ON SLANT HELICES AND GENERAL HELICES IN EUCLIDEAN n -SPACE. Yusuf YAYLI 1, Evren ZIPLAR 2. yayli@science.ankara.edu.tr. evrenziplar@yahoo.

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1 ON SLANT HELICES AND ENERAL HELICES IN EUCLIDEAN -SPACE Yusuf YAYLI Evre ZIPLAR Departmet of Mathematcs Faculty of Scece Uversty of Akara Tadoğa Akara Turkey Departmet of Mathematcs Faculty of Scece Uversty of Akara Tadoğa Akara Turkey Abstract I ths paper Eucldea -space E we vestgate the relato betwee slat helces ad sphercal helces Moreover E we show that a slat helx ad the taget dcatrx of the slat helx have the same axs (or drecto) Also we gve the mportat relatos betwee slat helces sphercal helces E ad geodesc curves o a helx hypersurface E Keywor: Slat helces; geeral helces; sphercal helces; taget dcatrx; helx hypersurfaces Mathematcs Subject Classfcato : 5A4 5B5 5C4 5C5 INTRODUCTION Slat helce s oe of the most mportat topcs of dfferetal geometry Izumya ad Takeuch have vestgated the may propertes of slat helces that the ormal les make a costat agle wth a fxed drecto Eucldea -space [7] Moreover they proved that a space curve s a slat helx f ad oly f the geodesc curvature of the prcpal ormal of the curve s a costat fucto [7] Besdes Izumya ad Takeuch descrbed a slat helx f ad oly f the fucto κ τ ( ) ( κ + τ ) κ s costat where κ s curvature ad τ s torso of the curve respectvely[7] Moterde obtas a geometrc characterzato of Salkowsk curves whose the ormal vector matas a costat agle wth a fxed drecto space []

2 O the other had Kula ad Yayl have vestgated sphercal mages of taget dcatrx of a slat helx Eucldea -space ad they proved that the sphercal mages are sphercal helx Eucldea -space[9] eeral helce whose tagets make a costat agle wth a fxed drecto s also cosderable subject of dfferetal geometry May geometers have studed o ths type of curves [54] I 845 de Sat Veat frst proved that a space curve s a geeral helx f ad oly f the rato of curvature to torso be costat[5] Moreover Al ad López cosder the geeralzato of the cocept of geeral helces the Eucldea - space E [] Also Yılmaz ad Turgut troduce a ew verso of Bshop frame ad troduce ew sphercal mages [7] I dfferetal geometry of surfaces a helx hypersurface E s defed by the property that taget plaes make a costat agle wth a fxed drecto [4] D Scala ad Ruz-Herádez have troduced the cocept of these surfaces [4] Ad AINstor has also troduced certa costat agle surfaces costructed o curves E [] Özkaldı ad Yayl gve some characterzato for a curve lyg o a surface for whch the ut ormal makes a costat agle wth a fxed drecto [] Oe of the ma purpose of ths work s to observe the relatos betwee slat helces ad sphercal helces Aother purpose of ths study s to gve a mportat relato betwee helx hypersurfaces ad sphercal helces PRELIMINARIES Defto Let α : Ι ΙR E be a arbtrary curve E Recall that the curve α s sad to be of ut speed ( or parametrzed by the arc-legth fucto s ) f α ( s) α ( s) where s the stadart scalar product the Eucldea space gve by E X Y x y for each X ( x x x ) Y ( y y y ) E

3 Let { s) V ( s) V ( )} V be the movg frame alog α where the vectors V are ( s mutually orthogoal vectors satsfyg V The Freet equatos for α are gve by V V k V V V k k k V k k V V V k V V Recall that the fuctos k (s) are called the -th curvatures of α [6] Defto A ut speed curve : Ι Ι R E s called geeral helx f ts taget vector V makes a costat agle wth a fxed drecto U [] Theorem Let : Ι E be a ut speed curve ad a geeral helx ut drecto of the helx : U cosθ [ V + V ] E The the Here V ( s) U cosθ ad [ k ( ) ] + O k the other had { V V } where [] V s the Freet frame of ad k k are curvatures of Defto A ut speed curve α : Ι ΙR E s called slat helx f ts ut prcpal ormal V makes a costat agle wth a fxed drecto L [] Theorem Let α : Ι E be a ut speed curve where E Defe the fuctos k k( s) [ k + ] k k 4 The α s a slat helx f ad oly f the fucto

