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Journal of Science and Arts Year 21, No. 1(54), pp. 171-198, 2021
ORIGINAL PAPER ON INVERSE OF A REGULAR SURFACE WITH RESPECT TO THE UNIT SPHERE S2 IN E3
SIDIKA TUL1, FERAY BAYAR2, AYHAN SARIOĞLUGİL1 ______Manuscript received: 16.11.2020; Accepted paper: 04.02.2021; Published online: 30.03.2021.
Abstract. The purpose of this paper, first, is to give a definition of the inverse surface of a given regular surface with respect to a unit sphere in E3. Second, some characteristic properties of the inverse surface are to express depending on the algebraic invariants of the original surface. In the last part of the study, we gave examples supporting our claims and plotted their graphics with the help of Maple software program. Keywords: inversion; surface; support function; principal curvatures; fundamental forms; Christoffel symbols of second kind.
1. INTRODUCTION
Let's begin this section by introducing the concept of inversion with erespect to a geometric object (circle and sphere). It is well known that the circle inversion is one of the most important transformations in egeometry. Inversion has a history ranging from synthetic geometry to differential geometry. But the history of the inversion transformations is complex and not clear. In the sixteenth century, inversely related points were known by French mathematician François Vieta. In 1749, Robert Simson, in his restoration of the work Plane Loci of Apollonius, included (on the basis of commentary made by Pappus) one of the basic theorems of the theory of inversion, namely that the inverse of a straight line or a circle is a straight line or a circle [1-3]. The nineteenth-century has great importance in view of the history of geometry. In this century, it has occurred important developments in geometry. In the early years of this century, French engineer and mathematician Jean-Victor Poncelet defined the inverse points with respect to a circle and called the reciprocal points. Swiss mathematician Jacob Steiner considered the inverse points in connection with the power of a point for a circle. The invention of the transformation of inversion is sometimes credited to Ludwig Immanuel Magnus, who published his work on the subject in 1831 [4]. A systematic development of inversion in a circle was first given by German mathematician and physici A - A . Julius Plücker approximated the problem analytically and showed that the inversion preserved the magnitude of the angles between lines and circles. From here, it can be said that the inversion is a conform transformation [5, 6]. On rhe other hand, conformal transformations of the plane appeared in Swiss mathematican Leonhard Euler's 1770 paper [2, 6].
1 O z M ı U v , F f S A , D f M , 55 9 S , T . E-mail: [email protected]; [email protected]. 2 Samsun University, Faculty of E , D f N S İ Mahallesi, Tekel Caddesi, 55420 O z ı , Samsun, Turkey. E-mail: [email protected]; [email protected].
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In this paper, Euler considered linear fractional transformations of the complex plane. In 1845, Sir William Thomson (Lord Kelvin) used inversion to give geometrical proofs of some difficult propositions in the mathematical theory of elasticity. In 1847, French mathematician Joseph Liouville called inversion the transformation by reciprocal radii [6, 7]. With the aim of getting further information about inversion geometry, the interested readers are referred to [8-19].
Definition 1: Let C be a circle with centered O and radius r , and let P be any point other than . If R is the point on the line OP that lies on the same side of as and satisfies the equation
2 OP OR OQ r2 (1) then we call the inverse of with respect to the circle C. Similarly, we can say that is the inverse of with respect to the circle C, (Fig. 1). On the other hand, if lies on the opposite side of from then we have
2 OP OR OQ r2 . (2)
The point is called the center of inversion, and is called the circle of inversion, [19]. From here we can write the following properties: 1. If P is inside in C then its inverse R is outside and vice versa. 2. Every point on C is its own inverse. 3. As P moves closer to the center of C, the inverse point R approaches infinity. In this case, the Euclidean plane with the added point of infinity is called inversive plane. Then inversion can be defined as a bijective transformation.
Figure 1. Circle inversion.
