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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.
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.
"A Short Preview of Free Statistical Software Packages for Teaching Statistics to Industrial Technology Majors" (PDF). Journal of Industrial Technology. 21 (2). Archived from the original (PDF) on October 25, 2005.
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For convex curves, the equality of total absolute curvature and total curvature follows from the fact that the curvature has a consistent sign. For closed curves that are not convex, the total absolute curvature is always greater than 2 π {\displaystyle 2\pi } , and its excess can be used as a measure of how far from convex the curve is.
Suppose () is a plane curve with curvature () which makes a convex curve when closed by the chord connecting its endpoints, and () is a curve of the same length with curvature (). Let d {\displaystyle d} denote the distance between the endpoints of C {\displaystyle C} and d ∗ {\displaystyle d^{*}} denote the distance between the endpoints of ...
This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwise.