page_title: Parallel curve
section_title: Normal fans
hierarchical_section_title: Parallel curve / Parallel curve to a curve with a corner / Normal fans
caption_attribution_description: English: Normal fan for defining parallel curves around a sharp corner.
context_page_description: 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.
  In computer-aided design the preferred term for a parallel curve is offset curve. 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 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 computation of parallel curves is involved in one of the fundamental drawing operations, called stroking, which is typically applied to polylines or polybeziers 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.
context_section_description: As described above, the parametric representation of a parallel curve, , to a given curver, , with distance  is:
   with the unit normal .
  At a sharp corner (), the normal to  given by  is discontinuous, meaning the one-sided limit of the normal from the left  is unequal to the limit from the right . Mathematically,
  .
  However, we can define a normal fan  that provides an interpolant between  and , and use  in place of  at the sharp corner:
  where .
  The resulting definition of the parallel curve  provides the desired behavior:
caption: Normal fan for defining parallel curves around a sharp corner
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