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Tubes

A tube of radius r of a set gamma is the set of points at a distance r from gamma. In particular, if gamma(t) is a regular space curve whose curvature does not vanish, then the normal vector and binormal vector are always perpendicular to gamma, and the circle N^^(t)costheta+B^^(t)sintheta is perpendicular to gamma at gamma(t). So as the circle moves around gamma, it traces out a tube, provided the tube radius r is small enough so that the tube is not self-intersecting. A formula for the tube around a curve is therefore given by

 S(t,theta)=gamma(t)+r[-N^^(t)costheta+B^^(t)sintheta],

for t over the range of the curve and theta in [0,2pi]. The illustrations above show tubes corresponding to a circle, helix, and two torus knots.

The surface generated by constructing a tube around a circle is known as a torus.


See also

Borromean Rings, Knot, Link, Torus

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References

Gray, A. "Tubes about Curves." §9.1 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 207-209, 1997. Bar-Natan, D. "Tube Plot Package." http://www.math.toronto.edu/~drorbn/KAtlas/Extras/TubePlot.m.

Referenced on Wolfram|Alpha

Tube

Cite this as:

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

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