DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

118 INDEXisometry, 107K, 48k-point contact, 54knot, 25Laplacian, 62line of curvature, 46linear fractional transformation, 89locally isometric, 39mean curvature, 48meridian, 37, 50metric, 71Meusnier’s Formula, 49minimal surface, 48, 52, 62, 104moving frame, 96normal curvature, 49normal field, 32normal plane, 17oriented, 80orthogonal, 107orthonormal, 107osculating circle, 21osculating plane, 17, 21osculating sphere, 21parabolic point, 48parallel, 37, 50, 64, 72, 89parallel translate, 65parametrizationregular, 1, 34parametrized by arclength, 7parametrized curve, 1pedal property, 106planar point, 48Poincaré disk, 94positively oriented, 107principal curvatures, 46principal directions, 46, 51principal normal vector, 11profile curve, 37pseudosphere, 49regular, 1regular parametrization, 34rotation index, 26ruled surface, 37developable, 41, 58, 62, 73ruling, 37second fundamental form, 45, 51shape operator, 44, 51smooth, 1, 34spherical coordinates, 35stereographic projection, 36support line, 32surface, 34surface area, 40surface of revolution, 37symmetric, 44tangent indicatrix, 24tangent plane, 38Theorema Egregium, 57, 98torsion, 11torus, 35total curvature, 24, 84total twist, 32tractrix, 5, 13triply orthogonal system, 53Tschebyschev net, 42twist, 32u-, v-curves, 34ultraparallels, 89umbilic, 48unit normal, 38unit tangent vector, 11variation, 101vector field, 64velocity, 1vertex, 29rectifying plane, 17reflection, 92