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2 Definition 2: For all P x11, y O , the function
fO: 22 (3) xy11 P f(), P 2 2 2 2 x1 y 1 x 1 y 1 is called an inversion transformation where [11, 14],
C Q x,:. y x2 y 2 r 2 2
Circle inversion can be used to study several well-known problems and theorems in b f A , S , F b ’ , T ’ , ’ , C b , ’ , others. In addition, inversion has important applications in areas of physics, engineering, astronomy, medicine, geometric modeling, etc. Circle inversion is generalizable in E3. Inversion in a sphere was considered by Italian G v T f v , . In this paper, Bellavitis showed that stereographic projection is the inversion of a sphere into a plane [6, 10, 20-24]. The Riemann sphere was introduced by Riemann in his 1857 paper [10]. In 1852, German mathematician August Ferdinand Mobius introduced the cross ratio of four points in the plane and studied the Mobius transformations of plane by Mobius in 1855. Liouville considered the Mobius transformations of 3-space in 1847. Subsequently, Mobius transforms were used by Henri Poincare, Eugenio Beltrami etc. to create a geometric model for non-Euclidean geometries [10, 24-28]. Because of the inversion transformation has a conformal structure in plane (or space), many scientists studied in this subject. For example; Considering the central conics in plane Allen [7], studied circles. Childress Ottens studied geometry of Dupin cyclides under the inversion transformation [28]. S ı ğ K ğ [29, 30] investigated inverse surfaces in E3. The, inverse surface surface was studied by Röthe [31]. The inversion of a point P in E3 with respect to a sphere centered at a point O with radius r is R such that
2 OP OR OQ r2 where the points P and R lie on the same side of OP ray. Then, we can give the definition of spherical inversion.
3 Definition 3: For all P x1,, y 1 z 1 O, the function
fO: 33 2 2 2 (4) r x1 r y 1 r z 1 P f(),, P 2 2 2 2 2 2 2 2 2 x1 y 1 z 1 x 1 y 1 z 1 x 1 y 1 z 1 is called an inversion transformation where (Fig. 2) [5, 8],
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2 2 2 2 2 3 Sr Q x,,:. y z x y z r
Figure 2. Spherical inversion.
This paper is organized as follows: Section 2 occurs of the basic concepts of the curves and surfaces theory to be used during throughout our study. In section 3, some characteristic properties of the inverse surface of a regular surface in E3 are given related to the support function and the length of the position vector of M . Finally, invers surfaces of some surfaces are found and their graphics are drawn with the help of the maple software program.
2. PRELIMINARIES
In this section, we will give the basic concepts of the differential geometry for later use. Let us consider the three dimensional Euclidean space E3 with the inner product g ,. Here, g is a metric and is called the Euclidean metric. This metric is given by
2 2 2 2 g ds dx dy dz (5) where x,, y z is the local coordinate system in E3. On the other hand, the vectoral product of 3 3 the vectors X xii e and Y yjj e is defined by i1 j1
e1 e 2 e 3
X Y x1 x 2 x 3 (6)
y1 y 2 y 3
3 where ei :1 İ 3 is a standard basis of E [32]. x i P Let M be a regular surface in E3 and let
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XUEXDE:()23 (7) uv, Xuv (,) xuvyuvzuv (,),(,),(,) be the parametric equation. Here
(8) XX0 uv
where XX () and XX ()are the partial derivates of X. From here, the unit u u v v normal vector of M is
1 (9) NXX uv. XXuv
Then we can say that
(10) T P Span X,. X M u v
On the other hand, the regular curve on M is defined by
3 XIME: (11)
s s X s where 2 : IE (12)
s s u s,. v s
If we keep the first parameter u constant, v X u, v is a curve on M. Similarly, if ν is constant, u X u, v is a curve on M. These curves called the parameter (or grid) curves of M. Here, the vectors Xu and Xv are the velocity vectors of the parameter curves (Fig. 3).
Figure 3. The curves on surfaces.
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Definition 4: Let M be a regular surface in E3. The support function of M is defined by
hM: (13)
P h P X, N where X is the position vector of M [32]. From here, it can be said that h is the length of the projection of X in direction of N (Fig. 4).
Figure 4. The support function of M.