4 s costat ad o-zero Moreover the costat C sec θ beg θ the agle that makes V wth the fxed drecto L that determes α [] C Theorem Let α : Ι E be a ut speed curve ad a slat helx ut drecto of the helx : Here V ( s) L cosθ ad L cosθ [ V ] k k s k ( ) [ + ] 4 k k O the other had { V V } V s the Freet frame of α [] E The the Defto 4 Let s defed by [] : Ι E be a ut speed curve H : Ι ΙR IR H k k { V[ H ] + H k} k + E Harmoc curvatures of Theorem 4 Let : Ι E be a ut speed curve ad a geeral helx ut drecto of the helx : Here Χ cosθ Χ cosθ [ V + H V + + H V ] V O the other had { V } V { H } H H are the harmoc curvatures of [] E The the V s the Freet frame of ad 4

5 Remark If the Freet frame of the taget dcatrx of a space curve α s { V T V N V } the B ad V V V T N B κ + τ ( κ t + τ b) κ + τ ( τ t + κ b) κ + τ κτ κ τ s the curvature of τ s the torso of where κ κ ( κ + τ ) κ { V t V V b} s the Freet frame of α ad κ s the curvature of α τ s the torso of α [8] SLANT HELICES AND SPHERICAL HELICES I the followg Theorem we gve the relato betwee slat helces sphercal helces o the ut hypersphere S - E E ad Theorem Let α : Ι ΙR E be a ut speed curve ( parametrzed by arclegth fucto s ) where S drecto L E ad let : Ι S E s the ut hypersphere be the taget dcatrx of the curve α E The the curve α s a slat helx wth E f ad oly f the curve s a geeral helx (sphercal helx) wth drecto L o S E I other wor α ad have the same drecto L Proof: The taget dcatrx of the curve α s defed by dα (s) We assume that the arclegth parameter of s s The we ca wrte d d Ad from the Freet equatos (see Defto ) 5

6 d k( s) V ( s) Thus from the last equato by takg orms o both sdes we obta k ( s) d ( k ( s) ) Hece we have V ( s) Now we assume that the curve α s a slat helx wth drecto L product betwee the ut prcpal ormal vector feld V wth L s V L cosθ E So the scalar d d O the other had we kow that V Hece we have < L > cosθ It follows that the curve s a geeral helx (sphercal helx) wth the drecto L o S E Coversely the curve s a geeral helx (sphercal helx) wth drecto L o S E d The we have < L > cosθ where s s the arclegth parameter of O the other had we kow that d V Hece we have V L cosθ So we deduce that the curve α s a slat helx wth drecto L proof E Ths completes the Ths above Theorem has the followg corollary Corollary Let α : Ι ΙR E be a slat helx wth ut speed (or parametrzed by arclegth fucto s ) ad let : Ι S E be the taget dcatrx of the curve α (parametrzed by arclegth fucto s ) where S s the ut hypersphere E We assume that the drecto of α s L cosθ [ V ] The the drecto L ca be expressed the forms 6

7 L cosθ [ V + V ] ad L cosθ [ V + H V + + H V ] where θ s costat Here for the curve α : k( s) k [ k + ] 4 k k ad { V V } V s the Freet frame of α Moreover for the curve : ad { V V } [ ( ) k + ] k V s the Freet frame of { H } curvatures of H H are the harmoc Proof: From the Theorem sce α ad have the same drecto L ad d V V the V L V L cosθ So we deduce that cosθ [ V ] cosθ [ V + V ] cosθ [ V + HV + + H V ] Ths completes the proof I the followg example t has bee obtaed a geeral helx o the ut hypersphere S E by usg a slat helx E 4 Example Let α( s ) ( s s s8s coss + cos8s ss) be a slat helx wth ut speed follows: E where π < π < s It s easly obta the curvatures as κ ( s) 4 s s ad τ ( s) 4 coss where κ s the curvature of α ad τ s the torso of α Now we are gog to fd out the taget dcatrx of the curve α : 7

8 d ( s) α ( coss cos8s s s s8s coss) That s (s) s a geeral helx o the hypersphere S E The slat helx α s show the followg Fgure ad the taget dcatrx of the curve α s show the followg Fgure where π < s < π Fgure Fgure 8

9 For Corollary we ca gve the followg example Example I ths example we are gog to fd out the drectos of the curves α ad defed Example The drecto of the slat helx α : where L cosθ [ V ] cosθ [ t + + b] κ κ ( s) τ ad { V t V V b} s the Freet frame of the slat helx α From the example we kow that κ ( s) 4s s ad τ ( s) 4coss So t s easly obta the ad as follows: 4 coss ad 4 ss O the other had from theorem we kow that ca deduce from the last equalty that cos θ 5 Fally the drecto of the slat helx α s foud out as 4 4 L ( coss) t + + ( ss) b The drecto of the taget dcatrx : + + sec θ Hece we U cosθ [ V + V ] cosθ [ T + B] where [ κ + ( ) ] τ { V T V N V B} ( s the Freet frame of From theorem we kow that ad So κ s the curvature of ad τ s the torso of ) ad κ O the other had τ from remark κ + τ ad κ κ κ τ κ τ τ Hece we obta κ( κ + τ ) (κ + τ ) Aga from remark T ad B ( τ t + κ b) κ τ κ τ κ + τ 9

