parallel curve

Given two curves, one is a parallel curve (also known as an offset curve) of the other if the points on the first curve are equidistant to the corresponding points in the direction of the second curve’s normal. Alternatively, a parallelMathworldPlanetmathPlanetmath of a curve can be defined as the envelope of congruent circles whose centers lie on the curve.

For a parametric curve in the plane defined by  F(u):=(x(u),y(u)),  its parallel curve  G(u):=(X(u),Y(u))  with offset t is defined by

X(u) =x(u)+ty(u)x(u)2+y(u)2
Y(u) =y(u)-tx(u)x(u)2+y(u)2

0.1 Examples

The most elementary example of parallel curves is given by the family of concentric circles

X(u) = tcosu
Y(u) = tsinu

Except for trivial cases such as circles and lines, parallel curves may be quite different from the original curve as the offset gets larger. An example of this is given by the catenary

x(u) = u
y(u) = coshu

From the definition, the family of parallel curves is then

X = u+tsinhu1+sinh2u=u+ttanhu
Y = coshu-t1+sinh2u=coshu-tcoshu

where t=0 correspond to the catenary.

Eliminating the parameter u from these equations; the latter gives  coshu=Y+Y2+4t2, i.e. u=arcoshY+Y2+4t2. Thus we obtain the implicit representation

arcoshY+Y2+4t2+ttanh(arcoshY+Y2+4t2)-X= 0
Title parallel curve
Canonical name ParallelCurve
Date of creation 2013-03-22 17:13:10
Last modified on 2013-03-22 17:13:10
Owner stitch (17269)
Last modified by stitch (17269)
Numerical id 21
Author stitch (17269)
Entry type Definition
Classification msc 51N05
Synonym offset curve
Related topic ParallellismInEuclideanPlane
Related topic NormalLine
Related topic HyperbolicFunctions