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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: =.
A space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).
A migrating wild-type Dictyostelium discoideum cell whose boundary is colored by curvature. Scale bar: 5 μm. In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane.
Immersed plane curves have a well-defined turning number, which can be defined as the total curvature divided by 2 π. This is invariant under regular homotopy, by the Whitney–Graustein theorem – topologically, it is the degree of the Gauss map , or equivalently the winding number of the unit tangent (which does not vanish) about the origin.
In differential geometry, the total absolute curvature of a smooth curve is a number defined by integrating the absolute value of the curvature around the curve. It is a dimensionless quantity that is invariant under similarity transformations of the curve, and that can be used to measure how far the curve is from being a convex curve. [1]
Suppose we are given a closed, oriented curve in the xy plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the winding number of the curve is equal to the total number of counterclockwise turns that the object makes around the origin.
The analogous definition applies in the case of the Monge patches of the other two forms. ... for any Monge patch (u, v) ↦ (u, v, h(u, v)) whose range includes p, n is a multiple of ( ∂h / ∂u , ∂h / ∂v , −1) as evaluated at the point (p 1, p 2). The analogous definition applies in the case of the Monge patches of the ...
This curve will in general have different curvatures for different normal planes at p. The principal curvatures at p, denoted k 1 and k 2, are the maximum and minimum values of this curvature. Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle. The curvature is taken to be positive if the curve ...