Definition 5: Let M be a regular surface in E3 and N be a unit normal vector of M. Then, the shape operator of M is defined by
STPTP: MM (14)
XSXDN X where D is the affine connection on M. Furthermore, the shape operator is linear and self- adjoint. Then the matrix of the shape oprator S is
det XXXXXXuu , u , v det uv , u , v 3 2 2 XXXXu v u v (15) S det XXXXXX , , det , , uv u v vv u v 2 2 3 XXXXu v u v
where TM P Sp X u, X v [32].
Definition 6: Let M be a regular surface in E3 and S be shape operator of M. Then, the q th fundamental form of M is defined by
Iq : T P T P C M , ,1 q 3, MM (16)
XYIXYSXY,,, qq 1 where N is the unit normal vector of M [33].
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For q 1, the map
ITPTPCM:,MM (17)
XYIXYXY,,, is called the first fundamental form of M. For q 2 , the map
II:, TMM P T P C M (18)
X, Y II X , Y S ( X ), Y is called the second fundamental form of M. For q 3, the map
III:, T P T P C M MM (19)
X, Y III X , Y S2 ( X ), Y is called the third fundamental form of M. Then, for the coefficients of the first, second and third fundamental forms we have [4],
g11 Xu,,,,, X u g 12 g 21 X u X v g 22 X v X v (20)
b11 Xu,,,,, N u b 12 b 21 X u N v b 22 X v N v (21)
n11 Nu,,,,,. N u n 12 n 21 N u N v n 22 N v N v (22)
From here, the first, second and third fundamental forms are [34]:
22 I g11 du 2 g 12 dudv g 22 dv (23)
22 II b11 du 2 b 12 dudv b 22 dv (24)
22 III n11 du 2. n 12 dudv n 22 dv (25)
Theorem 1: Let M be a regular surface in E3. Then the Christoffel symbols of the second kind are
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g g2 g g g g 1 22 11u 12 12 u 12 11 v 11 2 2W g g g g 11 22 11vu 12 22 12 21 2 2W g g2 g g g g 1 22 22u 22 12 v 12 22 v 22 2W 2 (26) g g 2 g g g g 2 11 11v 11 12 u 12 11 u 11 2W 2 g g g g 22 11 22uv 12 11 12 21 2W 2 g g 2 g g g g 2 11 22v 12 12 v 12 22 u 22 2W 2 where [32], 2 2 W g11 g 22 g 12 0. (27)
If g12 is identically zero, then we have
gg 12 11uv 11 11 , 11 22gg11 11 1 gg11 1 2 22 2 (28) vu , 12 21 12 21 22gg11 22 gg 12 22uv 22 22 , 22 22gg11 22
Definition 6: Let M be a regular surface in E3and S be shape operator of M. The characteristic values of are called the principal curvatures of M. Non-zero vectors corresponding to these characteristic values are called the principal directions of M. Then we have
S() X kX (29) where X is a principal direction corresponding to the principal curvature k [32].
Definition 7: Let C be a regular curve on M and T be a unit tangent vector of C. If S() T kT , then is called the line of curvature on M [15]. The equation of the lines of curvature is
dv22 dudv du
g11 g 12 g 22 0 (30)
b11 b 12 b 22
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where gij and bij ,1 i , j 2,arethe coefficients of the first and the second fundamental forms of M , respectively [32].
Defintion 8: Let C be a regular curve on M and T be a unit tangent vector of C. The tangent vector is called the asymptotic direction satifsfied by
S T, T II T , T 0 (31)
The differential equation of the asymptotic lines is
22 b11 du20 b 12 dudv b 22 dv (32) where is the coefficients of the second fundamental forms of M, respectively [32].
Theorem 2 (Gauss Equations): Let M be a regular surface in E3 . Then we have
12 Xuu 11 X u 11 X v b 11 N 12 Xuv X vu 12 X u 12 X v b 12 N (33) 12 Xvv 22 X u 22 X v b 22 N
k where bij and ij ,1 i , j , k 2,are the coefficients of the second fundamental form and, the Christoffel symbols of the second kind of M , respectively [32].
