Ruled surfaces are one of the most important topics of differential geometry. The

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1 CONSTANT ANGLE RULED SURFACES IN EUCLIDEAN SPACES Yuuf YAYLI Ere ZIPLAR Deparme of Mahemaic Faculy of Sciece Uieriy of Aara Tadoğa Aara Turey Deparme of Mahemaic Faculy of Sciece Uieriy of Aara Tadoğa Aara Turey Abrac I hi paper we udy he pecial cure ad ruled urface o helix hyperurface whoe age plae mae a coa agle wih a fixed direcio i Euclidea -pace E Beide we obere ome pecial ruled urface i ΙR ad gie requireme of beig deelopable of he ruled urface Alo we ieigae he helix urface geeraed by a plae cure i Euclidea -pace E Keyword: Helix urface; Coa agle urface;ruled urface Mahemaic Subjec Claificaio : 5A45A5 5A7 5A5B55C4 INTRODUCTION Ruled urface are oe of he mo impora opic of differeial geomery The urface were foud by Gapard Moge who wa a Frech mahemaicia ad ieor of decripie geomery Ad may geomeer hae ieigaed he may properie of hee urface i [467] Coa agle urface are coiderable ubjec of geomerythere are o may ype of hee urface Helix hyperurface i a id of coa agle urface A helix

2 hyperurface i Euclidea -pace i a urface whoe age plae mae a coa agle wih a fixed direciothe helix urface hae bee udied by Di Scala ad Ruiz- Herádez i [] Ad AINior ieigaed cerai coa agle urface coruced o cure i Euclidea -pace E [] Oe of he mai purpoe of hi udy i o obere he pecial cure ad ruled urface o a helix hyperurface i Euclidea -pace E Aoher purpoe of hi udy i o obai a helix urface by geeraed a plae cure i Euclidea -pace E PRELIMINARIES Defiiio Le Ι Ι R : α be a arbirary cure i E Recall ha he cure α i aid o be of ui peed ( or paramerized by he arc-legh fucio ) if ) ( ) ( α α where i he adar calar produc i he Euclidea pace E gie by i i x i y Y X for each E y y y Y x x x X ) ( ) ( Le { } ) ( ) ( ) ( be he moig frame alog α where he ecor i are muually orhogoal ecor aifyig i i The Free equaio for α are gie by Recall ha he fucio ) ( i are called he i -h curaure of α []

3 Defiiio Gie a hyperurface M ΙR ad a uiary ecor d i ay ha M i a helix wih repec o he fixed direcio d if for each Ι R we q M he agle bewee d ad T M i coa Noe ha he aboe defiiio i equiale o he fac q ha d ξ i coa fucio alog M where ξ i a ormal ecor field o M [] Theorem Le H ΙR be a orieable hyperurface i uiary ormal ecor field of H The θ ΙR ad le N be a ( i( θ ) N( x) + co( θ ) d ) f : H ΙR ΙR coa f ( x x + θ i a helix wih repec o he fixed direcio d i θ Ι R where x H ad ΙR Here d i he ecor ( ) Ι R uch ha d i orhogoal o N ad H [] Defiiio Le i ( i ) i We call α a be a ui peed cure wih ozero curaure E ad le { } deoe he Free frame of he cure of α -la helix if he -h ui ecor field mae a coa agle ϕ wih a fixed direcio Χ ha i π Χ co( ϕ) ϕ ϕ coa alog he cure where Χ i ui ecor field i E [] THE SPECIAL CURES ON THE HELIX HYPERSURFACES IN EUCLIDEAN -SPACE E Theorem Le M be a helix hyperurface wih he direcio d i E ad le M be a ui peed geodeic cure o M The he cure α i a -la helix wih he direcio d i E Proof: Le ξ be a ormal ecor field o M Sice M i a helix hyperurface wih repec o d d ξ coa Tha i he agle bewee d ad ξ i coa o eery poi of he urface M Ad α ( λξ α ( alog he cure α ice α i a geodeic cure o M Moreoer by uig he Free equaio α ( ) we

