodeic curve Suppoe we are given a 3-dimenional parameric curve i he arc lengh (regular and form a where 1 Surface pencil ha inerpolae, L L 1 2,,,, P a T b U c V a,, b, and c, a,, b, and c, are 1 C ). 1 2, L L, in which a a common curve i given in he parameric, L L, T T , (3.1) funcion. The value of he marching-cale funcion indicae, repecively, he exenion-like, flexion-like, and reorion-like effec caued by he poin uni hrough ime, aring from r. Remark 3.1 : Oberve ha chooing differen marching-cale funcion yield differen urface poeing a a common curve. Our goal i o find he neceary and ufficien condiion for which he curve ioparameric and geodeic on he urface on he urface parameer P, Secondly he curve 0 T, T 1 2 P,, here exi a parameer 0. Firly, a T, T 1 2 uch ha a, b, c, 0, L L, T T uch ha i an ioparameric curve r i an geodeic curve on he urface P, According o he geodeic heory [39], geodeic curvaure along geodeic. Thu, if we ge: where n, The normal vecor of 0 n(, ) 0 i. (3.2) k de r ', r '', n g here exi a vanihe // N(). (3.3) i he urface normal along he curve P P, can be wrien a n, r P, P, and N i a normal vecor of From Eqn. (2.1) and (2.3), he normal vecor can be expreed a c(, ) b(, ) b(, ) c(, ) n, a(, ) ( )co ( ) a(, ) ( )in ( ) T( ) Thu,. r a(, ) c(, ) c(, ) a(, ) a(, ) ( )in ( ) 1 b(, ) ( )co ( ) c(, ) ( )in ( ) U() b(, ) a(, ) a(, ) b(, ) 1 b(, ) ( )co ( ) c(, ) ( )in ( ) a(, ) ( )co ( ) V ( )

5 where n,, T, U, V c b b c 1, 0 (, 0 ) a(, 0 ) ( )co ( ) (, 0) (, 0) (, 0) a(, 0) ( )in ( ), a c c a 2, 0 (, 0 ) (, 0) a(, 0) ( )in ( ) (, 0) 1 (, 0) b(, 0) ( )co ( ) 0 c(, ) ( )in ( ), b a a b 3, 0 (, 0 ) 1 (, 0) b(, 0) ( )co ( ) c(, 0) ( )in ( ) (, 0) a(, 0) ( )co ( ) (, 0 ). Remark 3.2 : Becaue, a, 0 b, 0 c, 0 0, 0 T1, T2, L1 L2, (3.4) along he curve r, by he definiion of parial differeniaion we have According o remark above, we hould have 1, 0 0, c 2, 0 (, 0), b 3, 0 (, 0). a b c, 0, 0, 0 0, 0 T1, T2, L1 L2. (3.5) Thu, from (3.4) we obain,,, n U V from Eqn. (2.3), we ge n, co ( ), in ( ), N in ( ), co ( ), B( ) from Eqn. (3.3), we know ha r i a geodeic curve if and only if co ( ), in ( ), 0 (3.6) and in ( ), co ( ), 0 (3.7) From (3.6) and (3.7), we obain 1 in ( ) 3 0, 0 here uing (3.5),we have

6 b (, 0) 0 and c in Thu we have following heorem: b, co, 0 0 Theorem 3.3 : The neceary and ufficien condiion for he curve ioparameric and geodeic on he urface P, c b b,0,0 0. i a, b, c, 0, in co, (, ) 0. Corollary 3.4 : The ufficien condiion for he curve geodeic on he urface P, i o be boh (3.8) o be boh ioparameric and b, f, in, c, f, co, f, 0,. 4. Ruled urface pencil wih a common aympoic curve Theorem 4.1 : Given an arc-lengh curve, here exi a ruled urface poeing r a a common geodeic curve. Proof : Chooing marching-cale funcion a and a, g, b, in, c, co Eqn. (3.1) ake he following form of a ruled urface, in co P 0 g T U V which aifie Eqn. (3.8) inerpolaing r a a common geodeic curve. Corollary 4.2 : Ruled urface (3.11) i developable if and only Proof : and only if 0 g, in co P 0 g T U V (2.1) (2.4) give r, d, d 0, where d gt gt cou inu inv cov d g T in U co V gt g cou inv cou in cot in V in cot gt g co co U g in in V. Employing Eqn. (3.12) in he deerminan we ge g, which complee proof. (3.9) P, (3.10), (3.11) i o be. i developable if. Uing Eqn. (3.12) 5. Example of generaing urface wih a common geodeic curve Example 5.1. Le how ha in( ), co, be a uni peed curve. Then i i eay o

7 3 3 4 T co, in( ), , 5 5 If we chooe 4 5 +c and c= U ' co co( ), in co( ), co( ) V ' coin( ), in in( ), in( ) hen Eqn. (2.5) i aified. By inegraion, we obain Now, U in( ) in( ), co( ) co( ), in( ), V co( ) co( ), in( ) in( ), co( ) ,, T U V i an RMF ince i aifie Eqn. (2.2). If we ake 4, 0,, in( )(in 1),, co( 4 a b c )co 5 5 and 0, 2, hen Eqn. (3.5) i aified. Thu, we obain a member of he urface pencil wih a common geodeic curve r a P co( ) in 1 in co co co 1 (, ) in( ) in 1in in in co co co co, co in in, in 2 in 1 co 2 co where 0 2, 0 2 (Fig. 1).

8 In Eqn. (3.8), if we ake Fig. 1. P (, ) 1 a a member of urface pencil and i geodeic curve. g() 4 3 urface wih a common geodeic curve, hen we obain he following developable ruled a in( ) co( ) in in in co co co, (, ) P co( ) in( ) in co co co in in, where 2 2, 0 2 (Fig. 2). Fig. 2. P (, ) 2 a a member of he developable ruled urface pencil and i geodeic curve.

9 Example 5.2. Le If we ake co,in( ),0 T in,co( ),0, 1, 0.,, T U V U co, in( ),0 b, 1 co, c, in be a uni peed curve. I i obviou ha and V 0,0,1 i an RMF. By chooing marching-cale funcion a and 0 0, hen Eqn. (2.2) i aified and a, 0,, hen Eqn. (3.5) i aified. Thu, we immediaely obain a member of he urface pencil wih a common geodeic curve 3 P (,) co co,in co,in where 0 2, 0 2 (Fig. 3). a Fig. 3. P (, ) 3 a a member of urface pencil and i geodeic curve. For he ame curve le u find a ruled urface. In Eqn. (3.8), if we ake we obain he following developable ruled urface wih a common geodeic curve where 4 P (, ) co,in,, -1 1 (Fig. 4). g() 0, hen a

10 Fig. 4. P (, ) 4 If we ake g () = 0 and a a member of he developable ruled urface pencil and i geodeic curve. 0 3 common geodeic curve r () a hen, we obain he developable ruled urface wih a (,) P5 co 1,in 1, where 0 2, 0 2 (Fig. 5). Fig. 5. P (, ) 5 a a member of he developable ruled urface pencil and i geodeic curve.

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