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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.
where κ n−1 is last Frenet curvature (the torsion of the curve) and sgn is the signum function. The minimum total absolute curvature of any three-dimensional curve representing a given knot is an invariant of the knot. This invariant has the value 2 π for the unknot, but by the Fáry–Milnor theorem it is at least 4 π for any other knot. [2]
Note that changing F into –F would not change the curve defined by F(x, y) = 0, but it would change the sign of the numerator if the absolute value were omitted in the preceding formula. A point of the curve where F x = F y = 0 is a singular point, which means that the curve is not differentiable at this point, and thus that the curvature is ...
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 combinations thereof. [1] [2] [3]
A flow is a process in which the points of a space continuously change their locations or properties over time. More specifically, in a one-dimensional geometric flow such as the curve-shortening flow, the points undergoing the flow belong to a curve, and what changes is the shape of the curve, its embedding into the Euclidean plane determined by the locations of each of its points. [2]
The total absolute curvature of a smooth convex curve, | |, is at most . It is exactly for closed convex curves, equalling the total curvature of these curves, and of any simple closed curve. For convex curves, the equality of total absolute curvature and total curvature follows from the fact that the curvature has a consistent sign.
A parametric C r-curve or a C r-parametrization is a vector-valued function: that is r-times continuously differentiable (that is, the component functions of γ are continuously differentiable), where , {}, and I is a non-empty interval of real numbers.