ON INTEGRAL INVARIANTS OF RULED SURFACES GENERATED BY THE DARBOUX FRAMES OF THE TRANSVERSAL INTERSECTION CURVE OF TWO SURFACES IN E

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1 Joural of Sciece ad rts Year 16, o. (5), pp , 016 ORIGIL PPER O ITEGRL IVRITS OF RULED SURFCES GEERTED Y THE DROUX FRMES OF THE TRSVERSL ITERSECTIO CURVE OF TWO SURFCES I E EGI S 1, YH SRIOĞLUGİL 1 Mauscript receied: ; ccepted paper: ; Published olie: bstract. I this paper, the some characteristic properties of ruled surfaces which are eerated by the Darboux frame of the trasersal itersectio cure of two surfaces were ie i -dimesioal Euclidea space E. lso, the relatios betwee the iteral iariats of the closed ruled surfaces were showed. Fially, the examples for parametric-parametric ad imlicit-implicit surfaces were ie. Keywords. Trasersal itersectio cure, ruled surface, eodesic curature, Darboux frame. Mathematics Subject Classificatio. 504, ITRODUCTIO It is well kow that the curatures of a cure ie by the parametric equatio i - dimesioal Euclidea space ca be foud easily. If a cure is a itersectio cure of two surfaces, represeti the cure as parametric is usually imposible ad computi the curatures of this cure is hard. Itersectios of eometric structures are also importat i the represetatio of the desi of complex shapes or i computer aimatio. So some methods ad formulas are deeloped. Sice the surfaces hae parametric or implicit forms, there are three aspects for the itersectio of the surfaces. 1- Parametric- parametric - Implicit- implicit - Parametric- implicit. Here, the mai purpose is to determie the itersectio cure of the surfaces. So determii the eometric properties as the taet ector, the curature, the torsio is su ciet for determii the itersectio cure. Two surfaces ca itersect to each other as the trasersal or the taetial. If the ormal ector fields of the surfaces are liearly depedet at the itersectio poit, the itersectio is called taetial itersectio. If the ormal ector fields of the surfaces are liearly idepedet at the itersectio poits, the itersectio is called trasersal itersectio. The uit taet ector of the trasersal itersectio cure ca be easily computed ia the ectoral product of the the uit ormal ectors of the surfaces. So, there are may studies o the eometric properties of the trasersal itersectio cures. Willmore (1959) studied the Freet ector.elds of the trasersal itersectio cure for two implicit surfaces. The formulas related to the curatures of the itersectio cures for all types (parametric-parametric,...,etc.) were deeloped by Hartma (1996). léssio (006) 1 Odokuz Mayıs Uiersity, Faculty of Sciece d rts, Departmet of Mathematics, 5519 Kurupelit, Samsu, Turkey. ei.as@hotmail.com; sarioluil@mail.com. ISS:

2 11 itroduced a method to compute the Freet ector fields ad the curatures of the trasersal itersectio cures o implicit surfaces. léssio ad Guadalupe (007) ae results related to the eodesic curature ad the eodesic torsio of the itersectio cure of two space-like surfaces i Loretzia - space. léssio (009) studied the itersectio cure of three implicit 4 surfaces i IR by usi implicit fuctio theorem. Goldma (005) foud the curature ad the torsio of itersectio cure by usi the classical di eretial eometry methods. Ye ad Maekawa (1999) computed the curatures ad the Freet ectors of the taetial ad trasersal itersectio cure for the three cases of two surfaces. lso, they computed the ormal curature i the directio of the taet ector of the itersectio cure ad ae the formulas to compute hiher order deriaties. For three parametric surfaces i E 4, the curatures ad the Freet ectors of the itersectio cure were ie by Düldül M. (010). Çalışka ad Düldül. (010) studied the eodesic curature ad the eodesic torsio of the itersectio cure for impilicit-implicit ad parametric- parametric surfaces. lso, they ae the curature ad the curature ector of itersectio cure by usi the ormal ectors of surfaces. Sarıoğluil ad Tutar (007) studied the eodesic curature ad the fudametal forms of the reular surfaces i E. I this paper, the relatio betwee the Darboux frames of the itersectio cure at the itersectio poit for two surfaces was ie. lso, the relatios betwee the eodesic curatures, the eodesic torsios ad the ormal curatures were iestiated. The apex ales, the pitches ad tha dralls were computed for the closed ruled surfaces eerated by the Darboux frames ad the relatios betwee each other were showed. Fially, the examples were ie for the parametric-parametric, implicit-implicit surfaces.. PRELIMIRIES I this sectio; firstly, we will reiew some basic cocepts i E for later use. Let : I E be a differetiable cure with arc-leth parameter s ad tb,, be the Freet frame of at the poit s, where tss, s s, s bstss. The Freet formulas of are t 0 0t 0. (1) b 0 0 b If is a cure ad x is a eerator ector, the the ruled surface X s, has the followi parameter represetatio X s, s xs. amely, a ruled surface is a surface eerated by the motio of a straiht lie x alo. Furthermore, if is a closed cure, the this surfaces is called closed ruled surface. Moreoer, the drall P x, the apex ale x ad the pitch l x of the closed ruled surface are defied by det, xx, Px,,,, x Dx lx Vx x ()

