IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 1 Ser. III (Jan – Feb 2020), PP 52-58 www.iosrjournals.org
Differential geometry of Curvature
Shahina Akter1, Saraban Tahora2 and Saila Akter Emon3 1Department of Computer Science & Engineering, University of Information Technology and Sciences (UITS), Dhaka-1212, Bangladesh. 2Department of Civil Engineering, University of Information Technology and Sciences (UITS), Dhaka-1212, Bangladesh. 3Department of Civil Engineering, University of Information Technology and Sciences (UITS), Dhaka-1212, Bangladesh.
Abstract: In the present paper some aspects of surface, curvature, Dupins indicatrix, Weingarten Equations are studied. The importance of this study lies in the fact that Mean and Gaussian curvature is established with the help of the coefficient of first and second fundamental form. Keywords: Curvature, Curvature Directions and Quadratic form of curvature directions, Mean & Gaussian curvature, Dupin Indicatrix and Asymptotic Direction, Weingarten Equations. ------Date of Submission: 05-02-2020 Date of Acceptance: 20-02-2020 ------
I. Introduction Differential geometry is a branch of mathematics using calculus to study the geometric properties of curves and surfaces. It arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces [5]. The theory developed in this study originates from mathematicians of the 18th and 19th centuries, mainly; Euler (1707-1783), Monge (1746-1818) and Gauss (1777-1855).Mathematical study of curves and surfaces has been developed to answer some of the nagging and unanswered questions that appeared in calculus, such as the reasons for relationships between complex shapes and curves, series and analytic functions. Study of curvatures [2] is an important part of differential geometry. During study we have derived the equations of some topics such as surface, first and second fundamental forms, curvature, normal and principle curvature, Dupins indicatrix, asymptotic directions and Weingarten equations. This paper also deals with some important theorems and examples.
II. Curvature Definition: The locus of a point whose Cartesian coordinates (푥, 푦, 푧) are functions of a single parameter is called curve and the locus of a point whose Cartesian coordinates (푥, 푦, 푧) are functions of two independent parameters u, v (say) is defined as a surface. 푑푡 Definition: Let 퐶 be a curve on the surface passing through a point 푃 and 푡 is the unit tangent of 퐶. Let = 푘. 푑푠 We now decompose 푘 into a component 푘푛 in the direction of the normal 푁 and a component 푘푔 in the tangential direction to the surface. The vector 푘푛 is called the normal curvature vector and can be expressed in terms of the unit surface normal vector 푁.
where 푘푛 is called the normal curvature. Definition: The normal to the surface at the point 푃 is a line passing through 푃 and perpendicular to the tangent line at 푃. DOI: 10.9790/5728-1601035258 www.iosrjournals.org 52 | Page Differential geometry of Curvature
Theorem (Meusnier): If 휑 is the angle between the principal normal and the surface normal to a curve 퐶 at any point 푃, then 푅 = 푅푛 cos 휑, where 1 1 푅 = , 푅푛 = . 푘 푘푛 Proof: Let 푃 be a point (푢, 푣) on the surface 푋 = 푋(푢, 푣). Suppose 푛 and 푁 denote the principal normal to the curve on the surface and surface normal [1] at 푃 respectively. Since 휑 is the angle between 푛 and 푁 , therefore, cos 휑 = 푛. 푁 Now, suppose 푘 is the curvature vector of the given curve at 푃. Then 푘 = 푘푛 푘 = 푘푛 + 푘푔 푘푛 = 푘푛 . 푁 + 푘푔 [ 푘푛 = 푘푛 . 푁] 푘푛 . 푁 = 푘푛 . 푁 + 푘푔 . 푁 푘푛. 푁 = 푘푛 . 푁. 푁 + 푘푔 . 푁 푘 cos 휑 = 푘푛 + 0 1 1 cos 휑 = 푅 푅푛 푘 cos 휑 = 푘푛 푅 = 푅푛 cos 휑 which is called the Meusnier’s theorem. Remark We have 푘푛 = 푘 if and only if 휑 = 0. Thus the necessary and sufficient condition for the curvature of a curve at 푃 to be equal to the normal curvature at 푃 in the direction of that curve is that the principal normal to the curve is along the surface normal at the point. Definition: Let 푥 = 푥 푢, 푣 be the equation of the surface. The quadratic differential equation of the form 푑푠2 = 퐸(푑푢)2 + 2 퐹푑푢 푑푣 + 퐺(푑푣)2 where 퐸 = 푥푢 . 푥푢 , 퐹 = 푥푣. 푥푣, 퐺 = 푥푢 . 푥푣 is called first fundamental form. Definition: Let 푥 = 푥 푢, 푣 be the equation of the surface and 푁 = 푁 푢, 푣 be unit surface normal. Then the second fundamental form II is defined by II−푑푥. 푑푁 = 푒(푑푢)2 + 2푓푑푢푑푣 + 푔(푑푣)2 is called the second fundamental form.
