Uniformly interleaved decorative knots

thingiverse

Knots formed by intricate combinations of sines and cosines are common and versatile, enabling the creation of complex curves. Loop crossings can be skillfully woven into simple knots with carefully selected sin/cosine functions, but these approximations often require large amplitudes to prevent intersections. In these designs, the interleaving process is achieved by calculating crossing points and then employing a segmented cosine function where each half-wavelength corresponds to the 'length' of the curve between crossings. This results in uniform separation at crossings, allowing for the construction of numerous configurations with uniform interleaving. The term 'length' refers to the angular difference in the independent variable t rather than the actual curve length between crossings, yet this approach yields surprisingly effective results. The OpenSCAD scripts are generated by an interactive tool that currently supports several functions, including 3-term sin/cosine functions and Lissajous curves. Parameter values can be adjusted using sliders or input directly, crossing calculations are performed, and SVG and OpenSCAD files are automatically generated. The curves are represented as arrays of points in the OpenSCAD code, making it possible to modify properties such as rope radius, resolution, vertical scaling, and others. To generate these objects, users can visit http://kitwallace.co.uk/3d/functional-objects.xq, where they can also save and restore designs. For more information, please refer to the blog post at http://kitwallace.tumblr.com/post/169901554274/uniform-interleaving-of-parametric-knots.