4 obai λξ α ( ) where i fir curaure of α Thu from he la equaio by aig orm o boh ide we obai ξ or ξ - So d i coaalog he cure α ice d ξ coa I oher word he agle bewee d ad i coa alog he cure α Coequely he cure α i a -la helix wih he direcio d i E Corollary For he followig Theorem obaied Theorem: Le M be a cure o a coa agle urface M wih ui ormal N ad he fixed direcio If a cure α o M i a geodeic he α i a la helix wih he axi i E (ee [5]) Theorem Le M be a helix hyperurface i E ad le M a ui peed cure o M If he -h ui ecor field of α equal o ξ or - ξ where ξ i a ormal ecor field o M he α i a -la helix wih he direcio d i E Proof: Le d E be a fixed direcio of he helix hyperurface M Sice M i a helix hyperurface wih repec o d d ξ coa Tha i he agle bewee d ad ξ i coa o eery poi of he urface M Le he -h ui ecor field of α be equal o ξ or - ξ The d i coa alog he cure α ice d ξ coa Tha i he agle bewee d ad i coa alog he cure α Fially he cure α i a -la helix i E Theorem Le M be a helix hyperurface wih he direcio d i E ad le M ( α ( ) M Ι ) be a cure o he urface M If α i a lie of curaure o M he α { T} d Sp alog he cure α where T i age ecor field of Proof: Sice M i a helix hyperurface wih he direcio d 4

5 No α d coa alog he cure α where N i he ormal ecor field of M If we are aig he deriaie i each par of he equaliy wih repec o we obai : Sice α i a lie of curaure o M T operaor of he urface M So we hae ( No α) d ( N o α ) S( T ) λ where S i he hape T d Fially { T} d Sp alog he cure α 4 THE RULED SURFACES IN Ι R Defiiio 4 Le cure o H where H ΙR be a orieable hyperurface i ΙR ad le β be a The β : Ι ΙR H ΙR β ( ) - ( i( θ ) N( β ( ) ) co( d ) Φ ( β ( ) + + θ ) i a ruled urface wih dimeio o f θ i Ι R ( f θ wa defied i Theorem ) where θ coa N i a uiary ormal ecor field of H ad d i coa ecor a defied i Theorem The urface Φ will be called he ruled urface geeraed by he cure β Theorem 4 The ruled urface Φ ( defied aboe i deelopable if ad oly if he cure β i a lie of curaure o he urface H Proof: We aume ha β i a lie of curaure o H Le coider he urface ( β ( ) + ( i( θ ) N( β ( ) ) + co( θ ) d ) ( β ( ) ) co( d Φ wih rulig X ( ) i( θ ) N + θ ) ad direcrix β If we are aig he parial deriaie i each par of he equaliy wih repec o we obai: 5

6 Ad Φ Φ β + ( i( θ ))( N o β ) () S( T ) λt ice β i a lie of curaure o H where T i age ecor field of dn β ad S i he hape operaor of he urface H Beide ( N o β ) S( T) d Therefore ( No β ) λt ad by uig () we obai he equaliy Φ ( + λ i( θ )) β ( + λ i( θ )) T Hece he yem { Φ T} i liear depede Ad we ow ha a age plae alog a rulig i paed by Φ X () ad Φ Fially he age plae are parallel alog he rulig i( ) N ( β ( ) ) co( θ ) d Φ ( i deelopable θ + paig from he poi β () Tha i he urface We aume ha he ruled urface ( liear depede So from he equaliy we ge Φ Φ β + Φ i deelopable The he yem { } ( θ ))( No β ) T + ( i( θ) )( No β) i( Φ T i ( N o β ) λt Therefore ( No β ) S( T) λt where S i he hape operaor of he urface H Tha i β i a lie of curaure o H Corollary 4 Le H be he hyperphere Le β be a cure o - ( ): ( ) S x x x x f x xi f ΙR i H S where β : Ι ΙR H S β ( ) ΙR The he ruled urface ( ) Φ( β ( ) + i( θ ) β ( ) + co( θ) d ΙR i alway deelopable from Theorem 4 Becaue each cure o he hyperphere of curaure S i a lie 5 HELIX SURFACES GENERATED BY A PLANE CURE IN EUCLIDEAN - SPACE E 6

