curved drawing of the dual of the Herschel graph

The graph drawn above is the planar dual to the Herschel graph. The Herschel graph has nine quadrilateral faces, so its dual is a 9-vertex 4-regular planar graph, and therefore can be drawn as the arrangement graph of a system of smooth curves, or as in this case as the self-crossings of a single smooth curve.

Much of the research in graph drawing uses a drawing style in which edges are drawn as straight line segments, or as sequences of straight line segments articulated by "bends", corners where two line segments meet. But I think that, in many cases, smooth curves can create a more interesting aesthetic effect, that the extra freedom inherent in drawing curves makes it possible to spread out the features of the drawing more evenly, and that smooth curves are likely to be easier for the eye to follow than sharp bends.

No doubt Mark Lombardi would agree.




Comments:

patrickwonders:
2009-10-15T14:02:25Z
I would agree. The way you have the vertexes arranged here, you could have used straight lines. It would have been more crowded. Edges would be close enough together to make it tougher to follow. If you broke the curves up into three or four straight segments instead, the extra corners would be tougher to visually navigate.
phoe6:
2009-10-15T18:25:58Z
yeah, probably true, except for the cases when you want to represent graphs for any distant related problem (Shortest Path), one might be inclined to draw straight lines, again this just depends upon our visualization. I wonder how trees would be if it were curves. Trees better with straight lines. :)
11011110:
2009-10-15T18:32:42Z
Go look at some of Lombardi's art. Even the tree-like parts look pretty with curves.
patrickwonders:
2009-10-15T23:23:34Z
I've always liked the way trees look on the Poincare disk
patrickwonders:
2009-10-15T23:24:55Z
And, here's a less useful DAG rendered with a fish-eye effect.
11011110:
2009-10-15T23:34:32Z
Re Poincar: A good point. Although I tend to think of hyperbolic lines as being straight, really, if only you had eyes within a hyperbolic universe to see them that way. Your DAG picture is pretty, in a funhouse-mirror sort of way, but I wonder how effective it is at conveying the information represented by the DAG. I guess that's what you mean by "less useful"?
patrickwonders:
2009-10-16T00:10:30Z
True on both points. Alas, my eyes live very close to curvature zero. The DAG rendering was mostly toy. It has a huge number of edges and while my code made an effort to keep vertexes apart, it made no attempt to keep edges apart. And, the viewpoint was chosen by trial-and-error by me.