properties of parallel curves

  • Two plane curves are parallel curves of each other, if every normal of one curve is also a normal of the other curve (then one may show that the distance of the corresponding points of the curves is a ).

  • Two curves are parallel curves of each other, if they are the loci of the end pointsPlanetmathPlanetmath of a line segmentMathworldPlanetmath which moves perpendicularly to its own direction.

  • Every regular curve having a continuousMathworldPlanetmathPlanetmath curvature has an infiniteMathworldPlanetmath family of parallel curves.

  • The parallelismPlanetmathPlanetmath of curves is an equivalence relationMathworldPlanetmath.

  • The two parallel curves γ±a on both sides of a curve γ at the distance a form the envelope of the family of circles with center on γ and radius a.

  • If γ is a and closed curve with perimeterPlanetmathPlanetmath p, then the perimeter of γ±a is equal to  p±2πa  and the area between γ and the parallel curve is equal to  pa±πa2 (one must also assume that the parallel curve don’t intersect the evolute of γ).

Title properties of parallel curves
Canonical name PropertiesOfParallelCurves
Date of creation 2013-03-22 17:14:30
Last modified on 2013-03-22 17:14:30
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Topic
Classification msc 51N05