10 Cosequetly ıf we also cosder that drecto of the helx s foud out as κ ( s) 4ss ad τ ( s) 4coss the the 4 4 U ( coss) t + + ( ss) b where cos θ 5 Notce that the drectos of the curves α ad equal 4 HELIX HYPERSURFACES AND HELICES I ths secto we gve the mportat relatos betwee slat helces sphercal helces E ad geodesc curves o a helx hypersurface E Defto 4 ve a hypersurface M E ad a utary vector d say that M s a helx hypersurface wth respect to the fxed drecto d f costat fucto alog M where ξ s a ormal vector feld o M [4] E we d ξ s Theorem 4 Let M be a helx hypersurface wth the drecto d E ad let α : Ι ΙR M be a ut speed geodesc curve o M The the curve α s a slat helx wth the drecto d E [6] Proof: Let ξ be a ormal vector feld o M Sce M s a helx hypersurface wth respect to d d ξ costat That s the agle betwee d ad ξ s costat o every pot of the surface M Ad α (s) λξ (s) α alog the curve α sce α s a geodesc curve o M Moreover by usg the Freet equato α ( s ) V kv we obta λξ α ( s ) kv where k s frst curvature of α Thus from the last equato by takg orms o both sdes we obta ξ V or ξ -V So d V s costat alog the curve α sce d ξ costat I other wor the agle betwee d ad V s

11 costat alog the curve α Cosequetly the curve α s a slat helx wth the drecto d E Ths completes the proof Now we ca gve a ew Theorem by usg the above Theorem Theorem 4 Let α : Ι ΙR M E be a geodesc curve o M wth ut speed (or parametrzed by arclegth fucto s ) where M s a helx hypersurface wth respect to a fxed drecto d E The the taget dcatrx ) sphercal helx o ut the hypersphere S E α (s of the curve α (s) s a Proof: We assume that α s a geodesc curve o M The from Theorem 4 α s a slat helx wth the drecto d E O the other had from Theorem the taget dcatrx of the curve α s a sphercal helx wth the drecto d o the hypersphere S E Ths completes the proof Ths above Theorem has the followg corollary Corollary 4 Let M E be a helx hypersurface E The the taget dcators of all geodesc curves o the surface M are sphercal helces whose axes cocde REFERENCES [] Al AT López R Some characterzatos of cled curves Eucldea E space Nov Sad J Math Vol 4 No 9-7 [] Al AT Turgut M Apr 9 Some characterzatos of slat helces the Eucldea space E arxv: 9487v [mathd] [] Camcı Ç İlarsla K Kula L Hacısalhoğlu HH 9 Harmoc curvatures ad geeralzed helces E Chaos Soltos & Fractals [4] D Scala AJ Ruz-Herádez 9 Helx submafol of eucldea spaces Moatsh Math 57: 5-5

12 [5] luck H 966 Hgher curvatures of curves Eucldea space Amer Math Mothly [6] ök I Camcı Ç Hacısalhoğlu HH 9 V -slat helces Eucldea - space E Math Commu Vol 4 No pp 7-9 [7] Izumya S Takeuch N 4 New specal curves ad developable surfaces Turk J Math [8] Kula L Ekmekç N Yaylı Y İlarsla K Characterzatos of slat helces Eucldea -space Turk J Math [9] Kula L Yayl Y 5 O slat helx ad ts sphercal dcatrx Appled Mathematcs ad Computato [] Mlma RS Parker D 977 Elemets of dfferetal geometry Pretce- Hall Ic Eglewood Clffs New Jersey [] Moterde J 9 Salkowsk curves revsted: A famly of curves wth costat curvature ad o-costat torso Computer Aded eometrc Desg [] Nstor AI 9 Certa costat agle surfaces cosructed o curves arxv: 94475v [mathd] [] Özkaldı S Yayl Y Costat agle surfaces ad curves Iteratoal electroc joural of geometry Vol 4 No pp 7-78 [4] Scofeld PD 995 Curves of costat precesso Amer Math Mothly 5-57 [5] Struk DJ 988 Lectures o Classcal Dfferetal eometry New York: Dover [6] Yaylı Y Zıplar E Costat agle ruled surfaces Eucldea spaces Submtted [7] Yılmaz S Turgut M November A ew verso of Bshop frame ad a applcato to sphercal mages Joural of Mathematcal Aalyss ad Applcatos 7 () pg E