Relations between a dual unit vector and Frenet vectors of a dual curve

Transcription

1 Kuwit J. Si. 4 () pp , 6 Burk Şhiner *, Mehmet Önder Dept. of Mthemtis, Mnis Cell Byr University, Murdiye, Mnis, 454, Turkey * Corresponding uthor: burk.shiner@bu.edu.tr Abstrt In this pper, we generlize the notions of helix slnt helix in dul spe in more generl se suh s the dul ngle between fixed dul unit vetor Frenet vetors of dul urve is not onstnt. We study the motions of the lines orresponding to dul Frenet vetors dul Frenet instntneous rottion vetor with respet to fixed line. We obtin some hrteriztions for suh dul urves show tht these hrteriztions inlude the definitions hrteriztions of dul helix dul slnt helix in some speil ses. Key words: Dul ngle; dul Drboux slnt helix; dul helix; dul slnt helix; motions of Frenet vetors.. Introdution In Euliden spe, the speil urves re n importnt subjet of urve theory. Generlly suh urves re thought s the urves whose urvtures stisfy some onditions. The well-known of suh urves re helies slnt helies. A helix is defined by the property tht the tngent line of the urve mkes onstnt ngle with fixed stright line (the xis of the generl helix) (Blshke, 945; Struik, 988). Therefore, generl helix n be equivlently defined s one whose tngent inditrix is plnr urve. Of ourse, there exist more speil urves in the spe. One of them is slnt helix, whih ws first introdued by Izumiy & Tkeuhi (4) by the property tht the prinipl norml lines of urve mke onstnt ngle with fixed diretion in the Euliden -spe E. Lter, Zıplr et l. () introdued the Drboux helix, whih is defined by the property tht the Drboux vetor of spe urve mkes onstnt ngle with fixed diretion they gve the hrteriztions of this new speil urve. On the other h, the dul spe is more generl spe thn Euliden spe ontins it. So, the onepts of helix, slnt helix Drboux helix n be onsidered in dul spe. Dul helix dul slnt helix hve been defined by Lee et l. (). They hve defined these dul urves by similr wy s given in Euliden se obtined the hrteriztions of these urves in dul spe. Moreover, dul vetor nlysis hs importnt pplitions to differentil geometry kinemtis (Çöken et l., 9; Kbdyı & Yylı, ; Krdğ et l, 4; Önder & Uğurlu, ). In this pper, we onsider more generl se in dul spe for speil urves. We tke fixed dul unit vetor dul urve. We study the motions of dul Frenet vetors dul Frenet instntneous rottion vetor with respet to fixed dul unit vetor. These motions orrespond to the motions of lines determinte by these dul unit vetors. We obtin the reltions between these dul vetors. Moreover, we obtin some new hrteriztions for dul helix, dul slnt helix dul Drboux helix in some speil ses.. Preliminries A dul number, s introdued by W. Clifford, n be defined s n ordered pir of rel numbers (, ), is lled the rel prt is lled the dul prt of the dul number. Dul numbers my be formlly expressed s is the dul unit whih is subjeted to the following rules (Veldkmp, 976): We denote the set of dul numbers by ID, i.e., Two inner opertions equlity on ID re defined s follows (Blshke, 945; Hıslihoğlu, 98): (i) Addition : (, ) + ( bb, ) = ( + b, + b ) (ii) Multiplition : (, )( b, b ) = ( b, b + b).