Theorem 3 (Weingarten Equations): Let M be a regular surface in E3. Then the shape operator S of M is given in terms of the basis XXuv, by
b g b g b g b g SXNXX 12 12 11 22 11 12 12 11 u uWW22 u v (34) b g b g b g b g SXNXX 22 12 12 22 12 12 22 11 v vWW22 u v where and are the coefficients of the first and the second fundamental forms of M, respectively [32]. On the other hand, if the parameter curves of M are the curvature lines then we can write gb12 12 0 . Thus, by (31) we have
b11 NXuu g11 (35) b NX 22 vv g22
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and is called the Olinde Rodrigues curvature formula. Furthermore, from (26) we get
b11 k1 g11 (36) . b k 22 2 g22
Here, k1 and k2 are called the principal curvatures of M. The curvatures and are the principal values of the shape operator S . Thus, the matrix of the shape operrator
k1 0 S . (37) 0 k2
Then, the Gauss curvature and the mean curvature of M are defined by
Kdet( S ) k12 k (38) and 11 H iz() S k k (39) 2212
On the other hand, from the definition K and H, the principal curvaturesare the roots of the equation k2 20 Hk K (40) from which 2 k1 H H K . (41) 2 k2 H H K
Theorem 4: Let M be a regular surface in E3. Then the Gauss curvature and the mean curvature of M, respectively, are
2 b11 b 22 b 12 K 2 (42) g11 g 22 g 12 and
b11 g 222 b 12 g 12 b 22 g 11 H 2 (43) 2(g11 g 22 g 12 )
where gij and bij ,1 i , j 2,are the coefficients of the first and second fundametal forms of M, respectively [32].
Theorem 5: Let M be a regular surface in E3.Then we have
1 1. N X g X g X (44) uW 11 v 12 u
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1 2. N X g X g X (45) vW 12 v 22 u
3. NNXX uu (46)
4. NNXX vv (47)
6. Xvu N X g12 N (48)
7. Xuv N X g12 N (49)
8. Xvv N X g22 N (50) where N is the unit normal vector of M [34].
Theorem 6: Let M be a regular surface in E3.Then we have
X hN g g X g g X (51) W 2 v12 u 22 u u 12 v 11 v
where TM P Span X u,. X v
Proposition 1: Let M be a regular surface in E3.Then we have
2 2 h I (52) 1, where 22 I g2 g g v 11 u v 12 u 22 (53) , 2 g11 g 22 () g 22
I Here , is the Beltrami operator with respect to the coefficients of the first fundamental form of M.
3. ON INVERSE OF A REGULAR SURFACE WITH RESPECT TO A UNIT SPHERE S2 IN E3
In this section we will investigated the geometry of the inverse surface of a regular surface in 3-dimensional Euclidean space E3 with respect to the unit sphere S 2 . Then, let us starting with the definition of the inverse surface of a given surface with respect to the unit sphere S 2 .
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Definition 9: Let M be a regular surface in E3 and let S 2 be a unit sphere centered O . The surface M is called the inverse of with respect to the unt sphere S 2 defined by
1 XX (53) 2 where denotes the length of the position vector X of [26], (Fig. 5). From here, we can write 2 XX,. (54)
Differentiating (54) with respect to the parameters u and v we get
uuXX, . (55) vv XX,
Figure 5. The inverse surface of the surface with respect to the unit sphere S2 in E3.
Theorem 7: Let be inverse of the regular surface M in . Then the coefficients of the first fundamental form of are
1 gg 11 4 11 1 gg12 4 12 . (56) 1 gg22 4 22
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Proof: From (20) we may write
g Xuu, X 11 g12 Xuv,. X (57) g Xvv, X 22
Differentiating (53) with respect to the parameters u and v we get
u 1 XXXu 2 u 32 . (58) v 1 XXXv 2 32 v
Substituting (58) into (57) the desired result is found. Then, the following corollaries can be given.
Corollary 1: Let M be inverse of the regular surface M in E3 . Then we have 2 1 WW 2. (59) 4
Corollary 2: Let be inverse of the regular surface in . Then we have
1 II (60) 4 where I is the first fundamental forms of M.
Corollary 3: Let be inverse of the regular surface in .If the parameter curves of are orthogonal, then we have
1 gg 11 4 11 g12 0 . (61) 1 gg22 4 22
Theorem 8: Let be inverse of the regular surface in . Then we have
h NXN2 (62) 2 where N is the unit normal vector of .