III. Curvature Directions and Quadratic form of curvature directions We have 푁. 푡 = 0, we obtain by differentiation, 푑푁 푑푡 . 푡 + 푁. = 0 푑푠 푑푠 푑푡 푑푥 푑푥 푁. = − 푑푁 . 푡 = 푑푁 . [∵ 푡 = ] 푑푠 푑푠 푑푠 푑푠 푑푠 푑푥.푑푁 푑푥.푑푁 푘. 푁 = − = − [∵ 푑푠 2 = 푑푥. 푑푥] (푑푠)2 푑푥.푑푥 푑푥.푑푁 푘 . 푁 + 푘 . 푁 = − [∵ 푘 = 푘 + 푘 푎푛푑 푘 = 푘 . 푁] 푛 푔 푑푥.푑푥 푛 푔 푛 푛 푑푥.푑푁 푘 . 푁. 푁 + 0 = − = 퐼퐼 푛 푑푥.푑푥 퐼 2 2 푘 = 푒(푑푢 ) +2푓푑푢푑푣 +푔(푑푣) [Using first and second fundamental theorem] 푛 퐸(푑푢 )2+2퐹푑푢푑푣 +퐺(푑푣)2 푑푣 푑푣 2 푒+2푓 +푔 = 푑푢 푑푢 푑푣 푑푣 2 퐸+2퐹 +퐺 푑푢 푑푢 2 = 푒+2푓휆 +푔휆 [where 휆 = 푑푣 ] 퐸+2퐹휆 +퐺휆2 푑푢 here, 푑푣 gives the directions, i.e. 휆 determines the direction. 푑푢 Definition: The directions for which 푘푛 is either maximum or minimum are called curvature directions. 푘푛 = 푘푛 (휆) is a function of 휆, 2 Where 푘 = 푒+2푓휆 +푔휆 푛 퐸+2퐹휆+퐺휆2 퐸+2퐹휆+퐺휆2 2푓+2푔휆 − 푒+2푓휆 +푔휆2 (2퐹+2퐺휆) ∴ 푑푘푛 = 푑휆 퐸+2퐹휆+퐺휆2 2 For extreme values (maximum or minimum) of 푘푛 , 푑푘푛 should be zero 푑휆
DOI: 10.9790/5728-1601035258 www.iosrjournals.org 53 | Page Differential geometry of Curvature
푑푘푛 = 0 푑휆 Therefore, extreme values of 푘푛 , we have, 퐸 + 2퐹휆 + 퐺휆2 2푓 + 2푔휆 − 푒 + 2푓휆 + 푔휆2 2퐹 + 2퐺휆 = 0 퐸 + 2퐹휆 + 퐺휆2 푓 + 푔휆 = 푒 + 2푓휆 + 푔휆2 (퐹 + 퐺휆) 2 푓+푔휆 = 푒+2푓휆 +푔휆 퐹+퐺휆 퐸+2퐹휆+퐺휆2 푘 = 푒+푓휆 +휆(푓+푔휆 ) = 푓+푔휆 푛 퐸+퐹휆+휆(퐹+퐺휆) 퐹+퐺휆 푓+푔휆 = −휆(푓+푔휆 ) = 푒+푓휆 +휆(푓+푔휆 ) 퐹+퐺휆 −휆(퐹+퐺휆) 퐸+퐹휆+휆(퐹+퐺휆) = −휆(푓+푔휆 )+푒+푓휆 +휆(푓+푔휆 ) −휆(퐹+퐺휆)+퐸+퐹휆+휆(퐹+퐺휆) 푓+푔휆 = 푒+푓휆 퐹+퐺휆 퐸+퐹휆 Since 푘 = 푒+푓휆 +휆(푓+푔휆 ) = 푓+푔휆 and considering (3.01) & (3.02), we have 푛 퐸+퐹휆+휆(퐹+퐺휆) 퐹+퐺휆 푘 = 푓+푔휆 = 푒+푓휆 푛 퐹+퐺휆 퐸+퐹휆 ∴ 푒퐹 + 푒퐺휆 + 푓퐹휆 + 푓퐺휆2 = 푓퐸 + 푔퐸휆 + 푓퐹휆 + 푔푓휆2 Or, 푔퐹 − 퐹퐺 휆2 + 푔퐸 − 푒퐺 휆 + 푓퐸 − 푒퐹 = 0 This is a quadratic equation in 휆. Definition: The normal curvature of a surface which have greatest and least curvatures are called principal curvatures. Definition: The two perpendicular directions for which the values of 푘푛 take on maximum or minimum values are called principal directions. Definition: The normal curvatures in the curvature directions