ON THE INTEGRAL INVARIANTS OF KINEMATICALLY GENERATED RULED SURFACES *

Transcription

1 Iranan Journal of Scnc & Tchnology Transacton A ol 9 No A Prntd n Th Islamc Rpublc of Iran 5 Shraz Unvrsty ON TH INTGRAL INARIANTS OF KINMATICALLY GNRATD RULD SURFACS H B KARADAG AND S KLS Dpartmnt of Mathmatcs Faculty of Scncs and Arts Inönü Unvrsty 4469 Malatya Turky mal: hbkaradag@nonudutr Abstract In ths papr th dual ara vctor of a closd dual sphrcal curv s knmatcally gnratd and th dual Stnr vctor of a moton ar tnsvly studd by th mthods of dffrntal gomtry Jacob s Thorms known for ral curvs ar nvstgatd for closd dual curvs Th closd trajctory IR n whch th closd dual surfacs gnratd by an orntd ln ar fd n a movng rgd body n curvs from Study s transfrnc prncpl s studd Th ntgral nvarants of ths closd ruld surfacs ar calculatd by mans of th ara vctor Morovr som thorms rsults and ampls ar gvn Kywords Mchansm knmatcs ara vctors ruld surfacs nvarants and motons INTRODUCTION Th knmatc gomtry of th nfntsmal postons of a rgd body n spatal motons s not only mportant but ntrstng as wll In a spatal moton th trajctory of th orntd lns and ponts mbddd n a movng rgd body ar gnrally ruld surfacs and curvs rspctvly Thus th spatal gomtry of ruld surfacs and curvs s mportant n th study of ratonal dsgn problms n spatal mchansms As an ampl som charactrstc nvarants of ruld surfacs wr appld to a mchansm thory by A T Yang t al [] Also usng th gomtry of curvs and dvlopabl ruld surfacs som spatal dsgn problms wr nvstgatd by H Pottmann t al [] J A Schaaf t al [] and Wang t al [4] Rathr unpctdly dual numbrs hav bn appld to study th moton of a ln n spac; n IR thy vn sm to b th most approprat apparatus for ths purpos It was frst don by Study [5] and snc hs tm dual numbrs hav had an stablshd plac n knmatcs as a tool to solv problms dalng wth lns n spac ast ltratur on th subjct can b found n [6-8] Th applcaton of dual numbrs to th lns of th ucldan -spac s carrd out by th prncpl of transfrnc whch was formulatd by Study It allows a complt gnralzaton of th mathmatcal prsson for th sphrcal pont gomtry to th spatal ln gomtry by mans of dual numbr tnson rplacng all ordnary quantts by th corrspondng dual numbr quantts [9] Jacob [] showd that th ndcatr of a tangnt vctor of any ral closd sphrcal curv dvds th surfac ara of a unt sphr nto two qual parts In th sam papr h also showd that th ndcatr of th prncpal normal vctor of any closd spac curv also dvds th surfac ara of Rcvd by th dtor May 5 4 and n fnal rvsd form July 5 Corrspondng author

