Üzerinde Laguerre Eğrilerinin Tschebycheff Şebekesi Oluşturduğu Yüzeyler

Abstract

Bu çalışmada, önce, üzerinde üç Laguerre eğri ailesinin Tschebycheff şebe kesi oluşturduğu yüzeylerin birer dönel silindirden ibaret olduğu gösterilmiş ve bu yüzeylerin vektörel denklemi elde edilmiştir. Ayrıca, üzerinde iki Laguerre eğri ailesinin bir yarı- Tschebycheff şebekesi veya bir Tschebycheff şebekesi oluşturduğu açılabilir yüzeyler incelenmiş ve bu yüzeylerin, sırasıyla, birer koniden veya birer dönel silindirden ibaret olduğu gösterilmiştir. Daha sonra, açılabilir yüzeyleri özel hal olarak kabul eden regle yüzeyler ele alınmış ve üzerinde iki Laguerre eğri ailesinin Tschebycheff şebekesi oluş turduğu regle yüzeylerin birer dönel koniden; Laguerre eğrilerinin bir yarı- Tschebycheff şebekesi oluşturduğu regle yüzeylerin ise birer minimal regle yüzeyden ibaret olduğu ispatlanmıştır. Çalışmanın son bölümünde, bir yüzeyin merkez yüzeyleri ile invers yüzeyi nin Laguerre eğrilerinin oluşturduğu alt Tschebycheff şebekeleri incelenmiş ve aşağıdaki sonuçlar elde edilmiştir : Merkez yüzeylerinden birine ait iki Laguerre eğri ailesinin eşlenik Tscheby cheff şebekesi oluşturduğu bir yüzey ve bunun bir merkez yüzeyi bir genel silindirdir. Merkez yüzeyleri üzerindeki iki Laguerre eğri ailesinin birbirine karşı geldiği ve merkez yüzeylerin birisi üzerindeki iki Laguerre eğri ailesinin eşlenik yarı- Tschebycheff şebekesi oluşturduğu yüzey bir genel silindir, bu merkez yüzey ise bir konidir. Üzerindeki Laguerre eğrilerinin, invers yüzeyi üzerindeki Laguerre eğrilerine karşı geldiği, küreden farklı, -bir yüzey bir Dupin siklidinden ibarettir. Bu çeşit yüzeyler üzerindeki bir eğrilik çizgisi ailesinin, eğrilik çizgilerinden farklı bir Laguerre eğri ailesinin transversali olması halinde yüzey bir boru (hortum) yüzeyidir.

