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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 ...
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 ...
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The curvature is the norm of the derivative of T with respect to s. By using the above formula and the chain rule this derivative and its norm can be expressed in terms of γ′ and γ″ only, with the arc-length parameter s completely eliminated, giving the above formulas for the curvature.
The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a Levi-Civita connection (or covariant differentiation) and Lie bracket [,] by the following formula: (,) = [,].
Theorema egregium of Gauss states that the Gaussian curvature of a surface can be expressed solely in terms of the first fundamental form and its derivatives, so that K is in fact an intrinsic invariant of the surface. An explicit expression for the Gaussian curvature in terms of the first fundamental form is provided by the Brioschi formula.
A plane curve with non-vanishing curvature has zero torsion at all points. Conversely, if the torsion of a regular curve with non-vanishing curvature is identically zero, then this curve belongs to a fixed plane. The curvature and the torsion of a helix are constant. Conversely, any space curve whose curvature and torsion are both constant and ...
Alternatively, the sectional curvature can be characterized by the circumference of small circles. Let be a two-dimensional plane in .Let () for sufficiently small > denote the image under the exponential map at of the unit circle in , and let () denote the length of ().