CONSTANT ANGLE RULED SURFACES IN EUCLIDEAN SPACES Yuuf YAYLI Ere ZIPLAR Deparme of Mahemaic Faculy of Sciece Uieriy of Aara Tadoğa Aara Turey yayli@cieceaaraedur Deparme of Mahemaic Faculy of Sciece Uieriy of Aara Tadoğa Aara Turey ereziplar@yahoocom 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
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 []
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
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
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
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
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
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
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
The 4 ( u) i( u) + co( u) i( u) 5 5 α [ 5π ] 4 i( u) co( u) i( u) 5 5 4 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) + 5 5 y ( ) co( u) + 4 z ( )i( u) 5 5 Ad he urface geeraed by he cure α i how he followig Figure 5 5 - -
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