2 456 H B Karadag / S Kls th unt sphr nto two qual parts Thn Fnchl [] and Avaqumovc [] usng Jacob Thorms showd th unt sphrcal closd curv s th prncpal normal ndcatr of a closd spac curv f th closd sphrcal curv dvds th surfac ara of th unt sphr nto two qual parts Also Yapar [] showd that th sphrcal ndcatr of ach unt vctor lyng n th osculatng plan of a closd sphrcal curv whch s fd to th curv dvds th surfac ara of th unt sphr nto two qual parts Th angl and lngth of th ptch whch ar th ntgral nvarants of a closd ruld surfac ar vry mportant n th study of th gomtry of lns from th prspctvs of nstantanous spac knmatcs and mchansms In rcnt yars svral authors hav usd ths nvarants n thr nvstgatons concrnng th gnralzaton of som of th thorms of plan knmatcs to spatal knmatcs [ ] In ths study th ntgral nvarants of closd ruld surfacs knmatcally gnratd ar calculatd and Jacob s Thorms ar statd by mans of th ara vctor and som rlatons and thorms ar gvn PRLIMINARIS A dual numbr has th form a +ε a whr a and a ar ral numbrs and ε s th dual unt wth th proprty ε Th st of all dual numbrs s a commutatv rng ovr th ral numbrs fld and dnotd by ID [8] Th st ID {A (A A A ) : A ID; } s a modul ovr th rng ID whch s calld an ID-modul or dual spac W call lmnts of ID dual vctors A dual vctor A may b wrttn as Aa+ε a a p a whr a p and a ar ral vctors n IR Th nnr product of two dual vctors A and B s dfnd as ( ab ab ) A B ab + ε + Whr a b cosϕ and a b + ab ϕ snϕ ϕ π Th cross-product of two dual vctors A and B s gvn by ( a b + a b) A B a b + Lt Φ b th dual angl btwn th unt dual vctors A and B thn A B cosφ cosϕ εϕ snϕ whr Φ ϕ + εϕ ϕ π ϕ IR s a dual numbr Hr th ral numbrs ϕ and ϕ ar th angl and th mnmal dstanc btwn th two orntd lns A and B rspctvly Th gomtrc plac of th ponts satsfyng th qualty A () whn A (a ) s calld a unt dual sphr n ID-modul Study stablshd a thorm whch stats "thr s a on to on mappng btwn th dual ponts of a unt dual sphr and th orntd lns n IR " Accordng to Study s Thorm; a unt dual vctor Aa+ε a corrsponds to only on orntd ln n IR whr th ral part a shows th Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A Autumn 5

3 On th ntgral nvarants of 457 drcton of ths ln and th dual part a shows th vctoral momnt of th unt vctor a wth rspct to th orgn Lt a movng orthonormal trhdron { } b mad a closd spatal moton along a closd curv c ( ) ( n IR Durng a closd spatal moton an orntd ln fd n IR gnrats a closd trajctory surfac Th paramtrc quaton of a closd trajctory surfac formd wth -as can b prssd as follows: for all t u IR If w tak th movng orthonormal trhdron as Ψ ( t u) ( + u( Ψ( t u) Ψ( t + π u) () ( ( ( ( ( ( ) ( t thn th as ntrsct at th strcton pont of -gnrator of th closd ruld surfac gvn by quaton () In ths cas ( s th strcton pont; and ar calld cntral normal and cntral tangnt rspctvly Th structural quatons of closd spatal moton dscrbd abov ar j j w w ( w j ( j w ( j d () whr th dffrntal forms w and w ar th natural curvatur and th natural torson of -closd trajctory surfac rspctvly quaton () can b wrttn n th followng form d ( w ( ) () whr w w + w s th Darbou vctor of th moton If w thn th Pol vctor and th Stnr vctor ar gvn by: w P s w (4) w rspctvly whr w s th nstantanous angular vlocty of th moton and ntgraton s takn along th closd curv c() on fd spac n R Th lngth of th ptch (Öffnungsctrack) of an -closd trajctory surfac s dfnd by: : d u d (5) Th orthogonal trajctory of an -closd trajctory surfac startng from pont P O on th -gnrator ntrscts th sam gnrator at pont P whch s gnrally dffrnt from P O Thus P P O Lt us consdr a unt vctor on th ( ) m cosθ + snθ -plan such that an m-orntd ln gnrats a dvlopabl ruld surfac (tors) along th orthogonal trajctory of an -closd trajctory surfac durng th closd moton Thn th total Autumn 5 Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A

