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The total curvature of a closed curve is always an integer multiple of 2 π, where N is called the index of the curve or turning number – it is the winding number of the unit tangent vector about the origin, or equivalently the degree of the map to the unit circle assigning to each point of the curve, the unit velocity vector at that point.
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 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 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.
Their total margin of victory is barely a quarter of the plus-208 point differential racked up through nine games by the 2007 New England Patriots on their way to the NFL’s most recent ...
The curvature depends in a trivial way on the parameterization of the curve: if a regularly parameterization of a curve is reversed, the same set of points results, but its curvature is negated. [5] A smooth simple closed curve, with a regular parameterization, is convex if and only if its curvature has a consistent sign: always non-negative ...
Police in East Windsor, N.J., arrested a 24-year-old man on Dec. 23, and charged him with misusing the town’s 911 system for allegedly calling emergency dispatchers a total of 17 times.