Curvature of the Projection of a Trefoil Knot

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.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

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 .
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.