2 Burk Şhiner, Mehmet Önder 6 (iii) Equlity : (, ) = ( bb, ) = b, = b Sine division by pure dul numbers is not defined, the set ID of dul numbers with the bove opertions is ommuttive ring, not field. A dul number divided by dul number, with b, n be defined s (Blshke, 945). We n define the funtion f( ) of dul number by exping it formlly in Mlurin series with s vrible. Sine for n >, we obtin { (,, ) :, (,)} S = = ID = lled dul unit sphere (Blshke, 945; Hıslihoğlu, 98). It is well known tht the points of S (dul unit vetors) orrespond to the lines of IR. This orrespondene is known s E. Study mpping. A dul ngle, subtended by two oriented lines in spe s introdued by Study in 9, is defined s, is the projeted ngle is the shortest distne between the two lines (Blshke, 945). is urve in dul spe ID is lled dul spe urve re rel vlued urves in the rel spe IR. If the funtions, ( i ) re differentible then the dul spe urve f ( ) is derivtive of f( ) with respet to (Bottem & Roth, 979; Dimentberg, 965). In nlogy with dul numbers, dul vetor referred to n rbitrrily hosen origin n be defined s n ordered pir of vetors (, ),, IR. Also dul vetors n be expressed s =+ ε,, IR. We denote the set of dul vetors by ID, i.e., is differentible in ID. The rel prt of the dul spe urve is lled inditrix. The dul r-length of the dul spe urve from t to t is defined by { (,, ):,,,} ID i ID i = = = ID is module over the ring ID it is lled dul ε spe or ID-module. For ny dul vetors =+ b=b+ εb in ID, the slr produt the vetor produt re defined by ( ), b = b, + ε b, +, b ( ) b = b+ ε b + b, T is unit tngent vetor of the inditrix (Yüesn et l., 7). Let be dul spe urve with dul r length prmeter s. The dul unit tngent vetor of is defined by Differentiting T with respet to dul r length prmeter s we hve respetively (Veldkmp, 976; Hıslihoğlu, 98). The norm of dul vetor is given by = + ε, (Veldkmp, 976; Hıslihoğlu, 98). A dul vetor with norm is lled dul unit vetor. The set of dul unit vetors is denoted by, is lled dul urvture. We restrit tht is never pure dul number. The dul unit vetor is lled dul unit prinipl norml vetor of α. The dul unit vetor B defined by B = T N is lled dul unit binorml vetor of. The dul frme is lled moving dul Frenet frme long the dul spe urve in ID. For the urve,

3 6 the dul Frenet derivtive formule n be given in mtrix form s. The unit dul vetor d n be expressed s the liner ombintion of the elements of the dul Frenet trihedron s follows d= T + N + B (.) is lled the dul torsion of (Yüesn et l., 7). Then the dul Frenet instntneous rottion vetor (or dul Drboux vetor) of is defined by whih gives the derivtive formule s follows T = W T, N = W N, B = W B, d T T =. The unit dul Frenet instntneous ds rottion vetor of is defined by Let be unit speed dul urve. We n write this urve in nother prmetri representtion, we hve new dul Frenet equtions s follows:. Reltions between unit dul vetor the tngent vetor of dul urve In the dul spe, unit speed dul urve with urvture torsion is lled generl dul helix if there is some onstnt dul unit vetors d, so tht is onstnt long the urve is dul onstnt ngle between dul unit vetors Td, (Lee et l., ). From this definition it is ler tht both re rel onstnts. In this setion, we will onsider some more generl ses suh s one of the rel numbers is not onstnt, i.e., is not onstnt. Let be dul urve with Frenet frme d be onstnt unit dul vetor. If is the dul ngle between unit dul vetor d unit dul tngent vetor T, we n write. (.) i = i( s), ( i =,,) re differentible funtions. Seprting (.) into rel dul prts, we hve d= T+ N+ B (.) Additionlly, sine d is dul unit vetor, we hve (.4) + + = (.5) + + =. (.6) Of ourse it is more omplited to study this subjet under the ssumption tht both re not onstnts. Then, we onsider simpler wy study two ses s follows: Cse. Let be non-zero onstnt while is not onstnt. From E. Study mpping, it mens tht the lines orresponding to the vetor T mke sptil motion in the spe suh tht the ngle between the lines orresponding to the dul unit vetors d T is onstnt while the distne between these lines is not onstnt. Differentiting (.) with respet to onsidering (.6), we hve (.7) Seprting (.7) into rel dul prts, following two equtions re found s respetively. With the id of (.) (.4), we hve Differentiting (.7), we hve Seprting (.9) into rel dul prts, we hve (.8) (.9)