are called principal curvature. Definition: The lines in the curvature directions are called lines of curvature [8]. Theorem (Rodrigues Formula): In the direction of a principal direction, the vector 푑푁 is parallel to 푑푥 and is given by 푑푁 = −푘푛 푑푥 Where 푘푛 is the principal curvature in that direction. Proof: We know that the principal curvatures are given by 푘 = 푓+푔휆 = 푒+푓휆 , 휆 = 푑푣 푛 퐹+퐺휆 퐸+퐹휆 푑푢 Or, 퐹푘푛 + 퐺푘푛 휆 = 푓 + 푔휆 and 퐸푘푛 + 퐹푘푛 휆 = 푒 + 푓휆 Or, 퐸푘푛 − 푒 + 퐹푘푛 − 푓 휆 = 0 and 퐹푘푛 − 푓 + 퐺푘푛 − 푔 휆 = 0 Or, 퐸푘 − 푒 + 퐹푘 − 푓 푑푣 = 0 and 퐹푘 − 푓 + 퐺푘 − 푔 푑푣 = 0 푛 푛 푑푢 푛 푛 푑푢 Or, 퐸푘푛 − 푒 푑푢 + 퐹푘푛 − 푓 푑푣 = 0 and 퐹푘푛 − 푓 푑푢 + 퐺푘푛 − 푔 푑푣 = 0 Or, 푥푢 . 푥푢 푘푛 + 푥푢 . 푁푢 푑푢 + 푥푢 . 푥푣푘푛 + 푥푢 . 푁푣 푑푣 = 0 and 푥푢 . 푥푣푘푛 + 푥푣. 푁푢 푑푢 + 푥푣. 푥푣푘푛 + 푥푣. 푁푣 푑푣 = 0 Or, [ 푥푢 푘푛 + 푁푢 푑푢 + 푥푣푘푛 + 푁푣 푑푣]. 푥푢 = 0 and [ 푥푢 푘푛 + 푁푢 푑푢 + 푥푣푘푛 + 푁푣 푑푣]. 푥푣 = 0 Or, 푁푢 푑푢 + 푁푣푑푣 + 푘푛 푥푢 푑푢 + 푥푣푑푣 . 푥푢 = 0 and 푁푢 푑푢 + 푁푣푑푣 + 푘푛 푥푢 푑푢 + 푥푣푑푣 . 푥푣 = 0 Or, 푑푁 + 푘푛 푑푥 . 푥푢 = 0 and 푑푁 + 푘푛 푑푥 . 푥푣 = 0 Since, 푥푢 and 푥푣 are linearly independent [i.e. 푑푁 + 푘푛 푑푥 is the tangent plane], we must have, 푑푁 + 푘푛 푑푥 = 0 Or, 푑푁 = −푘푛 푑푥 ∴ 푑푁 = −푘푛 푑푥 which is called the Rodrigues formula. IV. Mean & Gaussian curvature 2 2 2 We know that the roots of 퐸퐺 − 퐹 푘푛 + 푒퐺 + 푔퐸 − 2푓퐹 푘푛 + 푒푔 − 푓 = 0 are called principal curvatures. Definition: The arithmetic mean of principal curvatures 푘푛1 and 푘푛2 is called the mean curvature [3] and is defined as 1 푒퐺 + 푔퐸 − 2푓퐹 푀 = 푘 + 푘 = 2 푛1 푛2 2(퐸퐺 − 퐹2) Definition: The product of principal curvatures 푘푛1 and 푘푛2 is called the Gaussian curvature and is defined as
DOI: 10.9790/5728-1601035258 www.iosrjournals.org 54 | Page Differential geometry of Curvature
푒푔 − 푓2 퐾 = 푘 푘 = 푛1 푛2 퐸퐺 − 퐹2 Theorem (Eulers): The normal curvature 푘푛 in any direction can be expressed as, 2 2 푘푛 = 푘푛1 cos 훼 + 푘푛2 sin 훼 Where 푘푛1 and 푘푛2 are the principal curvature and 훼 is the angle between the 푢 parametric curve and arbitrary directions. Proof : We know that the normal curvature is given by 푒(푑푢)2 + 2푓푑푢푑푣 + 푔(푑푣)2 푘 = 푛 퐸(푑푢)2 + 2퐹푑푢푑푣 + 퐺(푑푣)2 When the parametric lines coincide with curvature directions then we have 푓 = 0, 퐹 = 0.In this case 푒(푑푢)2 + 푔(푑푣)2 푘 = 푛 퐸(푑푢)2 + 퐺(푑푣)2