3 11 respectiely. Here, D ad V are Steier rotatio ector ad Steier traslatio ector, respectiely. The Steier traslatio ector V ad Steier rotatio ector D are ie as follows: () V dx t ds where D wt b (4) w t b (5) is called Darboux ector, (Fi 1). Fiure 1. Darboux ector of a cure. hae If the Freet ectors tbare,, the straith lies of the closed ruled surface, the we t ds lt ds Pt 0 0, l 0, P (6) lb 0 b ds 1 Pb (Hacısalioğlu, H.H., 198). Defiitio 1. Let M be a orieted surface i E ad be a uit speed cure o M If t is the uit taet ector of is the uit ormal ector of M ad t at the poit s the t,, is called the Darboux frame of at that poit. Thus, Darboux formulas are t 0 t 0 (7) 0 where is the ale betwee ad the uit pricipal ormal of d cos, si ad ds. Here are called the ormal curature, the eodesic curature ad the eodesic torsio of respectiely, (Kühel, W., 1950). Defiitio. Let M be a orieted surface i E ad t be the taet ector of M at the poit P. The, the alue of St, t is called the ormal curature of M i the directio of t, where S is the shape operator o M, (O.eill,., 1966). amely, the ormal curature of M at P M is the curature of the projectio of o the plae which is spaed by the ormal ector ad the taet ector of M at P M. ISS:

4 114 Defiitio. Let M be a orieted surface i E ad be a differetiable cure o M. If the ormal curature of is zero, the is called a asymptotic lie, (O.eill,., 1966). Defiitio 4. Let M be a orieted surface i E ad be a differetiable cure o M. The curature of the projectio cure o the taet plae of M alo is called the eodesic curature of ad deoted by (Graustei, W. C., 1966). Defiitio 5. Let M be a orieted surface i E ad be a differetiable cure o M. If the eodesic curature of is zero, the is called a eodesic cure, (Struik, D. J., 1961). Theorem 1. Let M be a orieted surface i E ad be a differetiable cure o M. The, the relatio betwee the curature, the eodesic curature ad the ormal curature of is ie as follows (O.eill,., 1966). Defiitio 6. Let M be a orieted surface i E ad be a differetiable cure o M. If the eodesic torsio of is zero, the is called a pricipal lie, (Struik, D. J., 1961). Theorem. Let M be a ruled surface i E ad : I M be the leadi cure of M. The relatio betwee the Darboux frame t,, ad the Freet frame tb,, of is ie as follows: t t 0 si cos (8) 0 cos si b where is the ale betwee ad, (Şeatalar, M., Diferesiyel Geometri 1978).. ITERSECTIO CURVE OF TWO SURFCES Let ad be two parametric surfaces which hae X u, ad X u, parametric represetatios. The uit ormal of the parametric surface X u, is X X u u X X ad the uit ormal of the implicit surface is f f The cures u us ad s defie the cure s X us, s o the parametric surface X u,. Similarly, the cures x xs, y ys, z zs defie the cure f xs, ys, zs 0 o the implicit surface f x, y, z 0. If the ormal ectors of ad.