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Proof: By (9) we may write
1 (63) NXX uv. XXuv
Computing vectoral product of the vectors X u and X v we obtain
uvW XXXXXXNuv 2 2 . (64) 5vu 5 4
On the other hand, using (50) we get
1 X X h g X g X g g N (65) uW 12 u 11 v u 11 u 12 and 1 X X h g X g X g g N (66) vW 12 v 22 u u 22 v 12 where h is the support function of M. Substituting (65) and (66) into (64) we obtain
h 2 g g X g g X 1 2 v12 u 22 u u 12 v 11 v XXuv 4 . (67) W 22 22 g g g W2 N v11 u v 12 u 22
Substituting (50),(51) and (52) into (67) and rearranging we get
Wh XXXNuv 422. (68) From here we obtain W XXuv. (69) 4
Combining (68) and (69) with (63) we have
h NXN2. 2
This is completed the proof.
Theorem 9: Let M be inverse of the regular surface M in E3 . Then we have
h h . (70) 2 where h is the support function of .
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Proof: By (13) we may write
h X,. N . (71)
Substituting (53) and (62) into (71) we get
h h . 2 This is completed the proof.
Theorem 10: Let M be inverse of the regular surface M in E3 . Then the coefficients of the second fundamental form of are
1 2 b11 4 2 hg11 b 11 1 2 b12 4 2 hg12 b 12 . (72)
1 2 b22 4 2 hg22 b 22
where gij and bij ,1 i , j 2,are the coefficients of the first and the second fundamental forms of M.
Proof: From (21) we may write
b11 Xu, N u b12 Xu, N v . (73) b22 Xv, N v
Differentiating (62) with respect to the parameters u and v we get
2 h Nu hu 22 h u X X u N u 32 . (74) 2 h Nv 32 hv 22 h v X X v N v
Substituting (58) and (74) into (73) the desired result is found. Then, the following corollaries can be given.
Corollary 4: Let be inverse of the regular surface in . Then we have
1 2 II 4 2 hI II (75) where I and II are the first and second fundamental forms of M.
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Corollary 5: Let M be inverse of the regular surface M in E3 . If the parameter curves of are the lines of curvature, then the parameter curves of are the lines of curvature, The opposite of this proposition is also true.
Theorem 11: Let be inverse of the regular surface in . Then the coefficients of the third fundamental form of are
h 2 n11 4 hg b n 4 11 11 11 h 2 n12 4 hg b n 4 12 12 12 . (76)
h 2 n22 4 hg b n 4 22 22 22
Proof: From (22) we may write
n11 Nuu, N n12 Nuv,. N (77) n22 Nvv, N
Substituting (74) into (77) the desired result is found. Then, the following corollaries can be given.
Corollary 6: Let be inverse of the regular surface in . Then we have
h 2 II4 4 hI II III (78) where I II and III are the first, second and third fundamental forms of
Corollary 7: Let be inverse of the regular surface in .If the parameter curves of are the lines of curvature, then we have
g12 b12 n 12 0. (79)
Then, we can give the following theorem
Theorem 11: Let be inverse of the regular surface in . Then the curvature lines of M sends the curvature lines of M.
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Proof: F ( 0) ff q ı f v f M is
dv22 dudv du
g11 g 12 g 22 0 (80)
b11 b 12 b 22 or
22 bg12 bg 11 du bg 22 bg 11 dudv bg 22 bg 12 dv 0 (81) 11 12 11 11 12 22
Substituting (61) and (72) into (81) and rearranging we find
22 bg12 11 bg 11 12 du bg 22 11 bg 11 22 dudv bg 22 12 bg 12 22 dv 0 (82)
Then, the solution spaces of the equations (81) and (82) are same. In other words, the inversion map preserves the curvature lines.
Theorem 12: Let be inverse of the regular surface M in E3 . Then the Gauss curvature of is K44 h2 h 2 H 4 K (83) where H and K are the mean and Gauss curvatures of M , respectively.
Proof: From (42) we may write 2 b11 b 22 b 12 K 2 . (84) g11 g 22 g 12
(61) and (72) are written in their places in (84) and if the obtained equation is arranged then we get K4 h2 4 h 2 H 4 K .
