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Given six points in P 3 with no four coplanar, there is a unique twisted cubic passing through them. The union of the tangent and secant lines (the secant variety ) of a twisted cubic C fill up P 3 and the lines are pairwise disjoint, except at points of the curve itself.
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.
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 Gaussian curvature K = κ 1 κ 2 and the mean curvature H = (κ 1 + κ 2)/2 can be computed as follows: =, = + (). Up to a sign, these quantities are independent of the parametrization used, and hence form important tools for analysing the geometry of the surface.
Thus the Gaussian curvature is an intrinsic invariant of a surface. Gauss presented the theorem in this manner (translated from Latin): Thus the formula of the preceding article leads itself to the remarkable Theorem. If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged.
The development of calculus in the seventeenth century provided a more systematic way of computing them. [3] Curvature of general surfaces was first studied by Euler. In 1760 [4] he proved a formula for the curvature of a plane section of a surface and in 1771 [5] he considered surfaces represented in a parametric form.
In geometry, the Cesàro equation of a plane curve is an equation relating the curvature (κ) at a point of the curve to the arc length (s) from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature (R) to arc length. (These are equivalent because R = 1 / κ .)
6.6 (3,1) Weyl curvature. 6.7 Volume form. ... Note that this transformation formula is for the mean curvature vector, and the formula for the mean curvature ...