Parallel curve

A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of parallel lines. It is sometimes called an offset but that often refers also to translation.

Alternatively, one can fix a circle and a point on the curve and take the envelope of the translations taking that point to the circle.

Tracing the center of a circle rolled along the curve (see roulette) would give one branch of a parallel.

A curve that is a parallel of itself is autoparallel. The involute of a circle is an example.

For a parametrically defined curve its parallel curve with distance a is defined by the following equations:

$X[x,y]=x+{\frac {ay'}{\sqrt {x'^{2}+y'^{2}}}}$

$Y[x,y]=y-{\frac {ax'}{\sqrt {x'^{2}+y'^{2}}}}$

External links

• Parallel curves on MathWorld