This is completed the proof.
Theorem 13: Let be inverse of the regular surface in . Then the mean curvature of is H 2 h 2 H (85) where is the mean curvature of .
Proof: From (43) we may write
b11 g222 b 12 g 12 b 22 g 11 H 2 . (86) 2 g g g 11 22 12
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If (61) and (72) are written in their places in (86) and the obtained equation is arranged, then we get
H 2. h 2 H This is completed the proof.
Theorem 14: Let M be inverse of the regular surface M in E3 . Then the principal curvatures of is 2 k1 2 h k1 (87) 2 k2 2 h k 2
where k1 and k2 are the principal curvatures of M , respectively.
Proof: From (41) we may write 2 k1 H H K (88) 2 k2 H H K
(83) and (85) are written in their places in (88) and if the obtained equation is arranged then we get 2 2 2 k1 2 h H H K 2h 22 H H K 2 2hk 1 (89) and 2 2 2 k2 2 h H H K 2h 22 H H K 2 2.hk 2 (90)
From (89) and (90) we have
2 k1 2 h k1
2 k2 2 h k 2
This is completed the proof.
Alternative Proof: Let the lines of curvature of the M are the parameter curves the we may write gb12 12 0. From (36) and (72) we get
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1 gg 11 4 11 g12 0 . 1 gg22 4 22 and
1 2 b11 4 2 hg11 b 11 b12 0 . 1 2 b22 4 2 hg22 b 22
Using the Olin de Rodrigues curvature formula we get
2 k1 2 h k1 . 2 k2 2 h k 2
Then we can give the corollary.
Corollary 8: Let M be inverse of the regular surface M in E3 . Then we have
2 k12 k k12 k (91)
Proof: The proof is clear.
Theorem 15: Let be inverse of the regular surface M in . Then the Christoffel symbols of the second kind of are
1 121 11 42 11 u g 12 W 11121 12 21 42 12 v g 12 W 1 121 22 42 22 g 22 W (92) 2 122 11 42 11 g 11 W 22122 12 21 42 12 u g 12 W 2 122 22 42 22 v g 12 W
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where
uvgg12 11 (93 ) . vugg12 22 and 2 2 W g11 g 22 g 12 . Proof: By (26) we may write
1 g22 g 11 2 g 12 g 12 g 12 g 11 u u v 11 2 2W 11g22 g 11 g 12 g 22 vu 12 21 2 2W 1 g22 g 222 g 22 g 12 g 12 g 22 u v v 22 2 2W (94) 2 g11 g 112 g 11 g 12 g 12 g 11 v u u 11 2 2W 22g11 g 22 g 12 g 11 uv 12 21 2 2W 2 g11 g 22 2 g 12 g 12 g 12 g 22 v v u 22 2 2W
Differentiating (56) with respect to the parameters u and v we get
1 g 4 g g 11 4 u 11 11 u u 1 g 4 g g 11 4 v 11 11 v v 1 g 4 g g 12 4 u 12 12 u u . (95) 1 g 4 g g 12 4 v 12 12 v v 1 g 4 g g 22 4 u 22 22 u u 1 g 4 g g 22 4 v 22 22 v v
Substituting (59) and (95) into (94) and rearranging we get
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1 121 11 42 11 u g 12 W 11121 12 21 42 12 v g 12 W 1 121 22 42 22 g 22 W 2 122 11 42 11 g 11 W 22122 12 21 42 12 u g 12 W 2 122 22 42 22 v g 12 W where
uvgg12 11 vugg12 22 and 2 2 W g11 g 22 g 12 .
This is completed the proof.
If the parameter curves of M are the lines of curvature, then we have g12 0. From here we get
v g11 u g22 . (96) 2 W g11 g 22
Substituting (85) into (80) and rearranging we get
1 121 u 11 4 11 11 121 v 12 21 4 12 1 121 u g22 22 4 22 g11 (97) 2 122 v g11 11 4 11 g 22 22 122 u 12 21 4 12 2 12 2 v 22 4 22
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k Here, ij ,1 i , j , k 2,are the Christoffel symbols of the second kind in (28).