4 Burk Şhiner, Mehmet Önder 6 Td, + f Bd, = (.) (.) Substituting (.7), (.), (.8) (.6) into (.6) gives (.) respetively. With the id of (.) (.4), following two equtions re found s = f (.) By substituting (.9), (.), (.5) (.6) into (.), we find (.) So we hve respetively. Substituting (.) into (.5), we find + f =±. (.) Differentiting (.) substituting (.) into the result, we hve Integrting (.4), we get f =± + ( f ) is onstnt m = hve f. (.4). Sine =, we =. (.5) is onstnt. Substituting (.5) in (.) gives =. (.6) + By substituting (.6) (.5) into (.), we get From (.), we know tht = + By using (.7) (.8), we hve. (.7) (.8) (.9) Now we n give the following theorem. Theorem.. Let be dul urve, d be onstnt unit dul vetor be the dul ngle between the unit dul vetors d T. If is onstnt then holds, It is known tht the dul urve is dul helix if only if is onstnt. Then from Theorem., we n give the following orollry. Corollry.. Let be dul urve, d be onstnt unit dul vetor, be the dul ngle between d T, be onstnt. Then is dul helix if only if the following differentil eqution holds Cse. Let be non-zero onstnt while is not onstnt. From E. Study mpping it mens tht the lines orresponding to the vetor T mke sptil motion in the spe suh tht the ngle between the lines orresponding to the dul unit vetors d T is not onstnt while the distne between these lines is onstnt. By differentiting (.), we hve (.) By seprting (.) into rel dul prts using (.) (.4), we hve (.)

5 6 By substituting (.8) (.) into (.5), we find (.4) (.5) Substituting (.8), (.9), (.), (.4) (.5) into (.6), we find Differentiting (.), we hve (.6) (.7) By seprting (.7) into rel dul prts using (.) (.4), we hve (.8) holds, From Theorem., we n give the following orollry. Corollry.. Let be dul urve, d be onstnt unit dul vetor, be the dul ngle between d T, be onstnt. Then is dul helix if only if (.9) From (.8), we find (.) re defined s given in (.8) (.5), respetively. Substituting (.9), (.5), (.6) (.) into (.9), we hve From (.) (.), we find (.) 4. Reltions between unit dul vetor the prinipl norml vetor of dul urve In dul spe, unit speed urve is lled dul slnt helix, if its unit dul prinipl norml vetor N mkes onstnt ngle with fixed diretion in unit dul vetor d ; tht is, is onstnt long the urve (Lee et l, ). From this definition it is ler tht both re rel onstnts. In this setion, we will onsider some more generl ses suh s one of the rel numbers is not onstnt, i.e., is not onstnt. Let be dul urve, d be onstnt unit dul vetor be the dul Frenet trihedron of. If is the dul ngle between the unit dul vetor d the unit dul prinipl norml vetor N, we n write (4.) Now we n give the following theorem. Theorem.. Let be dul urve, d be onstnt unit dul vetor be the dul ngle between the unit dul vetors d T. If is onstnt then. Of ourse it is more omplited to study this subjet under the ssumption tht both re not onstnts. Then, we onsider simpler wy study two ses s follows: Cse. Let be non-zero onstnt while is not onstnt. From E. Study mpping, it mens tht the lines orresponding to the vetor N mke sptil

6 Burk Şhiner, Mehmet Önder 64 motion in the spe suh tht the ngle between the lines orresponding to the dul unit vetors d N is onstnt while the distne between these lines is not onstnt. Differentiting (4.), we hve re defined s (4.8) (4.7), respetively. From (4.5) (4.9), we get (4.) Seprting (4.) into rel dul prts using (.) (.4), we hve + = (4.) f (4.4) Now we n give the following theorem. Theorem 4.. Let be dul urve, d be onstnt unit dul vetor be the dul ngle between the unit dul vetor d the unit dul prinipl norml vetor N. If is onstnt then Sine (4.) (.) re the sme, we find the following eqution whih we hve found in Setion s f ( + f ) =± m. m =. Integrting the bove eqution, we hve is n integrtion onstnt. The integrtion onstnt n be disppered with prmeter hnge. So we get (4.5) From Theorem 4. definitions of dul helix dul slnt helix, we n give the following orollries. Corollry 4.. Let be dul urve, d be onstnt unit dul vetor, be the dul ngle between d N, be onstnt. is dul helix if only if m =. From (4.), we hve known tht (4.6) Substituting (4.) (4.6) into (.5), we hve Corollry 4.. Let be dul urve, d be onstnt unit dul vetor, be the dul ngle between d N, be onstnt. is dul slnt helix if only if (4.7) f is defined s (4.5). From (4.4), we find (4.8) (4.9) Cse. Let be non-zero onstnt while is not onstnt. From E. Study mpping, it mens tht the lines orresponding to the vetor N mke sptil motion in the spe suh tht the distne between the lines orresponding to the dul unit vetors d N is onstnt while the ngle between these lines is not onstnt. Differentiting (4.), we hve

