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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 from a given curve.〔 〕 These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not. A parallel curve is also called an offset curve〔〔 and this is the preferred term in CAGD. (In other geometric contexts, the term offset can also refer also to translation.) Offset curves are important for example in numerically controlled machining, where they describe for example the shape of the cut made by a round cutting piece 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.〔http://www.slideshare.net/Mark_Kilgard/22pathrender, p. 28〕 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 adjacent figure using a parabola 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 offset surface.〔 Increasing a solid volume by a (constant) distance offset is sometimes called ''dilation''.〔http://www.cc.gatech.edu/~jarek/papers/localVolume.pdf, p. 3〕 The opposite operation is sometimes called ''shelling''. == Alternative definitions == 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Parallel curve」の詳細全文を読む スポンサード リンク
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