5 115 are liear idepedet (liear depedet) at the itersectio poits, the the itersectio cure is called trasersal itersectio cure (taetial itersectio cure), (Ye, X. ad Maekawa, T., 1999)..1. TRSVERSL ITERSECTIO CURVE OF PRMETRIC-PRMETRIC SURFCES Let ad be two reular surfaces which hae X u, ad X u, parametric represetatios, respectiely be the trasersal itersectio cure of ad with arcleth parameter, t be the uit taet ector at the itersectio poit P X u, X u, ad ad be the ormal ectors of ad at the poit P, respectiely. The, we hae X X X X, X X X X u u u u Sice ad itersect trasersally, ad is ot parallel at the poit P. lso, sice the uit taet ector t of the itersectio cure lies o the taet plaes of ad, (9) t. (10) Fiure. Itersectio cure of two surfaces. Let, t, ad, t, be Darboux frame at the poit P where t ad t. The, Darboux formolus are t 0 t 0 0 (11) ISS:

6 116 t 0 t 0 0 respectiely, (Çalışka, M. ad Düldül U.,., 010). (1) Theorem. Let ad be two reular surfaces which hae X u, ad Y p, q parametric represetatios, respectiely ad be the trasersal itersectio cure of ad. The, the eodesic torsio of with respect to the surface is 1 EM FL u E GL u F GM EG F (1) Here, E Xu, Xu, F Xu, X ad G= X, X are the coefficiets of the first fudametal form of the surface ad L X,,, ad, uu M X u X are the coefficiets of the secod fudametal form of the surface, (Çalışka, M. ad Düldül U.,., 010), where 1 u G t, Xu F t, X EG F (14) 1 E t, X F t, Xu EG F (Ye, X. ad Maekawa, T., 1999). Theorem 4. Let ad be two reular surfaces which hae X u, ad Y p, q parametric represetatios, respectiely ad be the trasersal itersectio cure of ad. The, the eodesic curature of with respect to the surface is E Eu Fu, u, t X t X u 1 EG F G Gu t, Xu F t, X Gu t, Xu E t, X u EG F u u (15) (Çalışka, M. ad Düldül U.,., 010). Here, 1 si EG F 1, t, Xu, X u t, Xuu u t, Xu u t, X si EG F u, t, X, X t, Xuu u t, Xu u t, X (16) where pp pq qq uu u Y p Y p q Y q X u X u X. Theorem 5. Let ad be two reular surfaces which hae X u, ad Y p, q parametric represetatios, respectiely ad be the trasersal itersectio cure of ad. The, the ormal curature of with respect to the surface is

7 117, (17) (Ye, X. ad Maekawa, T., 1999). L u Mu.. TRSVERSL ITERSECTIO CURVE OF IMPLICIT-IMPLICIT SURFCES Let f x, y, z 0 ad x, y, z 0 be the implicit equatios of the surfaces ad, respectiely ad be the trasersal itersectio cure of ad. The, the uit taet ector of is where t f f, (18) f f, are the ormal ector fields of ad, respectiely, (Çalışka, M. ad Düldül U.,., 010). Theorem 6. Let f x, y, z 0 ad x, y, z 0 be the implicit equatios of the surfaces ad, respectiely ad be the trasersal itersectio cure of ad. The, the eodesic torsio of with respect to the surface is (19) where 1 cot G F T, f f x fxx fxy f xz F fx fy fz, G x y z, T y, fxy fyy fyz z fxz fyz f zz (0) (Çalışka, M. ad Düldül U.,., 010). Theorem 7. Let f x, y, z 0 ad x, y, z 0 be the implicit equatios of the surfaces ad, respectiely ad be the trasersal itersectio cure of ad. The, the eodesic curature of with respect to the surface is 1 yz y z fx zx z x fy xy x y f z f (1) Here, x, y ad z ca be obtaied by the followi liear equaito systems: xx yy zz 0 f xx fyy fzz fxx x fyy y fzz z fxyxy fxzxz fyzyz xx yy zz xx x yy y zz z xyxy xzxz yzyz () (Çalışka, M. ad Düldül U.,., 010). ISS:

