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PARALLEL CURVE OF A CURVE


Notion studied by Leibniz in 1692.

 
The curves (G1) and (G2) are parallel if respective current points M1 and M2 can be determined so that .
For an initial curve  with current point , a parallel curve is a set (Ga) of points .
Cartesian parametrization: .
The parallel curves of an algebraic curve are algebraic.
Curvilinear abscissa at the current point on (Ga) (oriented by the same tangent vector as (G)): .
Radius of curvature: .
The length of an arc of (Ga) is equal to the length of the corresponding arc on (G0) minus a multiplied by the angle spanned by the tangent between the beginning and the end. For example, for an eight-like curve, the two curves have the same length. The length of (G0) is equal to the mean of the lengths of (Ga) and (G-a).
The area of the strip included between two corresponding arcs of (G0) and (Ga) is equal to the length of the median arc of (Ga/2) times a, under the condition that the strip does not intersect itself, nor does it overlap on the evolute of (G0).

Two curves are said to be parallel of one another if any curve normal to one is normal to the other; it can be proved that, then, the distance between two points with common normal is a constant, called parallelism distance. Do not mistake with the image of a curve under a translation.

Two curves are therefore parallel to one another if they are the loci of the ends of a segment line of constant length moving perpendicularly to its direction, which is equivalent to saying that the line carrying this segment rolls without slipping on its envelope.

See also at reptoria the generalisation of parallel curves by the crawling motion of a circle on a curve.

As well as for lines, the parallelism relation of plane curves is an equivalence relation.
An equivalence class is the set of all trajectories of points linked to a line rolling without slipping on a curve, which is the common evolute to all these curves.
The parallel curves of a curve are therefore the involutes of its evolute.

The parallel curves of a curve (G0) are the curves (Ga), parallel of index a of (G0), obtained by algebraically copying a "length" a from the points on (G0) on the oriented normal; in other words, they are the loci of the points M where  is the normal vector at M0. Since the parallelism relation is symmetrical, (G0) is also parallel to (Ga).

The reunion of (Ga) and (G-a) is the envelope of the circles with radius a centred on (G0); therefore, it is also the visible outline of a tube, the bore of which is projected along (G0).

If the curve (G0) is placed on a plane in a motion of circular translation with radius a with respect to a fixed plane, then the envelope in the fixed plane is, again, the reunion of (Ga) and (G-a).

The parallel curves of a curve (G0) can also be considered as the plane contour lines of an equal slope surface with directrix (G0).

Physical interpretation: if the curve (G0) is a light source, according to the Huygens principle, the "wavefronts" are the envelopes of the elementary circular wavelets emitted by all the points on the curve (G0); they are therefore exactly the curves parallel to (G0).

The singularities of parallel curves describe the evolute of the initial curve; with the previous physical interpretation, the evolute therefore represents the place where the light rays emitted by the curve are concentrated.

Examples:
    - the curves parallel to a line are the lines parallel to this line (!)
    - the involutes of a curve are parallel to one another.
    - the toroids are the parallel curves of the ellipse
    - the Cayley sextic is one of the parallel curves of the nephroid
    - The parallel curves of the parabola x² = 2p y are the curves parametrized by: 

    - It can happen that the curves (Ga) and (G-a) are equal. Then, the curve (Ga) is parallel to itself at distance 2a.
 
The red curve is parallel in two ways to the blue curve, and self-parallel.

 A similar notion is the notion of contour line of the function "distance (of a point on the plane) to the curve", called distance curve (or line). These contour lines are composed of portions of parallel curves and arcs of circles, and are interesting because they form a partition of the plane, as opposed to parallel curves.


In green, the parallel curves, and in red, the contour lines of the function "distance to the curve"; when they do not coincide, the latter are composed of arcs of circles.

See also the 3D parallel curves and the parallel surfaces.
 
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© Robert FERRÉOL 2017