7 Le be a plae cure i Euclidea -pace ormal he biormal of α by of a plae cure i E i coa u T ad α( u) E Ad we deoe he age pricipal B repeciely Noe ha biormal Defiiio 5 We ca obai a ruled urface by uig he plae cure α uch ha φ : U E ( u φ( u α( u) + ( i( θ ) ( u) + co( θ ) B) The ruled urface will be called a he urface geeraed by he cure α Theorem 5 The ruled urface φ : U E ( u φ( u α( u) + i a helix urface wih he direcio B i ( i( θ) ( u) + co( θ) B) E where θ i coa α i a plae cure ad B i a coa ecor which i perpedicular o he plae of he cure α Proof: We wa o how ha Z B i a coa fucio alog φ where Z i a ormal ecor field of φ Fir we are goig o fid a ormal ecor field Z To do hi we will compue he parial deriaie of φ wih repec o u ad Noe ha for he cure α ice α i a plaer cure i E φ ( u ( i( θ )T ad φ ( u i( θ ) + co( θ B () u ) ) Uig he equaliie i () a ormal o he urface φ i gie by φ φ Z u co( θ ) i( ) + θ B φ φ u So we hae Z B i( θ ) co Fially φ i a helix urface wih he direcio B i E 7

8 Thi complee he proof Corollary 5 The helix urface φ : U E ( u i alway deelopable φ( u α( u) + ( i( θ ) ( u) + co( θ ) B) Proof: We ow ha If de( T X X ) where X i( θ ) + co( θ ) B ad T age of α he φ i deelopable So we will compue de( T X X ) : T α X i( θ ) + co( θ ) B ad o we hae Thi complee he proof X i( ) T θ de( T X X ) Theorem 5 Le α( be a plae cure (o a raigh lie) wih ui peed i Euclidea -pace coider he helix urface (geeraed by he cure α () E We φ : U E ( φ( α( + ( i( θ ) ( + co( θ ) B) The he Gau curaure of φ i zero ad he mea curaure of φ : H co( θ) i( θ) where i he fir curaure of he cure α Proof: From corollary 5 he urface φ i deelopable So he Gau curaure of φ i zero 8

9 Now we are goig o proe ha co( θ) H i( θ) The yem { x x } i a orhoormal bai for he age pace of φ a he poi φ φ ( where x ad x φ φ ( φ i parial deriaie of φ wih repec o ad φ i parial deriaie of φ wih repec o ) Recall ha a ormal ecor field of φ i Z ( co( θ ) ( + i( θ ) B by Theorem 5 Ad we ow ha he mea curaure of φ a a poi φ ( : where S i he hape operaor of φ H ( φ ( ) S( x i ) x i So firly we will compue S x ) ad S x ) : ad S( x ) D Z D ( Z ( D i Z dz co( θ ) d i( x φ S φ φ θ φ φ S dz S( x ) D Z D Z d where D i adard coaria deriaie i Therefore we hae Fially S( x ) x where i( θ) Thi complee he proof H x φ E co( θ ) ad S ( x ) x i( θ ) S( x ) x co( θ ) i( i i i θ ) T ) Corollary 5 The urface φ defied aboe i miimal if ad oly if θ π where i( θ ) I ha cae (wheeer θ π ) he urface φ i a plae Example 5 Le he cure α (u) be a plae cure paramerized by he ecor fucio 9

10 The 4 ( u) i( u) + co( u) i( u) 5 5 α [ 5π ] 4 i( u) co( u) i( u) B 5 5 u where i he pricipal ormal ad B i he biormal of α repeciely So If we chooe π 6 θ ad [ π ] ha he parameric repreeaio: he helix urface geeraed by he cure α (u) x ( )i( u) y ( ) co( u) + 4 z ( )i( u) 5 5 Ad he urface geeraed by he cure α i how he followig Figure

11 REFERENCES [] AI Nior 9 Cerai coa agle urface coruced o cure arxi:94475 [mahdg] [] Di Scala AJ Ruiz-Herádez G 9 Helix ubmaifold of Euclidea pace Moah Mah 57: 5-5 [] Gö I Camcı Ç Hacıalihoğlu HH 9 -la helice i Euclidea - pace E Mah Commu ol 4 No pp 7-9 [4] Holdich A Lady ad gelema diary for year 858 [5] Özaldı S Yaylı Y Coa agle urface ad cure i E Ieraioal elecroic joural of geomery olume 4 No pp 7-78 [6] Sarıoğlugil A Tuar A 7 O ruled urface i Euclidea pace E I J Coemp Mah Sci ol o pp - [7] Seier J Ge were Berli 88-88