Split quaternions and time-like constant slope surfaces in Minkowski 3-space

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1 Slit quaterios ad time-like costat sloe surfaces i Mikowski -sace Murat Babaarsla a, * Yusuf Yayli b a Deartmet of Mathematics, Bozok Uiersity, 6600, Yozgat-Turkey b Deartmet of Mathematics, Akara Uiersity, 0600, Akara-Turkey Abstract I this aer, we roe that time-like costat sloe surfaces ca be rearametrized by usig rotatio matrices corresodig to uit time-like slit quaterios ad homothetic motios. Afterwards we gie some examles to illustrate our mai results by usig Mathematica. Mathematics Subject Classificatio. 40, 5A7, 5A5. Key words: Time-like costat sloe surface, slit quaterios, homothetic motio. *Corresodig Author: murat.babaarsla@bozok.edu.tr. Itroductio We ca fid a lot of methods to costruct rotatios, for examle; orthogoal matrices, Euler agles, Rodrigues rotatioal formulae which is a good solutio to rotatio about a arbitrary axis, see [7] ad quaterios. I betwee them, quaterios are more efficiet tools for describig rotatios about a arbitrary axis sice we ca obtai easily rotatio matrices by usig quaterios. Thus they are usually used i robotics, comuter grahics, aimatios, aerosace alicatios, fractals ad irtual reality (see [], []). I Comuter Aided Geometric Desig (CAGD), B-Slie ad NURBS cures are well kow geometric modelig tools. Also trigoometric cures ca costruct a lot of imortat

2 cures due to their trigoometric basis fuctios, for examle; circle, circular cylider, rotatioal surfaces ad helix which makes a costat agle with a fixed directio, see [] ad refereces therei. If we wat to gie a differet examle, certaily, we should thik costat sloe surface which makes a costat agle with the ositio ector. These surfaces ca be thought as a geeralizatio of helices which arise i DNA double, collage trile helix, ao-srigs, carbo ao-tubes, helical staircases, helical structures i fractal geometry ad so o (see [5]). Muteau [9] showed that a costat sloe surface ca be costructed by usig a uit seed cure o the Euclidea -shere ad a equiagular siral. We kow that a rotatioal surface ca be obtaied by usig a rotatio matrix. I [], we demostrated that costat sloe surfaces ca be obtaied by usig quaterios. After that i [], we foud the relatios betwee sace-like costat sloe surfaces [] ad slit quaterios i Mikowski -sace. Fu ad Wag [4] studied time-like costat sloe surfaces ad classified these surfaces i. They showed that ad oly if it ca be arametrized by S is a time-like costat sloe surface lyig i the time-like coe if x( u, ) u cosh cosh ( u) f ( ) sih ( u) f ( ) f ( ), (.) where, is a ositie costat agle fuctio, ( u) tah l u ad f is a uit seed sace- like cure o seudo-hyerbolic sace. S is a time-like costat sloe surface lyig i the sace-like coe if ad oly if it ca be arametrized by x( u, ) u si cos ( u) g( ) si ( u) g( ) g( ), (.) where, is a ositie costat agle fuctio satisfyig ( u) cot l u ad g is a uit seed time-like cure o seudo-shere Also,. S is a time-like costat sloe surface lyig i the sace-like coe if ad oly if it ca be arametrized by

3 x( u, ) u sih cosh ( u) h( ) sih ( u) h( ) h( ), (.) where, is a ositie costat agle fuctio, ( u) coth l u ad h is a uit seed sacelike cure o seudo-shere. I light of recet studies, the urose of this aer is to gie some relatios betwee slit quaterios ad time-like costat sloe surfaces i. We will show that time-like costat sloe surfaces ca be rearametrized by usig rotatio matrices corresodig to uit time-like slit quaterios ad homothetic motios. Also we will gie some examles to stregthe the both this aer ad [4] by usig Mathematica.. Prelimiaries I this sectio, we recall some imortat cocets ad formulas regardig slit quaterios ad semi-euclidea saces. A slit quaterio ca be writte as i j k, 4 where,,, 4 ad i, j, k are slit quaterio uits which satisfy the ocommutatie multilicatio rules i, j k, ij ji k, jk k j i ad ki i k j. Let us deote the algebra of slit quaterios by ad its atural basis by {, i, j, k }. A elemet of is called a slit quaterio. For a slit quaterio i j 4k, the cojugate of is defied by i j k 4 [6]. Scalar ad ector arts of a slit quaterio are deoted by S ad V i j 4 k, resectiely. The slit quaterio roduct of two slit quaterios (,,, 4) ad q q q q q4 where (,,, ) is defied as q q V, V V qv V V, (.) q q q