4 458 H B Karadag / S Kls chang of θ s calld th angl of ptch (Öffnungswnkl) of th -closd trajctory surfac and gvn by on of th followng forms λ d θ d s (6) : Th lngth of th ptch and th angl of th ptch ar wll-known ntgral nvarants of a closd trajctory surfac [7-9] Thr s a on-to-on corrspondnc btwn sphrcal curvs and spac curvs Hnc th structural quaton () s also vald for th sphrcal curvs Thn th sphrcal ara boundd by a closd sphrcal curv c() s gvn by f π ( ν ) s (7) whr th vctor s th poston vctor of th pont and ν s th rotaton numbr around pont of th Pol curv c(p) [-] Th ara vctor of a closd spac curv c() n R s dfnd by v : d (8) whr th ntgraton s takn along th closd curv c() Th projcton ara of a closd spac curv c() n th drcton of a unt vctor n whch s normal to th projcton plan s gvn as follows []: f n v n (9) TH INTGRAL INARIANTS AND TH ARA CTORS Lt K b a movng dual unt sphr gnratd by a dual orthonormal trhdron ( ( ( ( ( ( ( ( ) ( () and K b a fd dual unt sphr wth th sam cntr n th dual sphrcal closd moton dnotd by K / K ar: j ID Thn th dffrntal quatons of j j j j j d Ω j Ω ( w ( + ε w ( Ω ( Ω j ( ( ) Ω ( () whr th dffrntal forms Ω ( w ( + εw ( and Ω ( w ( + εw ( ar th dual natural curvatur and torson rspctvly Th dual Stnr vctor of th closd moton s dfnd by W S W P W w + εw () W whr W Ω + Ω and P ar nstantanous Darbou vctor and th dual pol vctor of th moton rspctvly As known from th Study s transfrnc prncpl th dual quaton () Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A Autumn 5

5 On th ntgral nvarants of 459 corrspond to th ral quaton () of a closd spatal moton n IR In ths sns th dffrntabl dual closd curv ( t IR s consdrd as a closd trajctory surfac n IR Lt us consdr a dffrntabl unt dual sphrcal closd curv c() ( (t + π ) ( t IR () W know from Study s transfrnc prncpl that th dual curv dfnd by () whch shows a unt dual sphrcal closd curv corrsponds to an -closd trajctory surfac gnratd by an - orntd ln fd n a movng rgd body n IR Thus th curv () s calld th unt dual sphrcal mag (or ndcatr) of an -closd trajctory surfac Th dual angl of th ptch of th closd ruld surfac ( s qual to th dual projcton of th gnrator on to th dual Stnr vctor of th moton K / K that s [6]: S π λ ε (4) A whr A a + εa th dual sphrcal surfac ara of th dual sphrcal mag of -closd trajctory surfac Lt c() b th dual sphrcal ndcatr on K of an arbtrary fd dual pont on K Th dual sphrcal ara F surroundd by th dual closd curv c() s F π ( ν ) S (5) Hr ν s th rotaton numbr of th rotaton of th cntrod c(p) at th pont and dnots th dual poston vctor of an arbtrary pont of th dual closd curv c() on K [6] Th dual ara vctor of an (-closd sphrcal curv can b dfnd by as an analogu to th dfnton n [] whr : d (6) d W s th dffrntal vlocty of an -dual pont fd of th movng sphr K From quatons () and (6) th dual ara vctor may b dvlopd as or S - S (7) S + (8) Ths statmnt shows that thr s a rlatonshp btwn th dual angl of th ptch of an -closd trajctory surfac and ts dual ara vctor On th othr hand f a scalar product s mad wth th vctor S on both sds of quaton (7) thn w may wrt S (9) whr s th dual angl of th ptch of th -trajctory surfac gnratd by th ara vctor of c()-closd sphrcal ndcatr of -closd trajctory surfac It follows from (8) that Autumn 5 Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A

6 46 H B Karadag / S Kls S () Thus wth th ad of (9) and () th dual angl of th ptch of th -unt ara vctor trajctory surfac s obtand as So w may gv th followng thorm S () Thorm Thr s th rlatonshp S + () btwn th dual angl of ptchs of and -closd trajctory surfacs By sparatng quaton () nto ral and dual parts w hav s λ + λ v and s s λ λ v In th cas of th as of th unt ara vctor and th Stnr vctor S ar prpndcular to ach othr w gt λ and Thus th followng rsult may b gvn v v Rsult Durng th closd sphrcal moton th as of dual ara vctor and th dual Stnr vctor S ar prpndcular to ach othr f and only f Also from (4) w hav f and only f av π and a v Thus th followng rsult can b gvn Rsult Th dual sphrcal ndcatr of th unt dual ara vctor dvds th masur of th sphrcal surfac ara nto two qual parts f and only f A ruld surfac Ψ ( t u) ( + uv ( s gvn by v (v (+ε v( whr v s th unt ara vctor and v v s th vctoral ara vctor of v wth rspct to th orgn pont Snc th sphrcal mag of v ( s th unt ara vctor th dual ara vctor ( also has unt magntud Thus th ruld surfac can b rprsntd by a dual curv on th surfac of a unt dual sphr Th dual arc-lngth of th ruld surfac ( s gvn by v ( + εd) v v whr d s th dstrbuton paramtr (drall) of ths ruld surfac v Accordng to Study's transfrnc prncpl th followng thorm can b gvn Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A Autumn 5

