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 can also be defined as a curve whose points are at a fixed normal distance of a given curve.
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.
A curve that is a parallel of itself is autoparallel. The involute of a circle is an example.
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.
For a parametrically defined curve, the following equations define one branch of its parallel curve with distance (the other branch is obtained with ):
As for parallel lines, a normal line to a curve is also normal to its parallels.
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.
The Archimedean spiral is often called a spiral with constant distance between its successive coils. Although this property is approximately true for the outer coils of the spiral, it is nowhere the case exactly, and near the center it is obviously wrong. However, there is a spiral which has exactly the geometric property of constant distance between successive coils in the sense of parallel curves, the involute of a circle .