v-variable curve u- variable curve 휋 2 푘푛2 푘푛1 u= constant curve v=constant curve
Let us suppose that for 푢- variable curve the principal curvature is 푘푛1 and for 푣- variable curve the principal curvature is 푘푛2 . Therefore 푘푛1 is obtained from 2 2 푘 = 푒(푑푢 ) +푔(푑푣) 푛 퐸(푑푢 )2+퐺(푑푣)2 By setting 푑푣 = 0, we get, 2 푘 = 푒(푑푢 ) = 푒 푛1 퐸(푑푢 )2 퐸 And 푘푛2 is obtained by setting 푑푢 = 0 2 푘 = 푔(푑푣) = 푔 푛2 퐺(푑푣)2 퐺 Therefore when the parametric lines coincide with curvature directions then we have
Let us suppose that any arbitrary direction makes angle 훼 with the parametric curve v=constant. 푑푥.훿푥 cos 훼 = = 퐸푑푢훿푢 +퐹 푑푢훿푣 +푑푣훿푢 +퐺푑푣훿푣 푑푥.훿푥 퐸(푑푢 )2+2퐹푑푢푑푣 +퐺(푑푣)2. 퐸(훿푢 )2+2퐹훿푢훿푣 +퐺(훿푣)2 Since, the parametric curves coincide with the curvature directions, we have 푓 = 0, 퐹 = 0 and cos 훼 = 퐸푑푢훿푢 +퐺푑푣훿푣 퐸(푑푢 )2+퐺(푑푣)2. 퐸(훿푢 )2+퐺(훿푣)2 Here, 훼 is the angle between 푣 = constant curve and any arbitrary direction. So, we set 훿푣 = 0 and cos 훼 = 퐸푑푢훿푢 = 퐸푑푢 퐸(푑푢 )2+퐺(푑푣)2. 퐸(훿푢 )2 퐸(푑푢 )2+퐺(푑푣)2 For 푢 = constant curve cos(휋 − 훼) = 퐸푑푢퐷푢 +퐺푑푣퐷푣 2 퐸(푑푢 )2+퐺(푑푣)2 . 퐸(퐷푢)2+퐺(퐷푣)2 Since 퐷푢 = 0 sin 훼 = 퐺푑푣퐷푣 = 퐺푑푣 퐸(푑푢 )2+퐺(푑푣)2 . 퐺(퐷푣)2 퐸(푑푢 )2+퐺(푑푣)2 Now, 2 2 푘 cos2 훼 = 푒 . 퐸(푑푢 ) = 푒(푑푢 ) 푛1 퐸 퐸(푑푢 )2+퐺(푑푣)2 퐸(푑푢 )2+퐺(푑푣)2 2 2 푘 sin2 훼 = 푔 . 퐺(푑푣) = 푔(푑푣) 푛2 퐺 퐸(푑푢 )2+퐺(푑푣)2 퐸(푑푢 )2+퐺(푑푣)2 2 2 푘 cos2 훼 + 푘 sin2 훼 = 푒(푑푢 ) +푔(푑푣) = 푘 푛1 푛2 퐸(푑푢 )2+퐺(푑푣)2 푛 2 2 푘푛 = 푘푛1 cos 훼 + 푘푛2 sin 훼 DOI: 10.9790/5728-1601035258 www.iosrjournals.org 55 | Page Differential geometry of Curvature where 훼 is the angle between any curvature direction and any arbitrary direction. This is called Euler’s theorem. 2 Definition: The points where 푒푔 − 푓 > 0 are called elliptic points. Here either 푘푛1 < 0 and 푘푛2 < 0 or 푘푛1 > 0 and 푘푛2 > 0. 