7 On th ntgral nvarants of 46 Thorm 4 In th lns spac th trajctory surfac of th closd sphrcal ndcatr gnratd by th unt dual ara vctor formd along a unt dual closd curv n a closd sphrcal moton s a dvlopabl ruld surfac On th othr hand th orntd dual projcton ara of a planar rgon whch occurd by takng orthogonal projcton onto a plan n th drcton of a fd unt vctor N of th curv c() s gvn by F n N Th poston vctor of th pont fd n th movng sphr K n trms of th dual orthonormal vctors and can b wrttn as ( ( t ) + ( + ( () whr and ar constant coordnats of Lt c( ) c( ) and c( ) b th closd dual sphrcal ndcatr of th dual orthonormal vctors and rspctvly Thus w can gv th followng rsult Rsult 5 Th dual ara vctor of th closd dual curv c() drawn on a fd unt sphr K by a fd pont of movng dual unt sphr K durng th closd sphrcal moton s + k < k k k whr ( ( t dt ) and ( ( k ( + k ( k ( ) dt Snc th moton s closd w gt ( t ) dt Thus w hav Rsult 6 Th dual Stnr vctor S of th moton n trms of th dual ara vctors s S and Th rlaton btwn th orthogonal projcton ara and th paralll projcton ara can b gvn by th followng proposton Proposton 7 Lt F n b th orntd dual projcton ara of th planar rgon formd by takng th orthogonal projcton of closd dual sphrcal curv c() onto th plan and F p b th orntd dual projcton ara of th planar rgon formd by paralll projctng of th closd dual sphrcal curv c() onto th sam planar rgon n th drcton of a unt dual vctor P Thn Autumn 5 Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A

8 46 H B Karadag / S Kls F cos Θ n F p whr Θ s th dual angl btwn two mag plans [9] Thus w can gv th followng thorm Thorm 8 Th orntd dual projcton ara F p of th planar rgon formd by paralll projcton of th dual closd curv c() drawn by a fd pont of th movng sphr K s F F + F p p ampl 9 Lt us consdr th dual pont ( ( cosφ( snφ( ) whr Φ ( θ ( + εθ ( Now Lt s calculat th orntd dual projcton ara of th dual closd sphrcal curv c() formd durng th closd sphrcal moton Snc ( w hav and k < k ( cosφ( snφ( ) ( cosθ ( εθ ( snθ ( snθ ( + εθ ( cosθ ( ) dθ dθ d - snθ ( - ε ( snθ ( + θ ( cosθ ( )cosθ ( + ε ( cosθ ( θ ( snθ ( ) + ε dt dt dθ d + ε dt Thus th dual ara vctor of closd sphrcal curv c() s π k p k ( ( dt (π ) Usng quatons (5) and (6) th orntd dual projcton ara of th closd dual sphrcal curv c() w gt F N ()(π ) π N n n Thus th orntd dual projcton ara s obtand as F π n From th Blaschk ara formula and quaton (7) th followng thorm can b gvn Thorm Lt c() b th dual sphrcal ndcatr of a fd pont and also c( ) c( ) and c( ) b th closd dual sphrcal ndcatrs of th dual orthonormal vctors and durng th closd sphrcal moton rspctvly Thn th dual sphrcal ara boundd by th closd sphrcal curv c() n trms of th dual sphrcal aras F F and F boundd by th closd sphrcal ndcatrs c( ) c( ) and c( ) s Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A Autumn 5

