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Differential Geometry of Curvature

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Differential Geometry of Curvature 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 .
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