Curvature of the Projection of a Trefoil Knot

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The trefoil knot is the simplest example of a nontrivial knot. This Demonstration shows the projection of a two-parameter version of the trefoil knot and plots the minimum (in blue) and maximum (in red) curvature points along the knot. The maximum and minimum osculating circles (also known as the kissing circles or the circles of curvature) are drawn as well.

Contributed by: Aaron Becker (February 2012)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The two-parameter projection of a trefoil knot has parametric equations

,

,

with . The parameter scales the knot size and controls the spread or compactness of the knot. The curvature is defined as the reciprocal of the radius of curvature of the path's osculating circle, and is given by .



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