Involute-Evolute Curve Couples in the Euclidean 4-Space

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1 Int J Open Problem Compt Math Vol o June 9 Involute-volute Curve Couple in the ulidean -Spae min Özyılmaz Süha Yılmaz ge Univerity Faulty of Siene Dept of Math ornova-izmir urkey Dokuz ylül Univerity ua duational Faulty ua-izmir urkey -mail: eminozyilmaz@egeedutr uhayilmaz@yahooom Abtrat In thi work for regular involute-evolute urve ouple it i proven that evolute Frenet apparatu an be formed by involute apparatu in four dimenional ulidean pae when involute urve ha ontant Frenet urvature y thi way another orthonormal frame of the ame pae i obtained via the method expreed in [5] Moreover it i oberved that evolute urve annot be an inlined urve In an analogou way we ue method of [6] Keyword: Claial Differential Geometry ulidean pae Involute- volute urve ouple Inlined urve Introdution he idea of a tring involute i due to C Huygen 658 who i alo known for hi work in opti He diovered involute while trying to build a more aurate lok ee [] he involute of a given urve i a well-known onept in ulidean- pae ee [] It i well-known that if a urve differentiable in an open interval at eah point a et of mutually orthogonal unit vetor an be ontruted And thee vetor are alled Frenet frame or moving frame vetor he rate of thee frame vetor along the urve define urvature of the urve he et whoe element are frame vetor urvature of a urve i alled Frenet apparatu of the urve In reent year the theory of degenerate ubmanifold ha been treated by reearher ome laial differential geometry topi have been extended to

involute apparatu in four dimenional ulidean pae when involute urve ha ontant Frenet urvature y thi way another orthonormal frame of the ame pae i obtained via the method expreed in [5] Moreover it i

2 69 Involute-volute Curve Couple Lorentz manifold For intane in [6] the author extended tudied paelike involute-evolute urve in Minkowki pae-time In the preented paper in an analogou way a in [6] we alulate Frenet apparatu of the evolute urve by apparatu of involute W-urve We ue the method expreed in [5] hereafter we prove that evolute of an involute annot be an inlined urve in four dimenional ulidean pae Preliminarie o meet the requirement in the next etion here the bai element of the theory of urve in the pae are briefly preented A more omplete elementary treatment an be found in [] Let α : I R be an arbitrary urve in the ulidean pae Reall that the urve α i aid to be of unit peed or parametrized by arlength funtion if α α where i the tard alar inner produt of given by a b ab ab ab ab for eah a a a a a b b b b b In partiular the norm of a vetor a i given by a a a Denote by { } the moving Frenet frame along the unit peed urve α hen the Frenet formula are given by ee [] σ σ Here are alled repetively the tangent the normal the binormal the trinormal vetor field of the urve And the funtion σ are alled repetively the firt the eond the third urvature of the urveα And reall that a regular urve i alled a W-urve if it ha ontant Frenet urvature [] Let α : I R be a regular urve If tangent vetor field of α form a ontant angle with unit vetor U thi urve i alled an inlined urve in Let ξ be unit peed regular urve in i an

ulidean pae Preliminarie o meet the requirement in the next etion here the bai element of the theory of urve in the pae are briefly preented A more omplete elementary treatment an be found in [] Let

3 min Özyılmaz Süha Yılmaz 7 involute of ξ if lie on the tangent line to ξ at ξ the tangent to ξ ξ are perpendiular for eah i an evolute of ξ if ξ i an involute of And thi urve ouple defined by ξ µ In the ame pae in [5] author preented a vetor produt a method to determine Frenet apparatu of the regular urve a follow Definition Let a a a a a b b b b b be vetor in he vetor produt of a b i defined by the determinant e e e e a a a a a b b b b b where e e e e e e e e e e e e e e e e e e e e heorem Let α αt be an arbitrary urve in Frenet apparatu of the α an be alulated by the following equation α α α α α α α α η 5 α η α 6

urve a follow Definition Let a a a a a b b b b b be vetor in he vetor produt of a b i defined by the determinant e e e e a a a a a b b b b b where e e e e

