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Ian Barnes
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Minimal personal web site | ||
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View it at <http://www.ianbarnes.me.uk/> | ||
View it at <https://www.ianbarnes.me.uk/> |
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--- | ||
layout: post | ||
title: Visualisation of periodic surfaces | ||
date: 2020-12-20T01:26:08+01:00 | ||
usemathjax: true | ||
--- | ||
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Here's an image I generated while I was writing my [PhD thesis][thesis]. This is | ||
one repeating unit of the surface $$ \cos X + \cos Y + \cos Z = 0 $$, which is | ||
similar in many ways to the Schwarz P-surface, a periodic minimal surface. | ||
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 | ||
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Here's another. This one approximates the Neovius minimal surface, and is | ||
the solution of the equation | ||
$$ | ||
3 ( \cos X + \cos Y + \cos Z ) + 4 \cos X \cos Y \cos Z = 0 | ||
$$ | ||
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 | ||
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These images were all generated by a Pascal program that divided the repeating | ||
unit cube into 20x20x20 smaller cubes. For each small cube, it computed the | ||
function values at the 8 vertices, and if there were any sign changes, | ||
approximated the positions of the zeros of the function along each edge by | ||
linear interpolation, then joined these points to form a polygon that | ||
approximates the part of the surface inside that little cube. This was done in | ||
order from back to front so that closer polygons overwrote and obscured those | ||
further away from the viewing point, thus avoiding the need for complex hidden | ||
surface removal algorithms. | ||
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Each image took about an hour to produce on [Macintosh IIci][], a pretty powerful | ||
desktop workstation back in 1989. | ||
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[thesis]: https://openresearch-repository.anu.edu.au/handle/1885/49316 | ||
[Macintosh IIci]: https://en.wikipedia.org/wiki/Macintosh_IIci |
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