9 On th ntgral nvarants of 46 F π ( ν ) + F whr ν s th rotaton numbr of moton 4 TH JACOBI THORMS Lt us hav a closd dual curv c() of class C on a unt dual sphr K n ID At th ntal tm assum that th unt dual sphr K corrspondng wth K to b K K whr K s a fd sphr and K s a movng sphr wth rspct to K Th curv c() dscrbs a closd dual sphrcal moton Lt us consdr th dual movng fram and b frmly lnkd to any pont ( of th curv c() Hr and ar tangnt prncpal normal and bnormal unt dual vctors rspctvly Whl drawng th closd dual sphrcal curv c() durng th dual closd sphrcal moton th nd ponts of vctors and on K also draw closd sphrcal curvs c( ) c( ) and c( ) on K rspctvly Now lt us carry ths vctors to th orgn pont of th unt dual sphr K Thus from quatons () () and (6) w hav th followng thorm Thorm 4 Lt c( ) c( ) and c( ) b th sphrcal ndcatrs of th unt dual vctors and durng th closd dual sphrcal moton rspctvly Th dual ara vctors of ths closd sphrcal ndcatrs ar S - S S S whr S W s th dual Stnr vctor of moton K / K If th prsson (4) s sparatd nto ts ral and dual parts w hav th followng qualts: (4) v v v s + λ s λ v v v s s λ + λ + (5) whr v v v and v v v ar ral and dual ara vctors rspctvly From thorm (4) w can gv th followng rsults Rsult 4 Th dual ara vctor + s qual to th sum of th dual ara vctors Rsult 4 Th unt dual vctor s prpndcular to th dual ara vctors As a spcal cas of quaton () w hav Thus w can gv th followng thorm S and and Autumn 5 Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A

10 464 H B Karadag / S Kls Thorm 4 4 Thr s th rlatonshp S btwn th dual angl of th ptchs of and -closd trajctory surfacs whr S s th dual Stnr vctor of th moton Also from quatons (4) and (4) w hav and S (6) (7) on th othr hand snc from quatons () (4) (6) and (7) w hav + S S (8) and S S (9) Thus w can gv th followng thorm Thorm 4 5 Thr ar th rlatons (8) and (9) btwn th dual angl of ptchs of and -closd trajctory surfacs On th othr hand th dual angls of ptch of ruld surfacs corrspondng to th closd dual sphrcal curvs c( ) c( ) and c( ) rspctvly ar: λ ε () λ ε Now lt us consdr th sphrcal ndcatr c( ) of th unt dual vctor formd durng th closd moton If th ara of th rgon surroundd by th curv c( ) dnotd by F thn from quatons (4) and (5) Snc th abov ara should b F π ( ν ) + λ ε () F π accordng to th Jacob Thorm w obtan Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A Autumn 5

11 On th ntgral nvarants of 465 πν + λ ε () whr s a dual numbr From quaton () accordng to th qualty of two dual numbrs w hav Thus w can gv th followng thorm λ πν () Thorm 4 6 Lt c() b a closd dual sphrcal curv on th unt dual sphr Lt A b th ruld surfac corrspondng to th sphrcal ndcatr of th tangnt vctor of th closd dual curv c() Lt th ral angl of th ptch and lngth of th ptch of th closd ruld surfac A b λ and rspctvly thn w hav λ v πν v + v From thorm (46) snc v + v w hav th followng rsult Rsult 4 7 Th orntd lns (; ) and (v ; v ) ar ntrsctd From quaton (4) w obtan th ntgral nvarants of th closd ruld surfac corrspondng to th sphrcal ndcatr c( ) of th unt dual vctor n th lns spac as th followng λ (4) If th ara of th rgon surroundd by th curv c( ) dnotd by (5) w obtan F thn from quatons (4) and F π ( ) (5) ν Snc th abov ara should b F π accordng to th Jacob Thorm w obtan ν Thus w can gv th followng thorm Thorm 4 8 In th ucldan -spac IR th closd ruld surfac corrspondng to th sphrcal ndcatr of th prncpal normal vctor of th closd dual curv c() s a con that s: λ Lt C b th closd ruld surfac corrspondng to th sphrcal ndcatr of th bnormal vctor of th closd dual curv c() Th ara of th sphrcal rgon surroundd by c( ) s F π ν λ ε (6) ( ) + In addton th lngth of th ptch s thorm v + v Thus w can gv th followng Thorm 4 9 In th lns spac th sphrcal ndcatr of a bnormal vctor of any closd dual sphrcal curv c() on th unt dual sphr corrsponds to a closd ruld surfac Th lngth of th ptch of ths ruld surfac only dpnds on th curv c() and Autumn 5 Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A