4 ad V, V q q q q 4 4 i j k V Vq 4 ( 4q q4 ) i + ( 4q q4 ) j + ( q q ) k q q q 4 If S 0, the is called as a ure slit quaterio []. Defiitio.. If is equied with the metric tesor u, u u, i i i i i i it is called semi-euclidea sace ad deoted by, where is called the idex of the metric, 0 ad u,. If 0, semi-euclidea sace called Mikowski -sace [0]. is reduced to. For, is Defiitio.. A ector w is called (i) sace-like if w, w 0 or w 0, (ii) time-like if w, w 0, (iii) light-like (ull) if w, w 0 ad w 0. The orm of a ector w is w, w. Two ectors w ad w i are said to be orthogoal if w, w 0 [0]. We ca defie seudo-shere ad seudo-hyerbolic sace i as follows: ad Also, for ad 0, (,..., ), i i i i (,..., ). i i i i is called a seudo-hyerbolic sace of. 0 4

5 It ca be show that Therefore we idetify with semi- Euclidea sace 4 [6]. 4. Thus we ca defie time-like, sace-like ad light-like slit quaterios as follows: Defiitio.. A slit quaterio is sace-like, time-like or light-like, if I 0, I 0 or I 0, resectiely, where I []. 4 Defiitio.4. The orm of (,,, 4 ) is defied as N 4. If N, the is called as a uit slit quaterio ad 0 / N is a uit slit quaterio for N 0. Also sace-like ad time-like slit quaterios hae multilicatie ierses haig the roerty ad they are costructed by I /. Light-like slit quaterios hae o ierses []. The set of sace-like slit quaterios is ot a grou because it is ot closed uder multilicatio. Therefore we are oly iterested i time-like slit quaterios. The ector art of ay time-like slit quaterio ca be sace-like or time-like. Trigoometric forms of them are as below: (i) Eery time-like slit quaterio with the sace-like ector art ca be writte as where w is a uit sace-like ector i N (cosh w sih ),. (ii) Eery time-like slit quaterio with the time-like ector art ca be writte as where w is a uit time-like ector i N (cos w si ), [8, ]. Uit time-like slit quaterios are used to costruct rotatios i If 4 : (,,, ) is a uit time-like slit quaterio, usig the trasformatio law ( ) i Rij j, j V V the corresodig rotatio matrix ca be foud as 5

6 4 4 4 R 4 4 4, where V is a slit quaterio. This is ossible with uit time-like slit quaterios. Also causal character of ector art of the uit time-like slit quaterio is imortat. If the ector art of is time-like or sace-like the the rotatio agle is sherical or hyerbolic, resectiely []. I, oe-arameter homothetic motio of a body is geerated by the trasformatio Y ha C X, 0 where X ad C are real matrices of tye ad h is a homothetic scale ad A SO(). A, h ad C are differetiable fuctios of C class of a arameter t (see []). (.). Slit quaterios ad time-like costat sloe surfaces lyig i the time-like coe A uit time-like slit quaterio with the sace-like ector art ( u, ) cosh( ( u) / ) sih( ( u) / ) f ( ) defies a -dimesioal surface o seudo-hyerbolic sace, where ( u) tah l u, is a ositie costat agle fuctio, f ( f, f, f ) ad f is a uit seed sace-like cure o. Thus, usig (.), the corresodig rotatio matrix ca be foud as cosh sih ( f f f ) sih f f sih f sih f f sih f R sih f f sih f cosh sih ( f f f ) sih f f sih f. sih f f sih f sih f f sih f cosh sih ( f f f ) (.) Now we gie the relatio betwee uit time-like slit quaterios with the sace-like ector arts ad time-like costat sloe surfaces lyig i the time-like coe. 6

