Topic
Plane curve
About: Plane curve is a research topic. Over the lifetime, 4176 publications have been published within this topic receiving 61183 citations. The topic is also known as: planar curve.
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TL;DR: It is shown that one can determine through the use of relatively simple numerical techniques whether a given arbitrary plane curve is open or closed, whether it is singly or multiply connected, and what area it encloses.
Abstract: A method is described which permits the encoding of arbitrary geometric configurations so as to facilitate their analysis and manipulation by means of a digital computer. It is shown that one can determine through the use of relatively simple numerical techniques whether a given arbitrary plane curve is open or closed, whether it is singly or multiply connected, and what area it encloses. Further, one can cause a given figure to be expanded, contracted, elongated, or rotated by an arbitrary amount. It is shown that there are a number of ways of encoding arbitrary geometric curves to facilitate such manipulations, each having its own particular advantages and disadvantages. One method, the so-called rectangular-array type of encoding, is discussed in detail. In this method the slope function is quantized into a set of eight standard slopes. This particular representation is one of the simplest and one that is most readily utilized with present-day computing and display equipment.
1,751 citations
TL;DR: An approximation algorithm is presented which uses an iterative method to produce polygons with a small—but not minimum—number of vertices that lie on the given curve that justifies the abandonment of the minimum-vertices criterion.
1,323 citations
TL;DR: Soient M et M' des varietes de Riemann et F:M→M' une application reguliere, l'equation de la chaleur contracte M a un point as discussed by the authors.
Abstract: Soient M et M' des varietes de Riemann et F:M→M' une application reguliere. Si M est une courbe convexe plongee dans le plan R 2 , l'equation de la chaleur contracte M a un point
1,264 citations
TL;DR: In this paper, a curvature rule of motion of plane curves is considered whereby any given point of a curve moves toward its center of curvature with a speed that is proportional to the curvature.
Abstract: To represent ideal grain boundary motion in two dimensions, a rule of motion of plane curves is considered whereby any given point of a curve moves toward its center of curvature with a speed that is proportional to the curvature. A general theorem is deduced concerning the change of area enclosed by such a curve. Three families of curves are found that obey the curvature rule of motion while undergoing the shape preserving transformations of uniform magnification, translation, and rotation respectively. Pieces of these curves represent the steady shapes of idealized grain boundaries under certain symmetrical conditions.
938 citations
Book•
01 Jan 2001TL;DR: The projective closure of algebraic curves and their equations are discussed in this article, along with a discussion of the implicit function theorem and the Harnack inequality of singularities.
Abstract: Introduction Affine algebraic curves and their equations The projective closure Tangents and singularities Polars and Hessian curves The dual curve and the Plucker formulas The ring of convergent power series Parametrizing the branches of a curve by Puiseux series Tangents and intersection multiplicities of germs of curves The Riemann surface of an algebraic curve The resultant Covering maps The implicit function theorem The Newton polygon A numerical invariant of singularities of curves Harnack's inequality Bibliography Subject index List of symbols.
710 citations