12 466 H B Karadag / S Kls v + v Now lt us consdr all th unt dual vctors frmly attachd to th curv whch ls n th osculatng plan of th closd sphrcal curv c() Lt U b on of ths vctors and Θ θ + εθ b th angl btwn th unt dual vctor U and th unt dual tangnt vctor Thus th vctor U can b wrttn as follows: U cosθ + sn Θ (7) If th unt dual vctor U s sparatd nto ts ral and dual parts thn w obtan u cosθ θ θ θ θ θ + θ + sn u cos + sn sn cos θ (8) In th lns spac lt U b th ruld surfac corrspondng to th unt dual sphrcal ndcatr of th unt dual vctor U Th dual angl of ptch of ths ruld surfac from quatons () (7) and (8) s obtand as follows: U US λ cosθ ελ θ snθ λ cos Θ Thus th ral angl of th ptch and th lngth of th ptch of th ruld surfac U corrsponds to closd sphrcal curv c(u) drawn by th unt dual vctor U durng th moton n th lns spac ar λu λ cosθ λ θ snθ u (9) On th othr hand from quatons (4) (5) and (7) th ara of th sphrcal rgon surroundd by th closd sphrcal curv c(u) s obtand as FU π ( ν ) + λ cosθ Snc ths ara should b π [] w hav By takng λ cosθ πν λ θ snθ π < θ < and θ w gt λ So w can gv th followng thorms: Thorm 4 Th ruld surfac corrspondng to th sphrcal ndcatr of th tangnt vctor of closd dual curv c() s a con that s: Thus w hav λ (4) λ ε Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A Autumn 5

13 On th ntgral nvarants of 467 Thorm 4 Lt and b th tangnt and th prncpal normal vctors of th closd curv c() rspctvly Th unt dual vctor s prpndcular to th ara vctor Substtutng qualts n th thorm (4) nto quaton (9) w can gv th followng thorm: Thorm 4 Lt U b th unt dual vctor whch ls n th osculatng plan of closd unt dual curv c() Thn th ruld surfac corrspondng to th sphrcal dual curv c(u) s a con that s λ u u ampl 4 Lt us consdr a unt closd sphrcal curv c() s gvn by π c( ) ( ( cosθ cos( t scθ )cosθ sn( t scθ )snθ θ (k + ) k IR Th dffrntal quatons of ths curv n matr form can b wrttn as ( ( scθ ( scθ ( ( ( Thus w gt s (t sc θ ) Usng quaton (8) th unt ara vctor of th unt closd curv c() s obtand as v ( snθ cos( t scθ )snθ sn( scθ ) cosθ ) t Lt th drcton v of a ln L b gvn by v ( snθ cos( scθ )snθ sn( scθ ) cosθ ) t t Thn w hav th paramtrc quaton of th ruld surfac gnratd by L: Ψ( t u) ( + u v ( ((cosθ u snθ )cos( t scθ )(cosθ u snθ )sn( t scθ )snθ + u cosθ ) u IR Th unt dual ara vctor functon rprsntng Ψ ( t u) s gvn by ~ ( v ε ( v ) v εv + + ( snθ cos( t scθ )snθ sn( t scθ ) cosθ ) + ε ( sn( t scθ ) cos( t scθ )) whr v v s th ara vctoral momnt of th unt ara vctor v Th dffrntal quatons of th unt dual closd sphrcal curv c( ) ( n matr form ar obtand as ( t ) ( t ) cscθ ( t ) cscθ ( t ) ( t ) θ kπ k IR ( t ) Autumn 5 Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A

14 468 H B Karadag / S Kls ~ whr t t tanθ s th dual arc-lngth of th dual closd curv c ( ) Thus w gt S ( t cscθ ) From quaton (4) th dual angl of th ptch of a -closd trajctory surfac s obtand as t From quatons (4) and (4) th dual angls of th ptch of ruld surfacs corrspondng to th closd dual sphrcal curvs c( ) c( ) and c( ) rspctvly ar obtand as t cscθ Also from thorm (44) and th quatons (6) and (7) th dual angls of th ptch of and -ara vctors trajctory surfacs ar found as t t csc θ csc θ rspctvly If th dstrbuton paramtr of th closd ruld surfac Ψ ( t u) s dnotd by d thn th dstrbuton paramtr d s obtand as v v d v Hnc th closd ruld surfac Ψ ( t u) s dvlopabl (s Fg ) Fg A dvlopabl ruld surfac Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A Autumn 5

