'El elipsoide de Monge y las líneas de curvatura'. The theory of plane, curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the. This textbook is the long-awaited English translation of Kobayashi’s classic on differential geometry acclaimed in Japan as an excellent undergraduate textbook. Let M be a surface in Euclidean space with second fundamental form I I ( X, Y ) : Cite journal requires |journal= ( help) For a minimal surface, the mean curvature is zero at every point. If at least one of the principal curvatures is zero at every point, then the Gaussian curvature will be 0 and the surface is a developable surface. The product k 1 k 2 of the two principal curvatures is the Gaussian curvature, K, and the average ( k 1 + k 2)/2 is the mean curvature, H. A systematic analysis of the principal curvatures and principal directions was undertaken by Gaston Darboux, using Darboux frames. From a modern perspective, this theorem follows from the spectral theorem because these directions are as the principal axes of a symmetric tensor-the second fundamental form. The directions in the normal plane where the curvature takes its maximum and minimum values are always perpendicular, if k 1 does not equal k 2, a result of Euler (1760), and are called principal directions. In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. The curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, and otherwise negative. Saddle surface with normal planes in directions of principal curvatures. Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle.

The principal curvatures at p, denoted k 1 and k 2, are the maximum and minimum values of this curvature. This curve will in general have different curvatures for different normal planes at p. A normal plane at p is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. Nevertheless, our main tools to understand and analyze these curved ob- jectsare(tangent)linesandplanesandthewaythosechangealongacurve, resp.surface.

They measure how the surface bends by different amounts in different directions at that point.Īt each point p of a differentiable surface in 3-dimensional Euclidean space one may choose a unit normal vector. CHAPTER 1 Plane and Space: Linear Algebra and Geometry The purpose of this course is the study ofcurvesandsurfaces, and those are, in gen- eral,curved. Its exact copy can be downloaded at arXiv.Ĭan be used as you want see the license below.Ĭreative Commons Attribution-ShareAlike 4.0 International License.Maximal and minimal curvature at a point of a surface Saddle surface with normal planes in directions of principal curvatures

Please email us comments/corrections/suggestions. We focus on the techniques that are absolutely essential for further study, keeping it problem-centered, elementary, visual, and virtually rigorous. We discuss smooth curves and surfaces - the main gate to differential geometry. They should be more than sufficient for a semester-long course. These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. What is differential geometry: curves and surfaces What is differential geometry: curves and surfaces by Anton Petrunin and Sergio Zamora Barrera, with artwork by Ana Cristina Chávez Cáliz

Differential geometry surfaces