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Prompted by the question What regular polygons can be constructed on the points of a regular orthogonal grid?:

A regular octagon can be approximated on a quad lattice (grid) to about $1\text{%}$ error by knowing that the length of the diagonal of a square is $\sqrt{2}$ (~$1.414$) times as long as its side. With that information we can draw a "regular" octagon by marking the four lattice points 7 orthogonal lengths from a center point and marking the four lattice points 5 diagonal lengths from the same center point.

Is there a general rule that can be applied to create close approximations of other regular polygons on a quad-lattice (triangle, pentagon, enneagon, decagon, dodecagon, etc.)?

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oosterwal
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    You can approximate any regular polygon to any desired (relative) tolerance simply by making it large enough compared to the grid. It's just a matter of rounding each corner to the nearest grid point ... – hmakholm left over Monica Sep 15 '11 at 20:34
  • @Henning Makholm: I understand that. The example I gave for the octagon is relatively simple. Is there a simple method for determining the optimal points for other polygons, especially those that are not multiples of 4? – oosterwal Sep 15 '11 at 20:39
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    The real challenge is to produce best-possible approximations relative to the size of the grid. This is related to simultaneous diophantine approximation. – Robert Israel Sep 15 '11 at 22:04
  • How do you measure the "closeness" of an approximation? – Peter Kagey Dec 21 '18 at 17:20
  • I think my answer at https://math.stackexchange.com/questions/2251555/drawing-approximated-regular-shapes-on-square-grid/3067129#3067129 might answer this question. – Timothy Jan 09 '19 at 17:29
  • @Timothy I don't see how it does, but please feel free to expand to an answer here ;) – Giovanni Funchal Jul 17 '19 at 13:33
  • BTW, this question is related to https://mathoverflow.net/questions/150938/regular-lattice-polygons – Giovanni Funchal Jul 17 '19 at 13:37

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