15 On th ntgral nvarants of CONCLUSIONS Th startng ponts of ths papr ar th dfntons of th ara vctor of a gvn closd spac curv and th projcton ara of ths curv n th drcton of a unt vctor gvn n [7] Usng th ara vctor of a closd dual sphrcal curv th ntgral nvarants of th ruld surfacs n th lns spac corrspondng to th closd sphrcal curv wth th Study transfrnc prncpl ar nvstgatd and Jacob s Thorms ar gvn wth a dffrnt mthod Ths closd curvs and ruld surfacs ar an mportant and ffctv tool n studyng spatal knmatcs It s hopd that ths study wll brng a dffrnt ntrprtaton to th studs n ths fld and wll contrbut to th study of ratonal dsgn problms of spac mchansms Acknowldgmnts- Th authors would lk to thank th anonymous rfrs for thr valuabl commnts on ths papr RFRNCS Yang AT Krso Y & Roth B (975) On a knmatcs thory for ruld surfac Procdng of th fourth world Congrss on th Thory Mach and Mch Nwcastl Upon Tyn (77-74) ngland Pottmann H Lü W & Ravan B (995) Ratonal ruld surfacs Tchncal Rport No Insttut Für Gomtr Tchnsch Uv Wn Schaaf J A & Ravan B (998) Gomtrc contnuty of ruld surfacs Computr Add Gomtrc Dsgn Wang D L Lu J & ao D Z (997) Knmatc dffrntal gomtry of a rgd body n spatal moton-(i II III) Mchansm and Machn Thory Study (89) on dn bwgungn und umlgungn Mathm Annaln Blaschk W (958) Anwndung dualr quatrnonn auf d knmatk Annals Acadma Scntarum Fnnca - 7 Yang A T & Frudnstn F (964) Applcaton of dual-numbr quatrnon algbra to th analyss of spatal mchansms Journal of Appld Mchancs -8 8 ldkamp G R (976) On th us of dual numbrs vctors and matrcs n nstantanous spatal knmatcs Mchansm and Machn Thory Bottma O & Roth B (979) Thortcal knmatcs Amstrdam North-Holland Jacob J (84) Übr nqu mrkwürdg kurvn thorm Schumastr Astr Nch 5- Fnchl W (94) Übr nm Jacobschn satz dr thor Tohoku Math Journal Avakumovc G (95) Übr gschlossn kurvn auf dr kugl Sbornk Radova Sýrpska Acadma Nauka 7-8 Yapar Z (979) Uzay knmatğnd Jacob Tormlr KTÜ TmBl Fak 4 Hoschk J (97) Lnngomtr Zürch - 5 Hoschk J (97) Intgralnvarantn von rglflachn ArchMath ol I Hacısalhoğlu H H (97) On th ptch of a closd ruld surfacs Mchansm and Machn Thory Müllr H R (98) Übr öffnungsma β knmatsch rzugtr gschlossnr rglflachn Abhandl Braunschw Wss Gs Günş R & Klş S (994) Closd sphrcal motons and Holdtch s Thorm Mch and MachThory 9(5) Autumn 5 Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A

16 47 H B Karadag / S Kls 9 Karadağ H B (994) On th closd sphrcal curvs and Jacob s Thorms PhD Thss Unvrsty of Inonu Müllr H R (98) rwtrung ds satzs von Hldtch für gschlossn raumkurvn Abhandl Braunschw Wss Gs 9-5 Pottmann H (987) Übr gschlossn spharsch kurvn l Math Pottman H (989) D flachnvktorn dr bahnkurvn gschlossnr raumlchr zwanglauf Btrag Zur Algbra und Gomtr Iranan Journal of Scnc & Tchnology Trans A olum 9 Numbr A Autumn 5