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Two definitions of a parallel curve: 1) envelope of a family of congruent circles, 2) by a fixed normal distance. The parallel curves of a circle (red) are circles, too. A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of parallel (straight) lines.
Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines.
In Euclidean geometry, parallel lines are always parallel in both directions, but hyperbolic lines are only parallel in one direction, and they diverge in the other direction. So given a line, and a point not on that line, there are two parallel lines, one in each direction.
Parallel curves. The curves (G 1) and (G 2) are parallel if we can determine current points M1 and M2, respectively, such that and . For an initial curve with current point , a parallel is a set (G a) of points where , the angle being the torsion angle (see the notations).
The parallel curves of a curve are the curves , parallel of index a of , obtained by algebraically copying a "length" a from the points on on the oriented normal; in other words, they are the loci of the points M = where is the normal vector at M 0.
Parallel Curves. Curves can also be parallel when they keep the same distance apart (called "equidistant"), like railroad tracks. The red curve is parallel to the blue curve in both these cases: Parallel Surfaces. Surfaces can also be parallel, like this: Lines and Planes.
Feb 9, 2018 · 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.
