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Animation of the curvature and the acceleration vector T′(s) As in the case of curves in two dimensions, the curvature of a regular space curve C in three dimensions (and higher) is the magnitude of the acceleration of a particle moving with unit speed along a curve. Thus if γ(s) is the arc-length parametrization of C then the unit tangent ...
A simple example of a regular surface is given by the 2-sphere {(x, y, z) | x 2 + y 2 + z 2 = 1}; this surface can be covered by six Monge patches (two of each of the three types given above), taking h(u, v) = ± (1 − u 2 − v 2) 1/2. It can also be covered by two local parametrizations, using stereographic projection.
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. 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. For a minimal surface, the mean curvature is zero at every ...
The Gauss map can be defined for hypersurfaces in R n as a map from a hypersurface to the unit sphere S n − 1 ⊆ R n.. For a general oriented k-submanifold of R n the Gauss map can also be defined, and its target space is the oriented Grassmannian ~,, i.e. the set of all oriented k-planes in R n.
The equivalence classes are called C r-curves and are central objects studied in the differential geometry of curves. Two parametric C r-curves, : and :, are said to be equivalent if and only if there exists a bijective C r-map φ : I 1 → I 2 such that : ′ and : (()) = (). γ 2 is then said to be a re-parametrization of γ 1.
For example, in the two-dimensional case, the normal line to a curve at a given point is the line perpendicular to the tangent line to the curve at the point. In the three-dimensional case a surface normal , or simply normal , to a surface at a point P is a vector that is perpendicular to the tangent plane to that surface at P .
Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...
Thus, if the Ricci curvature (,) is positive in the direction of a vector , the conical region in swept out by a tightly focused family of geodesic segments of length emanating from , with initial velocity inside a small cone about , will have smaller volume than the corresponding conical region in Euclidean space, at least provided that is ...