Parallel curve

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An ellipse (red), its evolute (blue) and some parallel curves (green). Note how the parallel curves have cusps when they touch the evolute

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 the offset curve but the term "offset" often refers also to translation. The term "offset curve" is used, e.g., in numerically controlled machining (and in other computer graphics applications), where it describes the shape of the cut made by a round cutting piece, which is "offset" from the trajectory of the cutter by a constant distance in the direction normal to the cutter trajectory at every point.

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:

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

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

When parallel curves are constructed they will have cusps when the distance from the curve matches the radius of curvature. These are the points where the curve touches the evolute.

If the initial curve is a boundary of a planar set and its parallel curve is without self-intersections, then the latter is the boundary of the Minkowski sum of the planar set and the disk of the given radius.