Parallel curve

A parallel of a <a href="/facts/Curve/xBQ0pTBW">curve</a> is the <a href="/facts/Envelope_(mathematics)/jr5DkCGD">envelope</a> of a family of <a href="/facts/Congruence_(geometry)/2tKJ6AhB">congruent</a> <a href="/facts/Circle/9fy5ggKn">circles</a> centered on the curve. 
It generalises the concept of <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel (straight) lines</a>. It can also be defined as a curve whose points are at a constant <a href="/facts/Normal_distance/P0aEswE4">normal distance</a> from a given curve. 
These two definitions are not entirely equivalent as the latter assumes <a href="/facts/Smoothness/mffL4ch2">smoothness</a>, whereas the former does not.
In <a href="/facts/Computer-aided_design/5gSXrmmN">computer-aided design</a> the preferred term for a parallel curve is offset curve. (In other geometric contexts, <a href="/facts/Offset_(disambiguation)/9GI6pwAR">the term offset</a> can also refer to <a href="/facts/Translation_(geometry)/KlpWmnfX">translation</a>.) Offset curves are important, for example, in <a href="/facts/Numerically_controlled/B2c7UbaX">numerically controlled</a> <a href="/facts/Machining/Yrjfj6IV">machining</a>, where they describe, for example, the shape of the cut made by a round cutting tool of a two-axis machine. The shape of the cut is offset from the trajectory of the cutter by a constant distance in the direction normal to the cutter trajectory at every point.
In the area of 2D <a href="/facts/Computer_graphics/lapBE8wv">computer graphics</a> known as <a href="/facts/Vector_graphics/y8ok7M2P">vector graphics</a>, the (approximate) computation of parallel curves is involved in one of the fundamental drawing operations, called stroking, which is typically applied to <a href="/facts/Polyline/GRQ6BMQS">polylines</a> or <a href="/facts/Polybezier/n0WASWTJ">polybeziers</a> (themselves called paths) in that field.
Except in the case of a line or <a href="/facts/Circle/9fy5ggKn">circle</a>, the parallel curves have a more complicated mathematical structure than the progenitor curve. For example, even if the progenitor curve is <a href="/facts/Smooth_function/mffL4ch2">smooth</a>, its offsets may not be so; this property is illustrated in the top figure, using a <a href="/facts/Sine_curve/mD1CB09M">sine curve</a> as progenitor curve. In general, even if a curve is <a href="/facts/Rational_curve/LnWsPmDP">rational</a>, its offsets may not be so. For example, the offsets of a parabola are rational curves, but the offsets of an <a href="/facts/Ellipse/1wUDnGxY">ellipse</a> or of a <a href="/facts/Hyperbola/6R6ER0mY">hyperbola</a> are not rational, even though these progenitor curves themselves are rational.
The notion also generalizes to 3D <a href="/facts/Surface_(mathematics)/whF08jvy">surfaces</a>, where it is called an offset surface or parallel surface. Increasing a <a href="/facts/Solid_(geometry)/tHH3JFoq">solid</a> volume by a (constant) distance offset is sometimes called dilation. The opposite operation is sometimes called shelling. Offset surfaces are important in <a href="/facts/Numerically_controlled/B2c7UbaX">numerically controlled</a> <a href="/facts/Machining/Yrjfj6IV">machining</a>, where they describe the shape of the cut made by a ball nose end mill of a three-axis machine. Other shapes of cutting bits can be modelled mathematically by general offset surfaces.

Parallel curve open-in-new

Parallel curve