Let S be a real surface of class C4 in Euclidean 3-space with vector equation x = x(u, v) and let C be a line on S. C is said to be a Laguerre line (G-line) if at every point P of C, the normal plane of S containing the tangent line PT to C, cuts the surface S in a line superosculated by its circle of curvature at P. From the above definition, one can see that a line C on S will be a Laguerre line, if and only if the relation C = pn - 2 pg Tg = 0, (/»" = -£) (1) holds all along C, where C is Laguerre's direction function and pn, pg, rg and S are, respectively, the normal curvature, the geodesic curvature, the geodesic torsion and the arc length of C. Let C make the angles 7* and 7** (7* + 7** = S) with the parametric lines v = const, and u - const, respectively. With the aid of the generalized Euler, Ossian-Bonnet and Liouville formulae which are respectively given by pn = [r* COS7* sin7** + r*m sirry* COS7** + (t* - t**) sury* sin7**]/sin6 Pg = [(y*+7r)sin7" + (5"-72*)sin7l/sin5. Tg = [t* COS7* sin7** + t** süry* COS7** + (r** - r*) snry* sin7**]/sin<5 the differential equation of the Laguerre lines in the most general coordinate system is obtained from (1) as (r; - 2g*t*) sinY* + 3 [r* - 2 tm(g" - S2)} sinV* sin7* + 3 [rj* - 2 i**(0* + <5i)]sin7" sinV + (r2** - 2 5-** r*) sin37* = 0, where r*, r** ; 5*, 5** ; t*, i** are, respectively, the normal curvatures, the geodesic curvatures and the geodesic torsions of the parameter curves and the indices 1 and 2 denote invariant differentiation. vi A mathematical problem that is said to correspond to the physical prob lem of applying a piece of cloth smoothly to a curved surface ("clothing" a given surface) was first formulated and treated by Tschebycheff. It is desired to spread a net - a fish net for example - over a surface in such a way that no cord is slack and all cords are unstretched. Instead of attacking this problem Tschebycheff treated a different problem in which a net of discrete meshes is regarded as a part of a family of regular parameter curves that cover the surface completely. Thus he raises the question whether surface parameters u, v can be introduced such that the line element takes the form ds2 = du2.+ 2 F du dv + dv2. A Tschebycheff net is, by definition, a set of parameter curves u = const., v = const, on a surface in terms of which the line element takes the form ds2 = du2 + 2 cos<5 du dv + dv2 or, equivalently, a net is a Tschebycheff net if the tangent vector field of either family of the net undergoes a parallel displacement in the sense of Levi- Ci vita along each curve of the other family. The notion of a Tschebycheff net can be generalized to three-dimensional nets. Let vi, V2, V3 be three linear independent, smooth vector fields denned on the surface S. The triple (vi, V2, V3) is called a 3-net on S. If every vector field belonging to this 3-net undergoes a parallel displacement, in the sense of Levi-Civita, along the integral curves of the remaining vector fields, then such a net is called a Tschebycheff net of the first kind. If every two-dimensional area element belonging to the net undergoes a parallel displacement along the integral curves of the remaining vector field, such a net is called a Tschebycheff net of the second kind. In this work, surfaces on which the Laguerre lines ( G-lines ) form a Tschebycheff net are studied. Some semi- Tschebycheff nets formed by these lines are also investigated. First of all, surfaces on which the three families of Laguerre lines form a Tschebycheff net of the first kind are investigated and these surfaces are completely determined. The fundamental magnitudes of such surfaces are shown to be E = G = 1, F - cos 8 = const., L = h\ - const., N = b2 = const., M = y 61 62. Then the vector equation of these surfaces is obtained as vn x(u, v) = < A2 {^cos [A(bu + v)]} ex + {^|sm[A(bu + v)]} e2 b, A2 +{i^1(l+i)6"+(l-i)"1}?3 ^°.^° sin25. bi. _ sin26.. bi. _ - - (cos^- u) ex + - - (sm-- u) e2 bx smû bi sinfl - (u cosS + v) e3, 61 ^ 0, b2 = 0 where, A2 = sin 5 (61 + &2 - 2\/&i62 cos <5) B = sin 5 (b2-bi). It is clear that these surfaces are cylinders of revolution. Next, the developable surfaces on which two families of Laguerre lines form a Tschebycheff net or a semi-Tschebycheff net are investigated and it is shown that these surfaces are respectively, a cylinder of revolution or a cone. Furthermore, it is proved that a cone of revolution is the only ruled surface on which the two families of Laguerre lines form a Tschebycheff net. Ruled surfaces having a semi-Tschebycheff formed by Laguerre lines are shown to be a minimal ruled surface. The last section of this work is devoted to the study of sub-Tschebycheff nets on the centro-surfaces and inverse surfaces of a given surface. The differential equation {Lu - 2 (L rh + M T2n)} du3 + 3{LV-2(L T{2 + M T22)} du2 dv +3 {Nu -2{MT\2 + N T212)} du dv2 + {Nv-2 (M r*2 + İV r22)} dv3 = 0 (2) of Laguerre lines of a surface 5, where L, M and N are the coefficients of the second fundamental form of S and T^ (i,j,k = 1,2) are its Christoffel symbols, is used to prove the following results concerning the centro-surfaces Si and 52 of S. If the two families of Laguerre lines of Si form a conjugate Tschebycheff net, then S and Si are a general cylinder. If the two families of Laguerre lines of Si and 52 correspond and if two families of Laguerre lines on Si form a conjugate semi-Tschebycheff net, then S is a general cylinder and Si is a cone. Vlll In the same chapter, the inverse surface S of S is considered. Let P a point on S and P a point on S. Then the vector equation of S is c2 r=-f, (|f| = r) where, r is the position vector of S and c is the radius of inversion. If the lines of curvature of S are taken as parametric curves, then the dif ferential equation of Laguerre lines of S becomes £3/2 r* du3 + 3 E V?r; du2 dv + 3 Gy/Er? du dv2 + G3'2 r~ dv3 = 0 (3) where r* and r** are the principal curvatures of S. Using the fact that the lines of curvature on S and S correspond, the dif ferential equation of the Laguerre lines of S is obtained as E (r2 r* + 2p)u du3 + 3E (r2 r* + 2p)v du2 dv +3 G (r2 r" + 2p)u du dv2 + G (r2 r** + 2p)v dv3 = 0, where p is the perpendicular distance from the centre of inversion to the tangent plane of S, measured in the sense of the unit normal of S. The Laguerre lines of S and S will correspond to each other, if and only if the conditions < (r2 r* + 2p)v = r£ (r2 r* + 2p)u r;(r2r** + 2p)u = rr(r2r* + 2p)u r; (r2 r** + 2p)" = K* (r2 r* + 2p)u r'v (r2 r** + 2p)u = <* (r2 r* + 2p)v r*v (r2 r*« + 2p)" = C (^2 ^* + 2p)v C C7"2 r" + 2p)v = C (r2 r** + 2p)u are satisfied. From the above conditions the following results are obtained. ix Surfaces, other than a sphere, whose Laguerre lines correspond to those on the inverse surface are Dupin cy elides. If the Laguerre lines of S correspond to those on its inverse surface S and if a family of lines of curvature of S are transversals to a family of Laguerre lines, then S is a pipe surface.