A parallel curve of a given (progenitor) curve is the envelope of a family of congruent (equal-radius) circles centered on the curve. It generalises the concept of parallel (straight) lines. It can also be defined as a curve whose points are at a constant normal distance from a given curve. These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not. In computer-aided design the preferred term for a parallel curve is offset curve. (In other geometric contexts, the term "offset" can also refer to a translation; however, a parallel curve may have a different shape than its progenitor.) Offset curves are important, for example, in numerically controlled (NC) machining, 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 computer graphics known as vector graphics, the (approximate) computation of parallel curves is involved in one of the fundamental drawing operations, called stroking, which is typically applied to polylines or polybeziers (themselves called paths) in that field.
Except in the case of a line or circle, the parallel curves have a more complicated mathematical structure than the progenitor curve. For example, even if the progenitor curve is smooth, its offsets may not be so; this property is illustrated in the top figure, using a sine curve as progenitor curve. In general, even if a curve is rational, its offsets may not be so. For example, the offsets of a parabola are rational curves, but the offsets of an ellipse or of a hyperbola are not rational, even though these progenitor curves themselves are rational. The notion also generalizes to 3D surfaces, where it is called an offset surface or parallel surface. Increasing a solid volume by a (constant) distance offset is sometimes called dilation (similar to the dilation image operation). The opposite operation is sometimes called shelling. Offset surfaces are important in NC, 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 of a parametrically given curve If there is a regular parametric representation x → = ( x ( t ) , y ( t ) ) {\displaystyle {\vec {x}}=(x(t),y(t))} of the given curve available, the second definition of a parallel curve (s. above) leads to the following parametric representation of the parallel curve with distance | d | {\displaystyle |d|} :
x → d ( t ) = x → ( t ) + d n → ( t ) {\displaystyle {\vec {x}}_{d}(t)={\vec {x}}(t)+d{\vec {n}}(t)} with the unit normal n → ( t ) {\displaystyle {\vec {n}}(t)} . In cartesian coordinates:
x d ( t ) = x ( t ) + d y ′ ( t ) x ′ ( t ) 2 + y ′ ( t ) 2 {\displaystyle x_{d}(t)=x(t)+{\frac {d\;y'(t)}{\sqrt {x'(t)^{2}+y'(t)^{2}}}}}
y d ( t ) = y ( t ) − d x ′ ( t ) x ′ ( t ) 2 + y ′ ( t ) 2 . {\displaystyle y_{d}(t)=y(t)-{\frac {d\;x'(t)}{\sqrt {x'(t)^{2}+y'(t)^{2}}}}\ .}
The distance parameter d {\displaystyle d} may be negative. In this case, one gets a parallel curve on the opposite side of the curve (see diagram on the parallel curves of a circle). One can easily check that a parallel curve of a line is a parallel line in the common sense, and the parallel curve of a circle is a concentric circle.
Geometric properties Source:
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