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 parallel 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  $\overrightarrow{F}(u):=(x(u),y(u))$,  its parallel curve $\overrightarrow{G}(u):=(X(u),Y(u))$  with offset $t$ is defined by

$X(u)$$=x(u)+{\displaystyle \frac{t{y}^{\prime }(u)}{\sqrt{x^{\prime }{(u)}^2+y^{\prime }{(u)}^2}}}$

$Y(u)$$=y(u)-{\displaystyle \frac{t{x}^{\prime }(u)}{\sqrt{x^{\prime }{(u)}^2+y^{\prime }{(u)}^2}}}$

0.1 Examples

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

$X(u)$$=$$t\cos u$

$Y(u)$$=$$t\sin u$

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)$$=$$\cosh u$

From the definition, the family of parallel curves is then

$X$$=$$u+{\displaystyle \frac{t\sinh u}{\sqrt{1+{\sinh }^{2}u}}}=u+t\tanh u$

$Y$$=$$\cosh u-{\displaystyle \frac{t}{\sqrt{1+{\sinh }^{2}u}}}=\cosh u-{\displaystyle \frac{t}{\cosh u}}$

where $t=0$ correspond to the catenary.

Eliminating the parameter $u$ from these equations; the latter gives $\cosh u=\frac{Y+\sqrt{Y^2+4t}}{2}$, i.e. $u=\mathrm{arcosh}\frac{Y+\sqrt{Y^2+4t}}{2}$. Thus we obtain the implicit representation

$$\mathrm{arcosh}\frac{Y+\sqrt{Y^2+4t}}{2}+t\tanh \left(\mathrm{arcosh}\frac{Y+\sqrt{Y^2+4t}}{2}\right)-X=\mathrm{\hspace{0.17em}0}$$