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Differential Geometry

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Differential Geometry Previous Next First Last Back Zoom To Fit FullScreen Quit Close . Differential Geometry Eberhard Malkowsky and Vesna Veliˇckovi´c Malkowsky Malkowsky, Veliˇckovi´c Mathematisches Institut Department of Mathematics Justus–Liebig Universit¨atGießen Faculty of Sciences Arndtstraße 2 and Mathematics D-35392 Gießen University of Niˇs Germany Viˇsegradska 33 18000 Niˇs, Serbia and Montenegro email: [email protected] [email protected] [email protected] 1 Previous Next First Last Back Zoom To Fit FullScreen Quit Close . CONTENTS CONTENTS Contents 1 Curves in Three Dimensional Euclidean Space 10 1.1 Points and vectors .......................................................... 11 1.2 Vector–valued functions of a real variable .............................................. 23 1.3 The general concept of curves .................................................... 27 1.4 The arc length of a curve ....................................................... 30 1.5 The vectors of the trihedron of a curve ............................................... 42 1.6 Frenet’s formulae ........................................................... 45 1.7 The geometric significance of curvature and torsion ........................................ 52 1.8 Osculating circles and spheres .................................................... 60 1.9 Involutes and evolutes ........................................................ 67 1.10 The fundamental theorem of curves ................................................. 87 1.11 Lines of constant slope ........................................................ 91 1.12 Spherical images of a curve ...................................................... 115 2 Surfaces in Three Dimensional Euclidean Space 123 2.1 Surfaces and curves on surfaces ................................................... 123 2.2 The tangent planes and normal vectors of a surface ........................................ 134 2.3 The arc length, angles and Gauss’s first fundamental coefficients ................................. 141 2.4 The curvature of curves on surfaces, geodesic and normal curvature ............................... 163 2.5 The normal and principal curvatures of curves on surfaces .................................... 176 2.6 The shape of a surface in the neighbourhood of a point ...................................... 223 2.7 Dupin’s indicatrix ........................................................... 241 2.8 Lines of curvature and asymptotic lines ............................................... 247 2.9 Triple orthogonal systems ...................................................... 280 2.10 The Weingarten equations for the derivatives ........................................... 289 2 Previous Next First Last Back Zoom To Fit FullScreen Quit Close . CONTENTS CONTENTS 3 The intrinsic geometry of surfaces 312 3.1 The Christoffel symbols ....................................................... 312 3.2 Geodesic lines ............................................................. 334 3.3 Geodesic lines on surfaces with orthogonal parameters ...................................... 345 3.4 Geodesic lines on surfaces of rotation ................................................ 354 3.5 The minimum property of geodesic lines .............................................. 365 3.6 Orthogonal and geodesic parameters ................................................ 365 3 Previous Next First Last Back Zoom To Fit FullScreen Quit Close . LIST OF FIGURES LIST OF FIGURES List of Figures 1 The Euclidean distance of two points and length of a vector ................................... 13 2 Vectors and position vectors ..................................................... 14 3 The diagonals in a parallelogram .................................................. 15 4 The heights in a triangle intersect in one point ........................................... 16 5 Angle between two vectors ~x and ~y and projection of ~y to ~x ................................... 18 6 Distance of a point from a plane ................................................... 22 7 The surface area of a parallelogram ................................................. 23 8 Parametric representation of a curve; a helix ............................................ 28 9 Neil’s parabola ............................................................ 31 10 Ordinary cycloids for λ = 0.5, λ = 1 and λ = 1.5 and their construction ............................. 35 11 Construction of an epicycloid .................................................... 36 12 An epicycloid and hypocylcoid .................................................... 39 13 Special cases of hypocycloids: astroids for r1/r = 4, λ = 1; ellipses for r1/r = 2 and λ arbitrary ................ 40 14 General cycloids of circles of radius 1/6 and λ = 0, 1, 2 rolling inside and outside of the catenary ~x(t) = {t, a cosh t/a} for a = 1/2 .......................................... 41 15 A logarithmic spiral with parameters t and s ............................................ 43 16 The vectors of a trihedron of a helix ................................................. 46 17 Epicycloids and their curvature ................................................... 52 18 The curvature as a measure of the deviation of a curve from a straight line ........................... 53 19 The torsion as a measure of the deviation of a curve from a plane ................................ 55 20 A helix with osculating, normal and rectifying planes ....................................... 57 21 Approximations of a helix in the osculating, normal and rectifying planes ............................ 59 22 A catenary with tangent vectors and osculating circles; an ellipse, osculating circles and the curve of its centres of curvature 61 23 A curve, the curve of its centres of curvature, and the osculating plane, circle and sphere at X(0) ............... 65 24 A helix, an osculating plane, circle, sphere and the curve of the centres of the osculating spheres ............... 66 25 A tangent surface ........................................................... 68 26 Definition of Involutes ........................................................ 69 4 Previous Next First Last Back Zoom To Fit FullScreen Quit Close . LIST OF FIGURES LIST OF FIGURES 27 An involute in the tangent surface of a curve ............................................ 70 28 Family of involutes of a curve on a sphere ............................................. 71 29 Involutes of a curve on a catenoid .................................................. 72 30 A helix and the lines of intersection of its tangents with planes orthogonal to its axis ...................... 75 31 Definition of a plane involute .................................................... 77 32 Plane involutes of helices ....................................................... 80 33 Plane involutes of a catenoid ..................................................... 81 34 Evolute of a curve on a catenoid ................................................... 84 35 A catenary and its evolute for the intervals [ − 2, 2] and [ − 1, 1], respectively .......................... 85 36 Evolute of helix ............................................................ 86 37 Spherical curves ............................................................ 92 38 Curves of constant slope with a given curvature .......................................... 95 39 Curves with κ(s) = δ/s and τ(s) = c · κ(s) ............................................. 100 40 The constant angle between a line of constant slope and a direction ............................... 103 41 Lines of constant slope ........................................................ 104 42 Lines of constant slope on a sphere and a catenoid ......................................... 105 43 A curve with κ(s) = s, hence φ(s) = s2/2 .............................................. 107 44 Planar curves with curvature κ(s) = 2, 1/s and 4s ......................................... 110 45 Rotated klothoids ........................................................... 111 46 Orthogonal projection of a line of constant slope on a paraboloid ................................. 113 47 Orthogonal projection of a line of constant slope on a sphere ................................... 116 48 The spherical images of the tangent, principal normal and binormal vectors of a helix ..................... 118 49 The spherical images of the tangent, principal normal and binormal vectors of a curve on the unit sphere .......... 119 50 The spherical images of the tangent vectors of lines of constant slope on a catenoid ...................... 122 51 Curves on a surface and in its parameter plane ........................................... 125 52 A surface, its parameter plane and u1– and u2–lines ........................................ 126 53 Polar coordinates ........................................................... 129 54 The x1x2–plane as a surface with the polar coordinates as parameters .............................. 130 55 Curve on a plane with different parametric representations .................................... 131 5 Previous Next First Last Back Zoom To Fit FullScreen Quit Close . LIST OF FIGURES LIST OF FIGURES 56 The spherical coordinates as natural parameters for spheres .................................... 135 57 Semi–spheres with the parametrizations of Example 2.7 (a) and (b) ............................... 136 58 Part of a sphere with the parameterization of Example 2.7 (c) .................................
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