7 Theorem.. Let x : S be a time-like costat sloe surface immersed i Mikowski - sace ad x lies i the time-like coe. The the time-like costat sloe surface S ca be rearametrized by x( u, ) ( u, ) ( u, ), where ( u, ) cosh ( u) sih ( u) f ( ) is a uit time-like slit quaterio with the sacelike ector art, ( u, ) u cosh f ( ) is a surface ad a ure slit quaterio i uit seed sace-like cure o. ad f is a Proof. Sice ( u, ) cosh ( u) sih ( u) f ( ) ad ( u, ) u cosh f ( ), we hae u cosh cosh ( u) sih ( u) f ( ) f ( ) ( u, ) ( u, ) cosh ( u) sih ( u) f ( ) u cosh f ( ) u cosh cosh ( u) f ( ) u cosh sih ( u) f ( ) f ( ). By usig (.), we hae f f f, f f f. (.) Sice f is a uit seed sace-like cure o Substitutig (.) ito (.), we obtai, we get f f f f. (.) ( u, ) ( u, ) u cosh cosh ( u) f ( ) sih ( u) f ( ) f ( ). Thus, usig (.), we coclude that This comletes the roof. ( u, ) ( u, ) x( u, ). From Theorem., we ca see that the time-like costat sloe surface S lyig i the time-like coe is the slit quaterio roduct of -dimesioal surfaces ( u, ) o ( u, ) i. ad We ow cosider the relatios amog rotatio matrices R, homothetic motios ad time-like costat sloe surfaces lyig i the time-like coe. We hae the followig results of Theorem.. 7

8 Corollary.. Let R be the rotatio matrix corresodig to the uit time-like slit quaterio with the sace-like ector art ( u, ). The the time-like costat sloe surface S ca be writte as x( u, ) R ( u, ). Corollary.. For the homothetic motio ( u, ) u cosh ( u, ), the time-like costat sloe surface S ca be rearametrized by x( u, ) ( u, ) f ( ). Therefore, we hae Thus, we ca gie the followig examle: x( u, ) u cosh R f ( ). (.4) Examle.4. We cosider a uit sace-like cure o f ( ) (cosh,0,sih ). If we take 7 ad tah 7, the homothetic motio is defied by ( u, ) u cosh 7 cosh(l u) sih(l u)sih, 0, sih(l u) cosh. Usig (.4), we hae the followig time-like costat sloe surface lyig i the time-like coe: lu lu lu cosh sih cosh sih(l u)cosh sih sih cosh x( u, ) ucosh7 sih(l u)cosh cosh(l u) sih(l u)sih 0 lu lu lu sih sih sih sih(l u)sih cosh sih cosh ad the Thus, the icture of x( u, ) u cosh R f ( ) : ucosh7cosh(l u)cosh x( u, ) ucosh7sih(l u). ucosh7cosh(l u)sih 8

9 Figure : Time-like costat sloe surface lyig i the time-like coe x( u, ) ucosh R f ( ), f ( ) (cosh,0,sih ), 7. We ca gie the followig remarks regardig Theorem. ad Corollary.. Remark.5. Theorem. says that the ositio ectors o the surface ( u, ) are rotated by ( u, ) through the hyerbolic agle ( u) about the sace-like axis sa{ f ( )}. Remark.6. Corollary. says that the ositio ector of the sace-like cure f ( ) is rotated by ( u, ) through the hyerbolic agle ( u) about the sace-like axis sa{ f ( )} ad exteded through the homothetic scale u cosh. 4. Slit quaterios ad time-like costat sloe surfaces lyig i the sacelike coe As Sectio, we cosider a uit time-like slit quaterio with the time-like ector art 9