4. EXAMPLES
In this section, we will find the equations of the inverse surface of some surfaces and draw the graphics of theirs by the Maple software program.
Example 4. 1: Let us consider the surface M
z( x 1)22 ( y 3) . (98)
Then the parametric equation of this surface is
X( u , v ) ( u 1, v 3, u22 v ) (99)
Substituting (98) into (54) we obtain
2(u 1) 2 ( v 3) 2 ( u 2 v 2 ). (100)
Then, by (53) the equation of the inverse surface M of is
1 22 X2 2 2 2 u 1, v 3, u v . (101) (u 1) ( v 3) ( u v )
푀 regular surface 푀 regular surface 푆2 unit sphere 푆2 unit sphere 푀 inverse surface
푀 inverse surface
Figure 4.1. The inverse surface of M with respect to s2.
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Example 4.2: Let us consider the surfsce
M... X ( u , v ) (2cos v ,2sin v , u ) . (102)
Substituting (102) into (54) we obtain
224.u (103)
Then, by (53) the equation of the inverse surface M of M is
22u . (104) M... X ( u , v ) 2 cos v , 2 sin v , 2 4u 4 u 4 u
푀 regular surface sssurface 푀 regular surface sssurface 푆2 unit sphere 푆2 unit sphere
푀 inverse surface 푀 inverse surface
Figure 4.2. The inverse surface of M with respect to s2. Example 4.3: Let us consider the surface
x 3cos u sin v y 3sin u sin v (105) zv 3cos
Substituting (105) into (54) we obtain
2 9 (106)
Then, by (53) the equation of the inverse surface of is
1 1 1 X cos u sin v , sin u sin v , cos v . 3 3 3
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푀 regular surface
푆2 unit sphere
푀 inverse surface
Figure 4.3. The inverse surface of M with respect to s2.
Example 4. 4: Let us consider the surface M
x(3 cos u )cos v y(3 cos u )sin v (107) zu sin
This is a torus surface. Substituting (107) into (54) we obtain
2 2(5 3cosu ) (108)
Combining (53), (107) and (108) the equation of the inverse surface M of is
(3 cosu )cos v (3 cos u )sin v sin u X ,,. 2(5 3cosu ) 2(5 3cos u ) 2(5 3cos u )
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푀 regu푀lar regu surfacelar surface
푆 2 unit푆 sphere2 unit sphere 푀 푀 inverse inverse surface surface
Figure 4.4. The inverse surface of M with respect to s2.
Example 4.5: Let us consider the surface
M... X ( u , v ) ( u 1, v 3, u22 v ). (99)
Substituting (99) into (54) we obtain
2(u 1) 2 ( v 3) 2 ( u 2 v 2 ). (100)
Thus, by (53) the equation of the inverse surface M of M is
1 22 M... X2 2 2 2 u 1, v 3, u v . (101) (u 1) ( v 3) ( u v )
Then we can easily seen that
2 gu11 14 g12 4 uv (102) 2 gv11 14 and 2 b11 22 14uv b12 0 (103) 2 b22 14uv22
Substituting (102) and (103) into (30) and rearranging we have
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uvdu2 u 2 v 2 dudv uvdv 2 0
or
udv vdu udu vdv 0 . (104)
So we have to solve the differential equations
udv vdu 0 (105) and
udu vdv 0. (106)
If these equations are solved, v cu, c const ., and v k u2 ,., k const are obtained. Thus, the curvature lines of M are
u u 1, cu 3, 1 c22 u (107) and
(u ) u 1, k u2 3, k (108)
Figure 4.5. (a) The curvature lines on M. Figure 4.5. (b) The curvature lines on M .
On the other hand, the curvature lines of are
1 22 (109) u 2 u 1, cu 3, 1 c u u122 cu 3 1 c22 u
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and
1 u u 1, k u2 3, k (110) 2 u132 k u22 k
5. CONCLUSION
In this study, it was discussed geometry of the inverse surface of a regular surface with respect to a unit sphere in E3. Some characteristic properties of inverse surface were obtained related to the similar characteristic properties of regular surface. Finally, examples supported our assertion are given and their graphs are drawn by Maple 18 software program.
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