8 118 Theorem 8. Let f x, y, z 0 ad x, y, z 0 be the implicit equatios of the surfaces ad, respectiely ad be the trasersal itersectio cure of ad. The, the ormal curature of with respect to the surface is f x f y f z x y z f f f x y z () (Ye, X. ad Maekawa, T., 1999)... TRSVERSL ITERSECTIO CURVE OF PRMETRIC-IMPLICIT SURFCES Let X u, ad x, y, z 0 be the parametric represetatio ad the implicit equatio of the surfaces ad, respectiely ad be the trasersal itersectio cure of ad. The, the ormal ector field of the surfaces ad are Xu X, X X u (4) respectiely ad the uit taet ector of is t u u X X X X, (5) (Çalışka, M. ad Düldül U.,., 010). Theorem 9. Let X u, ad x, y, z 0 be the parametric represetatio ad the implicit equatio of the surfaces ad, respectiely ad be the trasersal itersectio cure of ad. The, the eodesic torsio of with respect to the surface is 1 det( X,,,, u X X Xu ), (6) X X u (Çalışka, M. ad Düldül U.,., 010). 4. O ITEGRL IVRITS OF RULED SURFCES GEERTED Y THE DROUX FRMES OF THE TRSVERSL ITERSECTIO CURVE OF TWO SURFCES I E Theorem 10. Let be the trasersal itersectio cure of the surfaces ad, with arc leth parameter ad let, t, ad, t, be the Darboux frames of at the poit P, respectiely. The the relatio betwee the Darboux frames of the surfaces ad, is ie as follows: t t 0 cos si 0 si cos (7)

9 119 where is the ale betwee ad. Proof. Let be the trasersal itersectio cure of the surfaces ad. y usi, t, ad Freet frame tb,, ca be write (8), the Darboux frame t t 0 si cos 0 cos si b (8) where is the ale betwee ad. Similarly, by usi (8), the Darboux frame t,, ad Freet frame tb,, ca be write t t 0 si cos b 0 cos si (9) where is the ale betwee ad. Substituti (9) ito (8) we hae or t t 0 cos si 0 si cos (0) where t t 0 cos si 0 si cos Thus, the followi corollary ca be ie. (1) Corollary 1.. Let be the trasersal itersectio cure of the surfaces ad, with arc leth parameter ad let, t, ad, t, be the Darboux frames of at the poit P, respectiely. If ad are orthooal to each other alo the cure, the the relatio betwee the Darboux frames of ad is ie as follows: t t () Proof. The proof is clear. ISS:

10 10 Theorem 11. Let be the closed trasersal itersectio cure of the surfaces ad ad let, t, ad, t, be the Darboux frames of at the poit P, respectiely. The the relatios betwee the apex ales of ruled surfaces which are eerated by the Darboux frames of are ie as follows: t ds cos sicot cos si ta () Proof. Let be the ale betwee ad ad let be the ale betwee ad. The apex ale of ruled surface which is eerated by the uit taet ector of the trasersal itersectio cure is Dt,. Substituti (4) ito the last equatio we obtai t t ds The apex ale of ruled surface which is eerated by the uit ormal ector the surface is D,. of Substituti (4) ito the last equatio we obtai si ds. Substituti (6) ito the aboe equatio we et si. (4) b From the last equatio we hae b si (5) Similarly, the apex ale of ruled surface which is eerated by the uit ormal ector of the surface is b si. (6) From (5) ad (6) we et si si

11 11 Substituti (1) ito the last equatio we obtai cos si cot. The apex ale of ruled surface which is eerated by the uit ector surface is D,. of the Substituti (4) ito the last equatio we obtai cos ds Substituti (6) ito the aboe equatio we et From the last equatio we obtai cos b. (7) b cos (8) Similarly, the apex ale of ruled surface which is eerated by the uit ector of the surface is b. cos (9) From (8) ad (9) we et cos. cos Subtituti (1) ito the last equatio we obtai cos si ta. Thus, the followi corollary ca be write. Corollary. Let be the closed trasersal itersectio cure of the surfaces ad ad let, t, ad, t, be the Darboux frames of at the poit P, respectiely.. If ad are orthooal to each other alo the cure, the the relatio betwee the apex ales of ruled surfaces which are eerated by the Darboux frames of are ie as follows: ISS:

