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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 vector T(s) is given by
Newton's method uses curvature information (i.e. the second derivative) to take a more direct route. In calculus , Newton's method (also called Newton–Raphson ) is an iterative method for finding the roots of a differentiable function f {\displaystyle f} , which are solutions to the equation f ( x ) = 0 {\displaystyle f(x)=0} .
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R 3 , the Gauss map is a map N : X → S 2 (where S 2 is the unit sphere ) such that for each p in X , the function value N ( p ) is ...
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 ...
where the normal chosen affects the sign of the curvature. The sign of the curvature depends on the choice of normal: the curvature is positive if the surface curves "towards" the normal. The formula above holds for surfaces in 3D space defined in any manner, as long as the divergence of the unit normal may be calculated. Mean Curvature may ...
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 ...
In summary, this has shown that, given a regular surface S, the Gaussian curvature of S can be regarded as a real-valued function on S; relative to a choice of unit normal vector field on all of S, the two principal curvatures and the mean curvature are also real-valued functions on S.
The total curvature of a closed curve is always an integer multiple of 2 π, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point.