CS 791 Assignment 2 extensions

What follows are some suggested non-trivial extensions to the implementation component of Assignment 2. You are welcome (and indeed encouraged) to dream up your own ideas for extensions. These are presented very roughly in increasing order of complexity. The harder ones make fine topics for final projects in the course.

• Add a way to selectively disable the rendering of some grid cells, so that you can produce patterns that don't have a square footprint.
• Add a user interface for specifying break markers interactively. Even better, be able to draw the knot and edit the rendering parameters interactively too.
• Come up with some new rendering styles for knots. This could mean a few different things:
  • Change the shapes of the strands into something more interesting.
  • Leave the shape alone, but apply rendering effects to those shapes (multiple bands, new interlacing types, textures along strokes, etc.).
  • Develop some painting hacks to turn a simple postscript drawing into a rasterized image with texture, faux-3D, etc. See Glassner's Celtic knotwork articles from CG&A for inspiration.
• Assign different colours to each unique strand in a knotwork.
• Add zoomorphs, key patterns, or other Celtic ornaments. See Matthew Kaplan and Elaine Cohen, Computer generated Celtic design, 2003 Eurographics Workshop on Rendering.
• Produce knotworks on different grids (i.e., different tilings), perhaps using the technique of Christian Mercat. Consider disk-shaped patterns, phyllotaxes, aperiodic tilings, spherical designs, or anything else that grabs you.
• Produce attractive knotworks on 3D polygonal meshes.
• Implement the method of Cameron Browne to fit knotworks to arbitrary shapes. See Cameron Browne, Font Decoration by Automatic Mesh Fitting, RIDT'98.
• Find a space-filling 3D analogue of Celtic knots. Glassner touches on this topic in the third article of his series on Celtic knotwork. This paper by Carlo Séquin might also provide some inspiration.
• If you're interested in knot theory, add code to automatically determine the knot-theoretic properties of a knotwork. How many independent strands are there? Is it truly knotted, or merely an unknot? What patterns of break markers will guarantee a single strand? What's the simplest Celtic trefoil? Are Celtic knots a good way to present a large family of elementary knots in an attractive, intuitive way?