2 Definition: The points where 푒푔 − 푓 < 0 are called hyperbolic points. Here either 푘푛1 < 0 or 푘푛2 < 0 i.e. 푘푛1 and 푘푛2 have opposite sign. 2 Definition: The points where 푒푔 − 푓 = 0 are called parabolic points. Here either 푘푛1 = 0 or 푘푛2 = 0 also, 푘푛1 and 푘푛2 are both zero.
V. Dupin Indicatrix and Asymptotic Direction If the x-axis represents one of the curvature of the directions of curvature, then the distance of any point on the ellipse to the curve is the square root of the radius of the curvature in the corresponding direction on the surface. The Form of the Dupin Indicatrix for ellipse: Let us draw an ellipse where principal semi axes are 1 1 푅푛1 = and 푅푛2 = 퐾푛1 퐾푛2 2 2 ∴ The equation to the ellipse will be 푘1푥 + 푘2푦 = 1
2 2 Let A is at (푥1, 푦1) and 푂퐴 = 푥1 + 푦1 2 2 2 = 푥1 + 푥1 tan 훼 = 푥1 sec 훼 2 2 Again, 푘1푥1 + 푘2푦2 = 1 2 2 2 Or, 푘1푥1 + 푘2푥1 tan 훼 = 1 2 2 2 2 2 Or, 푘1푥1 cos 훼 + 푘2푥1 sin 훼 = cos 훼 2 2 cos 훼 Or, 푥1 = 2 2 푘1 cos 훼+푘2 sin 훼 cos 훼 Or, 푂퐴 = 푥1 sec 훼 = 2 2 . sec 훼 푘1 cos 훼+푘2 sin 훼 1 = 2 2 푘1 cos 훼+푘2 sin 훼
But according to Euler’s theorem, 2 2 푘푛 = 푘1 cos 훼 + 푘2 sin 훼 From above, 1 푂퐴 = = 푅푛 푘푛 Hence, 푂퐴 gives the normal curvature in the direction of 푂퐴, where the curvature directions are in the direction of the coordinate axes. In this case 푘푛1 > 0 and 푘푛2 > 0, i.e. we have considered the case for ellipse points. 2 2 The ellipse 푘푛1푥 + 푘푛2푦 = 1 is called Dupin Indicatrix. At the point where, 푘푛1 < 0 and 푘푛2 < 0, i.e. we have considered the case for hyperbolic points and we get the equation of hyperbola. 2 At the parabolic points, say 푘푛2 = 0, we get 푘푛1푥 = 1, 1 푥 = ± = ± 푅푛1 푘푛1 2 푘푛 = 푘푛1 cos 훼 The asymptotes of Dupin Indicatrix corresponds to the asymptotic directions on the surface, the axes of indicatrix to the directions of the curvature. Definition: A direction at a point on a surface for which 퐼퐼 = 푒푑푢2 + 2푓푑푢푑푣 + 푔푑푣2 = 0 is called an asymptotic direction.
