Lec¢¸tii de geometrie diferen¢¸tiala a curbelor si¢¸«© a...

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  • Lecţii de geometrie diferenţială a curbelor şi a suprafeţelor

    Paul A. Blaga

  • Cuprins

    I Curbe 9

    1 Curbe în spaţiu 11 1.1 Introducere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Curbe parametrizate (drumuri) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Definiţia curbei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Reprezentări analitice ale curbelor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    1.4.1 Curbe plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.4.2 Curbe în spaţiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

    1.5 Tangenta şi planul normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.1 Ecuaţiile tangentei şi planului normal (normalei) pentru diferite reprezentări ale

    curbelor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.6 Planul osculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.7 Curbura unei curbe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

    1.7.1 Semnificaţia geometrică a curburii . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.8 Reperul lui Frenet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

    1.8.1 Comportamentul reperului Frenet faţă de o schimbare de parametru . . . . . . . 35 1.9 Curbe orientate. Reperul Frenet al unei curbe orientate . . . . . . . . . . . . . . . . . . 36 1.10 Formulele lui Frenet. Torsiunea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

    1.10.1 Semnificaţia geometrică a torsiunii . . . . . . . . . . . . . . . . . . . . . . . . . 40 1.10.2 Alte aplicaţii ale formulelor lui Frenet . . . . . . . . . . . . . . . . . . . . . . . 41 1.10.3 Elici generale. Teorema lui Lancret . . . . . . . . . . . . . . . . . . . . . . . . 42 1.10.4 Curbe Bertrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

    1.11 Comportamentul local al unei curbe parametrizate în jurul unui punct biregular . . . . . 46 1.12 Contactul dintre o curbă în spaţiu şi un plan . . . . . . . . . . . . . . . . . . . . . . . . 47 1.13 Contactul dintre o curbă în spaţiu şi o sferă. Sfera osculatoare . . . . . . . . . . . . . . . 49 1.14 Teoreme de existenţă şi unicitate pentru curbe parametrizate . . . . . . . . . . . . . . . 51

    1.14.1 Comportamentul reperului lui Frenet la o deplasare . . . . . . . . . . . . . . . . 51 1.14.2 Teorema de unicitate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.14.3 Teorema de existenţă . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

    2 Curbe plane 55 2.1 Introducere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.2 Înfăşurători de curbe plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

    2.2.1 Curbe date printr-o ecuaţie implicită . . . . . . . . . . . . . . . . . . . . . . . . 57

  • 6 Cuprins

    2.2.2 Familii de curbe care depind de doi parametri . . . . . . . . . . . . . . . . . . . 58 2.2.3 Aplicaţie: evoluta unei curbe plane . . . . . . . . . . . . . . . . . . . . . . . . 58

    2.3 Curbura unei curbe plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3.1 Semnificaţia geometrică a curburii cu semn . . . . . . . . . . . . . . . . . . . . 62

    2.4 Centrul de curbură. Evoluta şi evolventa unei curbe plane . . . . . . . . . . . . . . . . . 63 2.5 Cercul osculator al unei curbe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.6 Teorema de existanţă şi unicitate pentru curbe plane . . . . . . . . . . . . . . . . . . . . 67

    3 Integrarea ecuaţiilor naturale ale unei curbe în spaţiu 69 3.1 Ecuaţia Riccati asociată cu ecuaţiile naturale ale unei curbe . . . . . . . . . . . . . . . . 69 3.2 Exemple de integrare a ecuaţiei naturale a unei curbe plane . . . . . . . . . . . . . . . . 70

    Problems 75

    II Surfaces 83

    4 General theory of surfaces 85 4.1 Parameterized surfaces (patches) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

    4.2.1 Representations of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 The equivalence of local parameterizations . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.4 Curves on a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.5 The tangent vector space, the tangent plane and the normal to a surface . . . . . . . . . . 91 4.6 The orientation of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.7 Differentiable maps on a surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.8 The differential of a smooth map between surfaces . . . . . . . . . . . . . . . . . . . . 98 4.9 The spherical map and the shape operator of an oriented surface . . . . . . . . . . . . . 100 4.10 The first fundamental form of a surface . . . . . . . . . . . . . . . . . . . . . . . . . . 102

    4.10.1 First applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 The length of a segment of curve on a surface . . . . . . . . . . . . . . . . . . . 103 The angle of two curves on a surface . . . . . . . . . . . . . . . . . . . . . . . . 104 The area of a parameterized surface . . . . . . . . . . . . . . . . . . . . . . . . 105

    4.11 The matrix of the shape operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.12 The second fundamental form of an oriented surface . . . . . . . . . . . . . . . . . . . 107 4.13 The normal curvature. The Meusnier’s theorem . . . . . . . . . . . . . . . . . . . . . . 109 4.14 Asymptotic directions and asymptotic lines on a surface . . . . . . . . . . . . . . . . . . 110 4.15 The classification of points on a surface . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.16 Principal directions and curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

    4.16.1 The determination of the lines of curvature . . . . . . . . . . . . . . . . . . . . 117 4.16.2 The computation of the curvatures of a surface . . . . . . . . . . . . . . . . . . 118

    4.17 The fundamental equations of a surface . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.17.2 The differentiation rules. Christoffel’s coefficients . . . . . . . . . . . . . . . . 119

    Christoffel’s and Weingarten’s coefficients in curvature coordinates . . . . . . . 121 4.17.3 The Gauss’ and Codazzi-Mainardi’s equations for a surface . . . . . . . . . . . 121 4.17.4 The fundamental theorem of surface theory . . . . . . . . . . . . . . . . . . . . 123

    4.18 The Gauss’ egregium theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 4.19 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

    4.19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.19.2 The Darboux frame. The geodesic curvature and geodesic torsion . . . . . . . . 128

  • Cuprins 7

    4.19.3 Geodesic lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Examples of geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

    4.19.4 Liouville surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

    5 Special classes of surfaces 135 5.1 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

    5.1.1 General ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 The parameterization of a ruled surface . . . . . . . . . . . . . . . . . . . . . . 135 The tangent plane and the first fundamental form of a ruled surface . . . . . . . 137

    5.1.2 The Gaussian curvature of a ruled surface . . . . . . . . . . . . . . . . . . . . . 138 5.1.3 Envelope of a family of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.1.4 Developable surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

    Developable surfaces as envelopes of a one-parameter family of planes. The regression edge of a developable surface . . . . . . . . . . . . . . . . 141

    5.1.5 Developable surfaces associated to the Frenet frame of a space curve . . . . . . . 143 The envelope of the family of osculating planes . . . . . . . . . . . . . . . . . . 143 The envelope of the family of normal planes (the polar surface) . . . . . . . . . 144 The envelope of the family of rectifying planes of a space curves . . . . . . . . . 145

    5.2 Minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.2.1 Definition and general properties . . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.2.2 Minimal surfaces of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.2.3 Ruled minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

    5.3 Surfaces of constant curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

    Problems 159

    Bibliography 167

  • 8 Cuprins

  • Partea I

    Curbe

  • CAPITOLUL 1

    Curbe în spaţiu

    1.1 Introducere

    Intuitiv, curbele