Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ Spl Curvs of D ll Sp Mhm Bkş Mhmu Ergü Alpr Osm Öğrmş Fır Uvrsy Fuly of S Dprm of Mhms 9 Elzığ Türky mks@fr.du.r mrgu@fr.du.r ogrmslpr@gml.om Asr. Spl urvs d hr hrrzos r o of h m r of mhms d physs. I h prs ppr w df Mhm urvs for -dmsol ll sp d vsg som hrrzo of. Kywords: ll Sp Mhm Curvs Fr Formul Iroduo I lssl dffrl gomry hr r my works rld wh Brrd d Mhm urvs [-5]. W s mos suds proprs of Brrd d Mhm urvs whh ssrs h s of lr rlo w urvurs. I r yrs mhms hv gu o vsg urvs d surfs ll sp [6-]. A sp urv Euld -sp s lld mhm urv f d oly f for som oss ssfs h followg rlo k k k whr k d k r urvur d orso rspvly. Our work s orgzd s follows: I so som s proprs of ll sp r gv whh wll usd h lr sos. I so w gv som proprs of Mhm urvs D ll sp. Th orrspodg uhor.
Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ Prlmrs hps://ss.googl.om/s/jmsjourl/ Th ll sp s D ompl projv sp P whh h solu fgur w f I I } osss of rl pl w h solu pl rl l { f w h solu l d wo ompl ojug pos I I f h solu pos. Th sudy of mhs of pl-prlll moos rdus o h sudy of gomry of D sp wh oords { y } r gv y h moo formul []. Ths gomry s lld D ll gomry. Dffrl gomry of h ll sp hs lrgly dvlopd [7]. I [] s pld h D ll gomry whh suds ll proprs vr udr moos of ojs sp s v mor ompl. I ddo s sd h hs gomry dsrd mor prsly s h sudy of hos proprs of D sp wh oords whh r vr udr h grl ll rsformos s follows: y z os os os s s s os os os s y s s z v os os s os s os s s os os os y s os z v os s s s os y os z v os d wh os os. os Som fudml proprs of urvs D ll sp s gv for h purpos of h rqurms h so A urv I R s gv s follows y z w y z w C smooh fuos d I whr. L urv whh s prmrzd y rlgh s d gv h followg oord form s s y s z s w s.
Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ I ff oords h ll slr produ w wo pos P s dfd y. f f { P P g For h vors d ll ross produ s dfd s follows: whr r h sdrd ss vors. I hs ppr w do h r produ of wo vors h ss of ll y h oo. L s w s z s y s s urv prmrzd y rlgh s. For Fr urv h Fr formuls gv s followg form. o W kow h r muully orhogol vor flds ssfyg quos.
Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ Mhm Curvs ll Sp I [5] Mhm urvs for Euld -sp r grlzd. I hs ppr w hv vsgd grlzo of h urvs D ll sp. Dfo.. A spl urv s lld grlzd Mhm urv f hr ss spl Fr urv suh h h frs orml l h po of s ludd h pl grd y h sod d h hrd orml l of h orrspodg po udr. Hr w do y jo from o. Th h urv s lld h grlzd Mhm m urv of. A grlzd Mhm m urv s gv y h mp : I whh ssfs h followg quo I.. Hr w do smooh fuo o I y. Th prmr should o rlgh of. Th rlgh of dfd y Whr d d d s h rlgh of. s gv y f w hv For smooh fuo f : I I f d d d d for I. Th rprso of urv s wh rlgh prmr : I For h jo rprmrzo of. : dfd y f h s gv y h followg quo
Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ f I whr s smooh fuo o I. Th w o d f d d d f f I. Thorm.. If spl Fr urv s grlzd Mhm urv h frs urvur fuo d sod urvur fuo ssfs h followg quo I. hr w do os umr wh. Proof. I h followg shm w show s grlzd Mhm urv d s grlzd Mhm m urv of. f : : I I W df smooh fuo. f y f d d d s h rlgh prmr of. I ddo s jo h s dfd y f. Th h urv s rprmrzd s followg form f I. whr : I R R s smooh fuo d { } d { } r orhogol vor flds log d rspvly. Dffrg oh sds of quo. wh rsp o w g f.. O h ohr hd s h frs orml l h h po of s lyg h pl grd y h sod d h hrd orml l of 5
Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ h orrspodg pos udr jo s od s follows h vor fld g f h f whr g d h r som smooh fuos o I R. Tkg o ou of h followg quo f g f h f d usg. w hv. Th w dompos h quo. s follows h s f f f whr f. Dffrg oh sds of h quo.6 wh rsp o w o f f..5.6 I.7 Usg f g f h f h off of quo.7 vshs h s. Th h proof s ompld. Thorm.. L spl Fr urv suh h s oos frs d sod urvur fuos ssfy h followg quo for ll I R. If h spl Fr urv followg form h s grlzd Mhm m urv of. gv y h 6
Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ Proof. Th rlgh prmr of s gv y h quo d d d L us ssum h I. h w o f I. Dffrg h quo f h w g wh rsp o f f. Th w s f I..8 Dffrg h ls quo wh rsp o s f f.9. From h ssumpo w o. Th h off of h quo.9 s zro. O s from h quo.8 h f s lr omo of d. I ddo from quo.9 f s gv y lr omo of d. O h ohrhd s spl Fr urv h h vor whh ssfs h followg lr omo of f d f. Thrfor h frs orml l ls h pl grd y h sod d hrd orml l of h orrspodg pos udr h jo whh s dfd y f f. Th proof s ompld. 7
Irol Jourl of Mhml Egrg d S ISSN : 77-698 Volum Issu Mrh hp://www.jms.om/ hps://ss.googl.om/s/jmsjourl/ Rmrk.. I D ll sp spl Fr urv wh urvur fuos d ssfyg s o lr h smooh urv gv y. s spl Fr urv. Th rvrs of Thorm. s sll gr puzzl for h uhors. Rfrs. H. Blgr M. Bkş d J. Ioguh Null Brrd urvs Mkowsk -sp d hr hrrzos No Mh. o. 7-. FN. Ekmk d K. İlrsl O Brrd urvs d hr hrrzo Dffr. om. Dy. Sys.lro vol. o.. R. Blum A rmrkl lss of Mhm urvs Cd. Mh. Bull. 9-8 966. H. Lu d F. Wg Mhm Prr urvs -sp Jourl of omry 88-6 8 5. H. Msud d S. Yorozu O grlzd Mhm urvs Euld - sp Eglsh Nhok Mh. J. o. -56 9 6. S.Ylmz Cosruo of h Fr-Srr frm of urv D ll sp d som pplos I. Jourl of h Physl Ss Vol. 58 pp. 8-89 7. O. RoshlD gomr Ds llsh Rums Brh dr Mh.-S. Sko m Forshugszrum rz Br. 56: - 986 8. A.O. Öğrms H.Özk M. Ergü Brrd urvs ll sp d hr hrrzos Krgujv J. Mh. : 9-7 9 9. A.O. Öğrms M. Ergü M. Bkş O h Hls h ll Sp Ir. J. S. Th. Trs. A S. : 77-8 7. S.Ersoy M.Akyğ d M.Tosu A No o Admssl Mhm Curvs ll Sp Irol J.Mh. Com. Vol. 88-9. IM. Yglom A Smpl No-Euld omry d Is Physl Bss Sprgr-Vrlg Nw York 979 8