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Normal Curvature
In the differential geometry of surfaces, a Darboux frame is a natural moving frame constructed on a surface. It is the analog of the Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space. It is named after French mathematician Jean Gaston Darboux. Darboux frame of an embedded curve Let ''S'' be an oriented surface in three-dimensional Euclidean space E3. The construction of Darboux frames on ''S'' first considers frames moving along a curve in ''S'', and then specializes when the curves move in the direction of the principal curvatures. Definition At each point of an oriented surface, one may attach a unit normal vector in a unique way, as soon as an orientation has been chosen for the normal at any particular fixed point. If is a curve in , parametrized by arc length, then the Darboux frame of is defined by : \mathbf(s) = \gamma'(s), (the ''unit tangent'') : \ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
