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This curve has total curvature 6 π, and index/turning number 3, though it only has winding number 2 about p.. In mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arc length:
The total curvature of a geodesic triangle equals the deviation of the sum of its angles from π. The sum of the angles of a triangle on a surface of positive curvature will exceed π, while the sum of the angles of a triangle on a surface of negative curvature will be less than π. On a surface of zero curvature, such as the Euclidean plane ...
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.
This is almost the same as the formula for the total curvature, but differs in using the absolute value instead of the signed curvature. [2] Because the total curvature of a simple closed curve in the Euclidean plane is always exactly 2 π, the total absolute curvature of a simple closed curve is also always at least 2 π.
The Jacobian, Gaussian curvature and mean curvature are = (+ +), =, = The total curvature is . Osserman proved that a complete minimal surface in R 3 {\displaystyle \mathbb {R} ^{3}} with total curvature − 4 π {\displaystyle -4\pi } is either the catenoid or the Enneper surface.
In differential geometry, Fenchel's theorem is an inequality on the total absolute curvature of a closed smooth space curve, stating that it is always at least . Equivalently, the average curvature is at least 2 π / L {\displaystyle 2\pi /L} , where L {\displaystyle L} is the length of the curve.
The first Frenet-Serret formula holds by the definition of the normal N and the curvature κ, and the third Frenet-Serret formula holds by the definition of the torsion τ. Thus what is needed is to show the second Frenet-Serret formula. Since T, N, B are orthogonal unit vectors with B = T × N, one also has T = N × B and N = B × T.
The Gauss map can be defined for hypersurfaces in R n as a map from a hypersurface to the unit sphere S n − 1 ⊆ R n.. For a general oriented k-submanifold of R n the Gauss map can also be defined, and its target space is the oriented Grassmannian ~,, i.e. the set of all oriented k-planes in R n.