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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.
There are a few reviews of free statistical software. There were two reviews in journals (but not peer reviewed), one by Zhu and Kuljaca [26] and another article by Grant that included mainly a brief review of R. [27] Zhu and Kuljaca outlined some useful characteristics of software, such as ease of use, having a number of statistical procedures and ability to develop new procedures.
Ooms, Marius (2009). "Trends in Applied Econometrics Software Development 1985–2008: An Analysis of Journal of Applied Econometrics Research Articles, Software Reviews, Data and Code". Palgrave Handbook of Econometrics. Vol. 2: Applied Econometrics. Palgrave Macmillan. pp. 1321– 1348. ISBN 978-1-4039-1800-0. Renfro, Charles G. (2004).
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.
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. This curve has total curvature 6π, and turning number 3.. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if ...
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 ...