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Show HN: Delaunay Triangulating Robert Delaunay's Eiffel Tower Paintings

modalduality.org

39 pointsmodalduality11 comments

If you're using Python anyway, you can use scipy's triangulation module to avoid having to compile triangle:

https://docs.scipy.org/doc/scipy-0.14.0/reference/generated/...

The triangulation can then be plotted fairly straightforwardly with matplotlib's triangulation routines: matplotlib.pyplot.triplot and matplotlib.pyplot.tripcolor (https://matplotlib.org/examples/pylab_examples/tripcolor_dem...).

I did some work [1] on constrained Delaunay triangulations similar to this where visually important, and constrained, edges were automatically determined by Laplace-like filters.

[1] https://femtondev.wordpress.com/

It seems to me that the Voronoi diagram [0], the counterpart of Delaunay triangulation, is probably more useful as a general approach to this idea.

The cellular nature is great for lots of applications.

[0] https://en.m.wikipedia.org/wiki/Voronoi_diagram

If your goal is to make it "cellular" you can get an even more cellular appearance by using the Delaunay triangle centroids instead of the circumcenters like Voronoi diagrams use. Side by side comparisons https://www.redblobgames.com/x/1721-voronoi-alternative/ (scroll halfway down). The polygons end up looking rounder. I also made a little toy where you can drag an image onto the page to fill a cellular polygon diagram https://www.redblobgames.com/x/1722-delaunay-map-import/ (cartoon images work pretty well but fine detail is lost)

As suggested in the discussion points, thin lines/stripes get muddled up with wider adjacent surfaces. Is this a problem with the goal of the exercise ? Maybe it needs to be more clearly defined. I suppose the idea is that the method and the style fit together nicely, and as a result the paintings are representable quite efficiently with low resolution triangulations. It already mostly fits the bill, except for some exceptions to the main shapes represented. Does the author suggest further work should be devoted to improving the efficiency of the depiction? I guess the thin lines are quite important in keeping the paintings structured but I'm not sure what's to gain from that exercise if straying too far from the original triangulation — I think you'd lose the amusing coincidence of these concepts applying nicely together.

The main idea is to explore the concept of skinny triangles being unaesthetic (and thus undesirable for triangulations). This is true for many tasks in computational geometry, but not so true in art. Bringing these two together shows jarring conflict where the Delaunay Triangulation and your brain's sense of aesthetics disagree.

It's not really supposed to be useful, just exploration in reduction of an already reduced art style: what is lost when we enforce that all shapes are triangles that tend to be more large-angle? For one, I think this destroys the perspective since one common trope is to make objects narrower as they are farther away.

Although the entire thing is mostly a joke based on their shared last names.

Thanks for your answer.

For what it's worth I think the result is incredibly adapted to the art style; I think you overestimate your conclusions and I don't think that is fair to extrapolate claims over the importance of skinny triangles in art in general.

This representation conveys most of what the originals had to offer, and has the added benefit of being a light content in a meaningful format. It's a good sign that your questions pulled me into the subject but I don't think there's much more to extrapolate from the experiment.

Now how about trying delaunay triangulation on animated content with the added constraint of optimising for fluidity? (Like the works on incorporating art styles to pictures and videos from last year that you may have noticed, and that were published in siggraph.)

The other day I was looking at this coding challenge to automatically select points and create a Voronoi of an image: https://codegolf.stackexchange.com/questions/50299/draw-an-i...

Of course half the fun here is the coincidence of the last names :)

Interesting experiment! Per the talking points, I think a good color strategy would be to sample the lightest and darkest points from each region, and then use them as endpoints in a gradient fill.

Super cool! Why does the Delaunay triangulation avoid skinny triangles?

https://en.wikipedia.org/wiki/Delaunay_triangulation

Delaunay triangulations maximize the minimum angle of all the angles of the triangles in the triangulation

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