10 ( u, ) cos( ( u) / ) si( ( u) / ) g( ), where ( u) cot l u, is a ositie costat agle fuctio satisfyig g ( g, g, g) ad g is a uit seed time-like cure o corresodig rotatio matrix ca be foud as. Thus, usig (.), the cos si ( g g g ) si g g si g si g g si g R si g g si g cos si ( g g g ) si g g si g, si g g si g si g g si g cos si ( g g g ) (4.) Now we gie the relatio betwee uit time-like quaterios with the time-like ector arts ad the time-like costat sloe surfaces lyig i the sace-like coe. Theorem 4.. Let x : S be a time-like costat sloe surface immersed i Mikowski - sace ad x lies i the sace-like coe. The the time-like costat sloe surface S ca be rearametrized by x( u, ) ( u, ) ( u, ), where ( u, ) cos ( u) si ( u) g( ) is a uit time-like slit quaterio with time-like ector art, ( u, ) u si g( ) is a surface ad a ure slit quaterio i seed time-like cure o. ad g is a uit Proof. Sice ( u, ) cos ( u) si ( u) g( ) ad ( u, ) u si g( ), we obtai u si cos ( u) si ( u) g( ) g( ) ( u, ) ( u, ) cos ( u) si ( u) g ( ) u si g( ) u si cos ( u) g( ) u si si ( u) g( ) g( ). By usig (.), we hae gg g, g g g. (4.) Sice g is a uit seed time-like cure o, we hae gg g g. (4.) 0

11 Substitutig (4.) ito (4.) gies ( u, ) ( u, ) u si cos ( u) g( ) si ( u) g( ) g( ). Therefore alyig (.), we coclude that Thus the roof is comleted. ( u, ) ( u, ) x( u, ). From Theorem 4., we ca see that the time-like costat sloe surface S lyig i the sace-like coe is the slit quaterio roduct of -dimesioal surfaces ( u, ) o ( u, ) i. ad Let us cosider the relatios amog rotatio matrices R, homothetic motios ad timelike costat sloe surfaces lyig i the sace-like coe. We hae the followig results of Theorem 4.. Corollary 4.. Let R be the rotatio matrix corresodig to the uit time-like slit quaterio with time-like ector art ( u, ). The the time-like costat sloe surface S ca be writte as x( u, ) R ( u, ). Corollary 4.. For the homothetic motio ( u, ) u si ( u, ), the time-like costat sloe surface S ca be rearametrized by x( u, ) ( u, ) g( ). Therefore, we hae Also, we ca gie a examle: x( u, ) u si R g( ). (4.4) Examle 4.4. We cosider a uit time-like cure o If we take / 4, the homothetic motio is defied by g( ) (sih,0,cosh ). ( u, ) u cos(l u) si(l u) cosh, 0, si(l u) sih. By usig (4.4), we hae the followig time-like costat sloe surface lyig i the sace-like coe:

12 lu lu lu lu cos si cosh si sih si sih sih x( u, ) u si(l u)sih cosh(l u) si(l u)cosh 0 lu lu lu si sih si(l u)cosh cos si cosh cosh ad the u cos(l u )sih x( u, ) usi(l u). u cos(l u )cosh Thus, the icture of x( u, ) u si R g( ) : Figure : Time-like costat sloe surface lyig i the sace-like coe x( u, ) usi R g( ), g( ) (sih,0,cosh ), / 4.

13 We ca gie the followig remarks regardig Theorem 4. ad Corollary 4.. Remark 4.5. Theorem 4. says that the ositio ectors o the surface ( u, ) are rotated by ( u, ) through the sherical agle ( u) about the time-like axis sa{ g ( )}. Remark 4.6. Corollary 4. says that the ositio ector of the time-like cure g( ) is rotated by ( u, ) through the sherical agle ( u) about the time-like axis sa{ g ( )} ad exteded through the homothetic scale u si. Also, we ca gie the relatio betwee uit time-like slit quaterios with the sace-like ector arts ad the time-like costat sloe surfaces lyig i the sace-like coe. We cosider a uit time-like slit quaterio with the sace-like ector art ( u, ) cosh( ( u) / ) sih( ( u) / ) h( ), where ( u) coth l u, is a ositie costat agle fuctio, h ( h, h, h ) ad h is a uit seed sace-like cure o as. Thus, usig (.), the corresodig rotatio matrix ca be foud cosh sih ( h h h ) sih h h sih h sih h h sih h R sih h h sih h cosh sih ( h h h ) sih h h sih h, sih h h sih h sih h h sih h cosh sih ( h h h) (4.5) Theorem 4.7. Let x : S be a time-like costat sloe surface immersed i Mikowski - sace ad x lies i the sace-like coe. The the time-like costat sloe surface S ca be rearametrized by x( u, ) ( u, ) ( u, ), where ( u, ) cosh ( u) sih ( u) h( ) is a uit time-like slit quaterio with sace-like ector art, ( u, ) u sih h( ) is a surface ad a ure slit quaterio i ad h is a uit

