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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 ...
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 ...
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ 1 and κ 2, at the given point: =.
The eigenvalues of S x are just the principal curvatures k 1 and k 2 at x. In particular the determinant of the shape operator at a point is the Gaussian curvature, but it also contains other information, since the mean curvature is half the trace of the shape operator. The mean curvature is an extrinsic invariant.
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 ...
Let K be the Gaussian curvature of M, and let k g be the geodesic curvature of ∂M. Then [2] [3] + = (), where dA is the element of area of the surface, and ds is the line element along the boundary of M. Here, χ(M) is the Euler characteristic of M.
Suppose that P is a point on γ where k ≠ 0. The corresponding center of curvature is the point Q at distance R along N, in the same direction if k is positive and in the opposite direction if k is negative. The circle with center at Q and with radius R is called the osculating circle to the curve γ at the point P.
The sectional curvature K(σ p) depends on a two-dimensional linear subspace σ p of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ p as a tangent plane at p , obtained from geodesics which start at p in the directions of σ p (in other words, the ...