12 1 t ds cot ta (40) Proof. The proof is clear. Theorem 1. Let be the closed trasersal itersectio cure of the surfaces ad ad let, t, ad, t, be the Darboux frames of at the poit P, respectiely. The the relatios betwee the pitches of ruled surfaces which are eerated by the Darboux frames of are ie as follows: l ds, l l l l 0 t (41) Proof. The pitch of ruled surface which is eerated by the uit taet ector of is lt V. t. (4) Substituti () ito the last equatio, we hae lt ds. Similarly, the pitches of ruled surfaces which are eerated by ectors,, ad are l l l l 0. Theorem 1. Let be the closed trasersal itersectio cure of the surfaces ad ad let, t, ad, t, be the Darboux frames of at the poit P, respectiely. Let be the eodesic torsio of with respect to the surface ad let be the eodesic torsio of with respect to the surface. The, the relatio betwee the eodesic torsios is ie as follows: d ds. (4) Proof. From (7) we kow that cos si Differetiati this equatio, we obtai si cos cos si.. (44)

13 1 From the Darboux formulas, we et t cos si t si si cos cos. Multiplyi the last equatio with, we hae d ds (46) Theorem 14. Let be trasersal itersectio cure of surfaces ad ad let, t, ad,, t be the Darboux frames of at the poit P, respectiely. Let ad be the eodesic curature ad the ormal curature of with respect to the surface, respectiely ad let ad be the eodesic curature ad the ormal curature of with respect to the surface, respectiely. The, the relatios betwee the eodesic curatures ad the eodesic torsios are ie as follows: cos si cos si (47) Proof. From (7) we kow that cos si. (48) Differetiati this equatio we obtai From the Darboux formulas we et si cos cos si. t cos si t si si cos cos. Multiplyi the last equatio with t, we hae cos si. (50) lso, from (7) we kow that si cos. (51) Differetiati this equatio we obtai t si cos t cos cos si si. (5) Multiplyi the last equatio with t, we hae ISS:

14 14 cos si. (5) Corollary. From (50) ad (5) it ca be easily see that. (54) Corollary 4. Let be trasersal itersectio cure of surfaces ad. If is the eodesic cure for the surfaces ad, the the relatio betwee ormal curatures of is. (55) Corollary 5. Let be trasersal itersectio cure of surfaces ad. If is the asymptotic cure for the surfaces ad, the the relatio betwee eodesic curatures of is. (56) Corollary 6. Let be trasersal itersectio cure of surfaces ad. If ad are orthooal to each other alo the itersectio cure, the the relatio betwee the eodesic curatures ad the eodesic torsios are (57) Theorem 15. Let be trasersal itersectio cure of surfaces ad ad let, t, ad, t, be the Darboux frames of at the poit P, respectiely. If the ale betwee ad is costat alo the cure, the the relatios betwee the dralls of ruled surfaces which are eerated by the Darboux frames of are ie as follows: P P P P (58) Proof. The drall of ruled surface which is eerated by the uit taet ector of is P t det, tt, t 0 the drall of ruled surface which is eerated by the uit ormal ector of the surface is P det,, (59) ad the drall of ruled surface which is eerated by the uit ector of the surface is

15 15 P det,, (60) Similarly, the dralls of ruled surfaces which are eerated by ad are ad P P (61) (6) From here, we et 1 1 P P. Substituti (4) ad (54) ito the last equatio we obtai 1 1 P P d ds d ds. Sice is costat, it is see that 1 1 P P P P P P,. (6) 5. EXMPLES 5.1. TRSVERSL ITERSECTIO CURVE OF PRMETRIC-PRMETRIC SURFCES. Example 1. Let the surface be the sphere ie by X u, cosucos,si ucos,si, 0 u, 0 ad the surface be the cylider ie by ISS:

