Uniformly interleaved decorative knots

Knots created by combinations of sines and cosines are common and generalise to produce curves which can be quite complex. Loop crossings can be interleaved in simple knots with carefully chosen sin/cosine functions but these are only approximations and often need large amplitudes to ensure crossing don't intersect..

In these designs, the interleaving is done by computing the crossing points and then using a segmented cosine function in which each half-wavelength is equal to the 'length' of the curve between crossings. This gives a uniform separation at crossings so that many more configurations can be constructed with uniform interleaving. 'length' here is measured by the angular difference in the independent variable t rather than the curve length between crossings but it seems to work rather well.

The OpenSCAD scripts are generated by an interactive tool which currently supports several functions including 3 term sin/cosine functions and Lissajous curves.The parameter values can be adjusted using sliders or input directly, crossings computed and SVG and OpenSCAD generated. The curves are represented as an array of points in the OpenSCAD code so the shape cannot be changed there but the rope radius, resolution, vertical scaling and other properties can be changed.

To generate these objects, try http://kitwallace.co.uk/3d/functional-objects.xq where you can also save and restore designs..

See my blog post http://kitwallace.tumblr.com/post/169901554274/uniform-interleaving-of-parametric-knots for details