DOI: 10.9790/5728-1601035258 www.iosrjournals.org 56 | Page Differential geometry of Curvature
Since, 푘 = 퐼퐼 and I is positive definite, the asymptotic directions are also the directions in which 푘 = 0. At 푛 퐼 푛 an elliptic point there are no asymptotic directions, at a hyperbolic point there are two distinct asymptotic directions, at a parabolic point there is one asymptotic direction. Theorem : The curvature directions bisect the asymptotic directions.
Proof. Consider the Dupin indicatrix at the parabolic points. We have = 푅푛 . If the straight line OA be the asymptote of the hyperbola then A is at infinity and 푂퐴 = ∞.
1 푘푛 = = 0 푂퐴 ∴ The asymptotes of this hyperbola give the asymptote directions. But the coordinate axes directions are in the direction of the coordinate axes. Therefore, the curvature directions bisect the asymptotic directions. Theorem: Binormal of asymptotic line is the normal to the surface. Proof. We know, 푑푥.푑푁 푘 = − 푛 푑푥.푑푥 푑푥.푑푁 = − (푑푠)2 푑푥 = − . 푑푁 푑푠 푑푠 = −푡. 푑푁 푑푠 But for asymptotic curve, 푘푛 = 0 ∴ for asymptotic curve, −푡. 푑푁 = 0 푑푠 Again, 푡. 푁 = 0 푑푡 . 푁 + 푡. 푑푁 = 0 푑푠 푑푠 Or, 푘 . 푁 + 푡. 푑푁 = 0 푛 푑푠 Since, 푡. 푑푁 = 0 , we have for asymptotic curve, 푘푛. 푁 = 0 푑푠 Therefore, for asymptotic curve either 푘 = 0 or, 푛. 푁 = 0. ∴ The straight line on a surface are asymptotic lines. Since 푡. 푁 = 0 and 푛. 푁 = 0. 푁 is perpendicular to 푡 and 푛 ,so 푁 is parallel to 푏 . Hence, 푁 = 푏 Hence, for curved asymptotic lines, surface normal coincides with the binormal of the asymptotic lines.
