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Parallel Curves


ParallelCurves

Parallel curves, frequently called "offset curves" in computer graphics applications, are curves which are displaced from a base curve by a constant offset, either positive or negative, in the direction of the curve's normal. The two branches of the parallel curve a distance k away from a parametrically represented base curve (f(t),g(t)) are

x=f+/-(kg^')/(sqrt(f^('2)+g^('2)))
(1)
y=g∓(kf^')/(sqrt(f^('2)+g^('2))),
(2)

where f^'=df/dt and g^'=dg/dt. The above figure shows curves parallel to a circle, ellipse, and 3-petalled rose curve, where the base curves are indicated in red.


See also

Circle Parallel Curves, Ellipse Parallel Curves, Parallel, Parallel Lines

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References

Gray, A. "Parallel Curves." §5.7 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 115-117, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 42-43, 1972.Yates, R. C. "Parallel Curves." A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 155-159, 1952.

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Parallel Curves

Cite this as:

Weisstein, Eric W. "Parallel Curves." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ParallelCurves.html

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