16 Y p, q cos p, si p, q, 0 u, q respectiely (Fi. ). Let us fid the eodesic curatures ad the ormal curatures of the itersectio cure at the poit 1 1 P X, Y,,, 4 4 ad show that theorem 14 is satisfied. Fiure. Itersectio cure of parametric-parametric surfaces. The partial deriaties of the surfaces at the poit P are Xu,,0, X,,, Xuu,,0, Xu,,0, X,,. From here, the uit ormal of surface at the poit P ad the coefficiets of the first ad secod fudametal form are 1 1 1,,, E L, G 1, E 1, F Eu F Gu G M 0, respectiely. Similarly, the partial deriaties of the surfaces at the poit P are 1,0,0, 0,0,1, 0, 1 Yp Yq Ypp,0, Ypq Yqq 0. From here, the uit ormal of the surface at the poit P ad the coefficiets of the first ad secod fudametal form are 1 1 0,1,0, e, 1, l, f ep eq fp fq p q m 0, 4

17 17 respectiely. Sice the uit taet ector is (16), it is hold t 6,0, at the poit P ad from (14) ad 6 6 u, p, q, 0,0,, u, p, q Substituti the last equaito ito (15) ad (17) we et 5 4,, 1,. 9 9 Usi this equaito, it ca be easily see that the theorem 14 is satisfied. 5.. TRSVERSL ITERSECTIO CURVE OF IMPLICIT-IMPLICIT SURFCES Example. Let ad be the surfaces are ie by f x, y, z z xy 0ad x, y, z x y z respectiely. Let us fid the eodesic curatures ad the ormal curatures of the itersectio cure at the poit P 1,, ad show that theorem 14 is satisfied. Fiure 4. Itersectio cure of implicit-implicit surfaces. The ormal ectors of the surfaces ad at the poit P are, 1,1,, 4,1 f respectiely. Moreoer, at the poit P we et ad f f f f f 0, f 1 xx xz yy yz zz xy ISS:

18 18 0,. xy xz yz zz xx yy From () we hae x z 0 x y z 0 x 4 y z. 5 Soli this liear equatio system, we obtai 4 x, y, z Substituti x, y ad zi (1) ad (), we et ,, 0, Usi this equaito, it ca be easily see that the theorem 14 is satis.ed. REFERECES [1] léssio, O., TEM Ted. Mat. pl. Comput., 7(), 169, 006. [] léssio, O., Guadalupe, I.V., Hadroic Joural, 0(), 15, 007. [] léssio, O., Comput. ided Geom. Des., 6(4), 455, 009. [4] laschke, W., Diferesiyel Geometri Dersleri, İstabul Üiersitesi Yayıları (Traslated by Ord. Prof. Dr. K. Erim), [5] Çalışka, M., Düldül, U.., cta Uiersitatis pulesis, 4, 161, 010. [6] Düldül, M., Computer ided Geometric Desi, 7, 118, 010. [7] Graustei, W.C., Differetial Geometry, Doer Publucatios, Ic., ew York, [8] Goldma, R., Computer ided Geometric Desi,, 6, 005. [9] Hacısalihoğlu, H.H., Yüksek oyutlu Uzaylarda Döüşümler e Geometriler, İöü Üiersitesi Temel ililer Fakültesi Yayıları, Mat. o:1, İstabul, [10] Hacısalihoğlu, H.H., Diferesiyel Geometri, İöü Üiersitesi Fe-Edebiyat Fakültesi Yayıları, Malatya, 198. [11] Hartma, E., The Visual Computer, 1, 181, [1] Kühel, W., Differetial Geometry, Cures-Surfaces-Maifolds (Secod Editio- Traslated by ruce Hut), merica Mathematical Society, US, [1] O.eill,., Elemetary Differetial Geometry, cademic Press, ew York, [14] Sarıoğluil,., Tutar,., It. J. Cotemp. Math. Sci., (1), 1, 007. [15] Struik, D.J., Itroductio to Differetial Geometry, Oxford Uiersity Press, [16] Seatalar, M., Diferesiyel Geometri (Eğriler e Yüzeyler Teorisi) İstabul Delet Mühedislik e Mimarlık kademisi Yayıları, İstabul, [17] Ye, X., Maekawa, T., Computer ided Geometric Desi, 16, 767, [18] Willmore, T.J., Itroductio to Differetial Geometry, Oxford Uiersity Press, Idia,