7 65 (4.) Seprting (4.) into rel dul prts using (.) (.4), we get From (4.) (4.), we get, respetively, Now we n give the following theorem. (4.) (4.) Theorem 4.. Let be dul urve, d be onstnt unit dul vetor be the dul ngle between the unit dul vetor d the unit dul norml vetor N. If is onstnt then From Theorem 4. definitions of dul helix dul slnt helix, we n give the following orollries. Corollry 4.. Let be dul urve, d be onstnt unit dul vetor, be the dul ngle between d N, be onstnt. is dul helix if only if 5. Reltions between unit dul vetor the binorml vetor of dul urve In the dul spe unit speed urve is lled dul B -slnt helix, if its unit dul binorml vetor B mkes onstnt ngle with fixed unit dul diretion d ; tht is, is onstnt long the urve (Lee et l., ). From this definition it is ler tht both re rel onstnts. In this setion, we will onsider the reltion between the unit dul vetor d unit dul binorml vetor B, in the se tht the dul ngle between them is not onstnt. Let be dul urve, d be onstnt unit dul vetor be the dul Frenet trihedron of. If is the dul ngle between the unit dul vetors d B, we n write (5.) Cse. Let be non-zero onstnt while is not onstnt. From E. Study mpping, it mens tht the lines orresponding to the vetor B mke sptil motion in the spe suh tht the ngle between the lines orresponding to the dul unit vetors d B is onstnt while the distne between these lines is not onstnt. Differentiting (5.), we hve (5.) Seprting (5.) into rel dul prts using (.) (.4), we get (5.) Assuming tht f is not equl to zero, from (5.), we hve =. (5.4) Differentiting (5.) seprting the result into rel dul prts, we hve f f + f = (5.5) Corollry 4.4. Let be dul urve, d be onstnt unit dul vetor, be the dul ngle between d N, be onstnt. is dul slnt helix if only if By substituting (5.4) into (5.5), we get (5.6) = f (5.7)

8 Burk Şhiner, Mehmet Önder 66 Sine (5.7) equls to (.) =, we get f = (5.8) is onstnt. Using (.), we find =. (5.9) + Substituting (5.8) (5.9) into (5.7), we get = + From (5.), we hve known Using (5.9) (5.), we find. (5.) (5.) (5.) Cse. Let be non-zero onstnt while is not onstnt. From E. Study mpping, it mens tht the lines orresponding to the vetor B mke sptil motion in the spe suh tht the distne between the lines orresponding to the dul unit vetors d B is onstnt while the ngle between these lines is not onstnt. By differentiting (5.), we hve (5.5) Seprting (5.5) into rel dul prts using (.) (.4), we get (5.6) Substituting (5.4), (5.9), (5.) (5.) into (.6), we hve (5.) Substituting (5.4), (5.8), (5.9), (5.), (5.) (5.) into (5.6) onsidering tht f =, we find From (5.6), we find By substituting (5.8) into (5.7), we hve (5.7) (5.8) From (5.8) (5.4), we get (5.4) From (5.8) (5.9), we find (5.9) Now we n give the following theorem. Theorem 5.. Let be dul urve, d be onstnt unit dul vetor be the dul ngle between unit dul vetor d unit dul binorml vetor B. If is onstnt then Now we n give the following theorem. Theorem 5.. Let be dul urve, d be onstnt unit dul vetor be the dul ngle between the unit dul vetor d the unit dul binorml vetor B. If is onstnt then holds, From Theorem 5., we n give the following orollry. Corollry 5.. Let be dul urve, d be onstnt unit dul vetor, be the dul ngle between d B, be onstnt. Then is dul helix if only if the following differentil eqution holds From Theorem 5., we n give the following orollry. Corollry 5.. Let be dul urve, d be onstnt unit dul vetor, be the dul ngle between d B, be onstnt. is dul helix if only if