4 7 Involute-volute Curve Couple α α α α 7 α α α α α α IV σ α α where η i taken ± to make determinant of [ ] matrix 8 9 Main Reult heorem Let be involute of ξ ξ be a W-urve in he Frenet apparatu of { σ } an be formed by apparatu of ξ { σ } Proof From definition of involute-evolute urve ouple we know ξ µ Differentiating both ide with repet to we obtain d d d µ µ d d d Definition of uh kind urve yield [] herefore we have dµ Hene µ hen we write d ξ d d aking the norm of both ide here denote derivative aording to 5 6

definition of involute-evolute urve ouple we know ξ µ Differentiating both ide with repet to we obtain d d d µ µ d d d

5 min Özyılmaz Süha Yılmaz 7 Conidering the preented method we alulate differentiation of four time We write repetively 7 [ ] { } σ 8 [ ] [ ] [ ] { } σ σ 9 Conidering we form [ ] Sine we have the prinipal normal the firt urvature of the urve [ ] Uing vetor produt of we get [ ] σ σ We obtain trinormal vetor [ ] [ ] 9 σ σ σ σ η y thi way we eaily have the eond urvature [ ] [ ] 9 σ σ 5 Conidering 9 we have the third urvature of the urve a

firt urvature of the urve [ ] Uing vetor produt of we get [ ] σ σ We obtain trinormal vetor [ ] [ ] 9 σ σ σ σ

6 7 Involute-volute Curve Couple [ ] σ σ σ 6 [ σ ] [ σ ] 9 Finally the vetor produt of follow that η { σ } 7 A where A [ σ ] [ σ ] 9 heorem Let ξ be unit peed regular urve in be involute of ξ he evolute annot be an inlined urve Proof y the definition of inlined urve we may write U oϕ where U i a ontant vetor ϕ i a ontant angle Conidering 5 we eaily have U oϕ 9 Differentiating 9 we obtain U herefore we may write U U Let u deompoe U a U u u One more differentiating of uing Frenet equation we have U whih i a ontradition hu evolute annot be an inlined urve Conluion Conidering obtained equation we expre the following reult i an orthonormal frame of Corollary { } Corollary In the ae ξ i a W-urve uffie it to ay that evolute may not be a W-urve 8 5 Open Problem In thi work we invetigate relation between involute-evolute urve ouple Frenet apparatu in the ulidean -pae In the exiting literature one an eek detail about ertr urve in the ulidean -pae Uing method of [5] with

herefore we may write U U Let u deompoe U a U u u One more differentiating of uing Frenet equation we have U whih i a ontradition hu evolute annot be an inlined urve Conluion Conidering obtained

7 min Özyılmaz Süha Yılmaz 7 an analogou way relation among Frenet apparatu of ertr urve may be determined Referene [] C oyer A Hitory of Mathemati ew York: Wiley968 [] H Gluk Higher Curvature of Curve in ulidean Spae Amer Math Monthly Vol pp [] HH Haıalihoğlu Differential Geometry urkih Ankara Univerity of Faulty of Siene [] K İlarlan Ö oyaıoğlu Poition Vetor of a Spae-like W-urve in Minkowki Spae ull Korean Math So Vol [5] A Magden Charaterization of Some Speial Curve in PhD diertation Dept Math Atatürk Univ rzurum urkey 99 [6] M urgut S Yilmaz On he Frenet Frame A Charaterization of pae-like Involute-volute Curve Couple in Minkowki Spae-time Int Math Forum Vol o

Ö oyaıoğlu Poition Vetor of a Spae-like W-urve in Minkowki Spae ull Korean Math So Vol 7 9-8 [5] A Magden Charaterization of Some Speial Curve in PhD diertation Dept Math

Smarandache Curves in Minkowski Space-time

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