14 seed sace-like cure o. Proof. Sice ( u, ) cosh ( u) sih ( u) h( ) ad ( u, ) u sih h( ), we obtai u sih cosh ( u) sih ( u) h( ) h( ) ( u, ) ( u, ) cosh ( u) sih ( u) h ( ) u sih h( ) u sih cosh ( u) h( ) u sih sih ( u) h( ) h( ). By usig (.), we get hh h, h h h. (4.6) Sice h is a uit seed sace-like cure o Substitutig (4.7) ito (4.6) gies, we hae hh h h. (4.7) ( u, ) ( u, ) u sih cosh ( u) h( ) sih ( u) h( ) h( ). Therefore alyig (.) we coclude that This comletes the roof. ( u, ) ( u, ) x( u, ). We hae the followig results of Theorem 4.7. Corollary 4.8. Let R be the rotatio matrix corresodig to the uit time-like slit quaterio with sace-like ector art ( u, ). The the time-like costat sloe surface S ca be writte as x( u, ) R ( u, ). Corollary 4.9. For the homothetic motio ( u, ) u sih ( u, ), the time-like costat sloe surface S ca be rearametrized by x( u, ) ( u, ) h( ). Therefore, we hae We ca gie the followig examle: x( u, ) u sih R h( ). (4.8) Examle 4.0. Let us cosider a uit sace-like cure o h( ) (0,cos,si ). 4 defied by

15 Takig 7 ad coth 7, the homothetic motio is equal to ( u, ) u sih 7 cosh(l u) 0,sih(l u) si, sih(l u) cos. By usig (4.8), we hae the followig time-like costat sloe surface lyig i the sace-like coe: ad the cosh(l u) sih(l u)cos sih(l u)si 0 lu lu lu cos si lu lu lu sih(l u)si sih si cosh sih cos x( u, ) usih7 sih(l u)cos cosh sih cos sih si usih7sih(l u) x( u, ) usih7cosh(l u)cos. usih7cosh(l u)si Thus, we ca draw the icture of x( u, ) u sih R h( ) as follows: Figure : Time-like costat sloe surface lyig i the sace-like coe x( u, ) u sih R h( ), h( ) (0,cos,si ), 7. 5

16 We ca gie the followig remarks regardig Theorem 4.7 ad Corollary 4.9. Remark 4.. Theorem 4.7 says that the ositio ectors o the surface ( u, ) are rotated by ( u, ) through the hyerbolic agle ( u) about the sace-like axis sa{ h ( )}. Remark 4.0. Corollary 4.9 says that the ositio ector of the sace-like cure h( ) is rotated by ( u, ) through the hyerbolic agle ( u) about the sace-like axis sa{ h ( )} ad exteded through the homothetic scale u sih. Refereces [] BABAARSLAN. M., YAYLI, Y., A ew aroach to costat sloe surfaces with quaterios, ISRN Geometry, Vol. 0, Article ID 658, 8 ages, Doi:0.540/0/658, 0. [] BABAARSLAN. M., YAYLI, Y., Slit quaterios ad sace-like costat sloe surfaces i Mikowski -sace, It. J. Geom.,, (0) -. [] FU, Y., YANG, D., O costat sloe sace-like surfaces i -dimesioal Mikowski - sace, J. Math. Aal. Al., 85, (0) [4] FU, Y., WANG, X., Classificatio of time-like costat sloe surfaces i -dimesioal Mikowski sace. Results. Math., 0, Doi: 0.007/s [5] ILARSLAN, K., BOYACIOGLU, O., Positio ectors of a time-like ad a ull helix i Mikowski -sace, Chaos, Solitos Fractals, 8, (008) [6] INOGUCHI, J., Time-like surfaces of costat mea curature i Mikowski -sace, Tokyo J. Math., (998)

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