VI. Weingarten Equations These equations were first investigated by J. Weingarten. The Weingarten equations express the derivatives of the vectors 푥푢 , 푥푣 and 푁 as linear combinations of these vectors with coefficients which are functions of the first and second fundamental coefficients. Theorem: On a surface 푥 = 푥(푢, 푣) , the vectors 푥푢 , 푥푣 , 푁푢 and 푁푣 lie on the tangent plane, then the Weingarten equations will be of the form 푁 = 푓퐸−푒퐺 푥 + 푒퐹−푓퐸 푥 푢 퐸퐺−퐹2 푢 퐸퐺−퐹2 푣 푁 = 푔퐹 −푓퐺 푥 + 푓퐹−푔퐸 푥 푣 퐸퐺−퐹2 푢 퐸퐺−퐹2 푣
DOI: 10.9790/5728-1601035258 www.iosrjournals.org 57 | Page Differential geometry of Curvature
Proof: We know 푥푢 , 푥푣 are linearly independent and lie on the tangential plane. Since, 푁푢 and 푁푣 lie on the tangential plane, then 푁푢 and 푁푣 can be written as a linear combination of 푥푢 , 푥푣 . 푁푢 = 푎1푥푢 + 푎2푥푣 ; 푁푣 = 푏1푥푢 + 푏2푥푣 ∴ 푁푢 . 푥푢 = 푎1푥푢 . 푥푢 + 푎2푥푣 . 푥푢 Or, −푒 = 푎1 퐸 + 푎2 퐹 ∴ 푁푢 . 푥푣 = 푎1푥푢 . 푥푣 + 푎2푥푣 . 푥푣 Or, −푓 = 푎1 퐹 + 푎2 퐺 Or, 푎1 퐸 + 푎2 퐹 + 푒 = 0 푎1 퐹 + 푎2 퐺 + 푓 = 0 푎1 = 푎2 = 1 푓퐸−푒퐺 푒퐹−푓퐸 퐸퐺−퐹2 Or, 푎 = 푓퐸−푒퐺 , 푎 = 푒퐹−푓퐸 1 퐸퐺−퐹2 2 퐸퐺−퐹2 Again, 푁푣 . 푥푢 = 푏1푥푢 . 푥푢 + 푏2푥푣 . 푥푢 Or, −푓 = 푏1 퐸 + 푏2 퐹 ∴ 푁푣 . 푥푣 = 푏1푥푢 . 푥푣 + 푏2푥푣 . 푥푣 Or, −푔 = 푏1 퐹 + 푏2 퐺 Or, 푏1 퐸 + 푏2 퐹 + 푓 = 0 푏1 퐹 + 푏2 퐺 + 푔 = 0 푏1 = 푏2 = 1 푔퐹−푓퐺 푓퐹−푔퐸 퐸퐺−퐹2 푏 = 푔퐹−푓퐺 , 푏 = 푓퐹−푔퐸 1 퐸퐺−퐹2 2 퐸퐺−퐹2 Therefore, 푁 = 푓퐸−푒퐺 푥 + 푒퐹−푓퐸 푥 (4.01) 푢 퐸퐺−퐹2 푢 퐸퐺−퐹2 푣 푁 = 푔퐹 −푓퐺 푥 + 푓퐹−푔퐸 푥 (4.02) 푣 퐸퐺−퐹2 푢 퐸퐺−퐹2 푣 The equations (4.01) and (4.02) are called Weingarten equations. Theorem: (Beltrami Enneper) The torsion of the asymptotic lines through point of surface [7] is 휏 = ± −푘 . Where k is the Gaussian curvature at the point. Proof: We know, for asymptotic curve, 푑푥.푑푁 푘 = − = 퐼퐼 = 0 푛 푑푥.푑푥 퐼 Hence, for asymptotic curve, 퐼퐼 = 0 and 푁 = 푏 . Again, we have, 푑푁. 푑푁 − 2푀퐼퐼 + 푘퐼 = 0 Where, 퐼, 퐼퐼 and 퐼퐼퐼 = first, second and third fundamental form, 푀 = Mean curvature & 푘 = Gaussian curvature. Or, 푑푁. 푑푁 + 푘퐼 = 0 [∵ 퐼퐼 = 0] Or, 푑푁. 푑푁 + 푘 푑푥. 푑푥 = 0 Or, 푑푏. 푑푏 + 푘 푑푠 2 = 0 [∵ 푑푥. 푑푥 = 푑푠 2] 푑푏 푑푏 Or, . + 푘 = 0 푑푠 푑푠 푑푏 푑푏 푑푏 Or, 휏2 + 푘 = 0 [∵ = −휏푛 , . = 휏2 푎푠 푛. 푛 = 1] 푑푠 푑푠 푑푠 Or, 휏2 = −푘 ∴ 휏 = ± −푘
VII. Conclusion Here we have discussed curvature elaborately. We have solved some examples and also problems related to this topic.
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DOI: 10.9790/5728-1601035258 www.iosrjournals.org 58 | Page