Curvature Of Plane Curves Research Articles

Uniform approximation of curvature for smooth classes of plane curves

We consider problems of approximating the curvature of plane curves from smooth classes by the curvature of elements of smooth finite-dimensional function spaces (trigonometric polynomials, splines with equidistant knots) in the uniform norm.

Integrable Equations Arising from Motions of Plane Curves. II

Integrable equations satisfied by the curvature of plane curves or curves on the real line under inextensible motions in some Klein geometries are identified. These include the Euclidean, similarity, and projective geometries on the real line, and restricted conformal, conformal, and projective geometries in the plane. Together with Chou and Qu [Physica D 162 (2002), 9–33], we determine inextensible motions and their associated integrable equations in all Klein geometries in the plane. The relations between several pairs of these geometries provide a natural geometric explanation of the existence of transformations of Miura and Cole-Hopf type.

Calculus: the elements

The Problem of Calculus Integral and Derivative Differentiation and Integration Differential Equations of Rectilinear Motion The Differential Equation of Intrinsic Growth: The Exponential and Logarithmic Functions Length and Curvature of Plane Curves and Representation of Functions by Infinite Taylor Series.

Convexity and the Average Curvature of Plane Curves

The average curvature of a rectifiable closed curve in R2 is its total absolute curvature divided by its length. If a rectifiable closed curve in R2 is contained in the interior of a convex set D then its average curvature is at least as large as the average curvature of the simple closed curve ∂ D which bounds the convex set.

A geometric inequality for the total curvature of plane curves

A geometric inequality for the total curvature of plane curves

Total central curvature of curves

Total central curvature refers to the measure of curvedness of a space curve contained in a ball (bounded by a sphere) obtained by averaging the total absolute curvatures of the image curves under central projection from all points on the sphere. The major object of this paper is to show that this total curvature coincides with the classical total absolute curvature of the original space curve. This result generalizes immediately to curves in n-space. As a corollary we show that a curve on S in E with total absolute curvature 4 in E can be unknotted in S3. We begin by studying, from an elementary standpoint, the specialization of this theorem to plane curves, and illustrate at the same time the methods toe be used in the general case. 1. Total central curvature of plane curves. Let f: S --, E be a continuous map of the circle S into the plane. A local support line to ] at x is a line containing x and bounding a closed half-plane which contains the image of a neighborhood of x in S1. Let r(]) be the number of local support lines to ] passing through the point p of E.