Ad
related to: curvature k calculator calculus equation generator 2
Search results
Results From The WOW.Com Content Network
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 ...
Negative Ricci curvature implies there are no nontrivial (nonzero) Killing fields. Nonpositive Ricci curvature implies that any Killing field is parallel. i.e. covariant derivative along any vector field is identically zero. If the sectional curvature is positive and the dimension of M is even, a Killing field must have a zero.
The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface ...
An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. [2] The osculating circle provides a way to understand the local behavior of a curve and is commonly used in differential geometry and calculus.
Dubins proved his result in 1957. In 1974 Harold H. Johnson proved Dubins' result by applying Pontryagin's maximum principle. [4] In particular, Harold H. Johnson presented necessary and sufficient conditions for a plane curve, which has bounded piecewise continuous curvature and prescribed initial and terminal points and directions, to have minimal length.
The variation formula computations above define the principal symbol of the mapping which sends a pseudo-Riemannian metric to its Riemann tensor, Ricci tensor, or scalar curvature.
The evolute will have a cusp at the center of the circle. The sign of the second derivative of curvature determines whether the curve has a local minimum or maximum of curvature. All closed curves will have at least four vertices, two minima and two maxima (the four-vertex theorem). In general a curve will not have 4th-order contact with any ...
Liouville's equation can be used to prove the following classification results for surfaces: 7] A surface in the Euclidean 3-space with metric dl 2 = g(z, _)dzd _, and with constant scalar curvature K is locally isometric to: the sphere if K > 0; the Euclidean plane if K = 0; the Lobachevskian plane if K < 0.