9 67 Theorem 6.. Let be dul urve, d be onstnt unit dul vetor be non-onstnt dul ngle between unit dul vetor d the unit dul Frenet instntneous rottion vetor W, then 6. Reltions between unit dul vetor the dul Frenet instntneous rottion vetor of dul urve In this setion, we will onsider the reltion between fixed unit dul vetor d dul unit Frenet instntneous rottion vetor W. First, we give the following definition. Definition 6.. In the dul spe unit speed urve is lled dul Drboux-slnt helix if its unit dul Drboux vetor (dul Frenet instntneous rottion vetor) W mkes onstnt dul ngle with fixed unit dul diretion d ; tht is, is onstnt long the urve. Then we hve re onstnts. From this definition it is ler tht both re rel onstnts. Now, we will onsider more generl se tht the dul ngle between these dul unit vetors is not onstnt. Then (6.) is not onstnt. Seprting (6.) into rel dul prts, we hve (6.) holds, From Theorem 6., we n give the following orollry. Corollry 6.. Let unit dul vetor d W. Then, be dul urve, d be onstnt be dul ngle between is dul helix if only if Let now onsider the speil se tht Differentiting (6.), we hve f =. is onstnt. Sine f n not be found in the se of =, we will ssume tht f =. So, we get f = (6.4) is onstnt. By substituting (6.4) into (6.), we find From (6.4) (6.5), we hve (6.5) (6.) From (6.) (6.), we find Now we n give the following theorem. Theorem 6.. Let be dul urve, d be onstnt unit dul vetor be dul ngle between unit dul vetors d W. If is onstnt then Thus, we get holds, is onstnt. From Theorem 6., we n give the following orollry.

10 Burk Şhiner, Mehmet Önder 68 Corollry 6.. Let be dul urve, d be onstnt unit dul vetor, be dul ngle between d W be onstnt. Then, is dul helix if only if is onstnt. Furthermore, if is onstnt in Theorem 6., we hve tht is dul Drboux slnt helix. Then we hve the following orollry. Corollry 6.. If holds, 7. Conlusions is dul Drboux slnt helix, then is rel onstnt. The study of motion of line in spe is n importnt reserh re in kinemtis. This study n be hieved more effiiently by using dul vetor lgebr obtined results n be interpreted. This pper is n exmple of suh studies of speil motions. In this pper, the motions of two lines orresponding to fixed unit dul vetor dul Frenet vetors of dul urve re studied in detil. A generl se of this motion suh tht the dul ngle between dul unit vetors is not onstnt is studied. Lter, some speil ses re onsidered hrteriztions for dul helix, dul slnt helix dul Drboux slnt helix re obtined. Çöken, A.C., Ekii, C., Koyusufoğlu, İ. & Görgülü, A. (9). Formuls for dul-split quternioni urves. KJSE- Kuwit Journl of Siene & Engineering,. 6(A):-4. Dimentberg, F.M. (965). The srew lulus its pplition in mehnis. Izdt, Nuk, Moskow, USSR. English trnsltion: AD6899, Cleringhouse for Federl Sientifi Tehnil Informtion. Hıslihoğlu, H.H. (98). Hreket geometrisi ve kuterniyonlr teorisi. Gzi Üniversitesi, Fen-Edebiyt Fkültesi, Ankr. Izumiy, S. & Tkeuhi, N. (4). New speil urves developble surfes. Turkish Journl of Mthemtis, 8:5-6. Kbdyı, H. & Yylı, Y. (). De Moivre s formul for dul quternions. Kuwit Journl of Siene & Engineering, 8(A):5-. Krdğ, H.B., Kılıç, E. & Krdğ, M. (4). On the developble ruled surfes kinemtilly generted in Minkowski -spe. Kuwit Journl of Siene, 4():-4. Lee, J.W., Choi, J.H. & Jin, D.H. (). Slnt dul Mnnheim prtner urves in the dul spe. Interntionl Journl of Contemporry Mthemtil Sienes, 6(): Önder, M. & Uğurlu, H.H. (). Norml spheril urves in dul spe. Mediterrnen Journl of Mthemtis, : Struik, D.J. (988). Letures on lssil differentil geometry. nd ed. Addison Wesley, Dover. Veldkmp, G.R. (976). On the use of dul numbers, vetors mtries in instntneous sptil kinemtis. Mehnism Mhine Theory, :4-56. Yüesn, A., Ayyıldız, N. & Çöken, A.C. (7). On retifying dul spe urves. Revist Mthemti Complutense, (): Zıplr, E., Şenol, A. & Yylı, Y. (). On Drboux helies in Euliden -spe. Globl Journl of Siene Frontier Reserh Mthemtis Deision Sienes, ():7-8. Submitted : 9//4 Revised : //5 Aepted : 7//5 Referenes Blshke, W. (945). Vorlesungen uber differentil geometrie, Bd. Dover Publ., New York. Bottem, O. & Roth, B. (979). Theoretil kinemtis. North-Holl Publ. Co. Amsterdm. Pp. 556.

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