Tuesday, October 26, 2010

Union, Tangencies and Colors

Union, Tangencies and Colors

I like the picture below very much.

What you see here are, more or less, pink, green and blue. The colors are chosen as such that they always form distinct colors, regardless of how you combine them. There are of course other combinations with these properties, but... this is at least one of them.

When the pink layer is slightly moved to the right, you see better how the image is constructed:

The pink (transparent) layer is the union of the two underlying (multi-)polygons, the green one and the blue one. It has one hole, one kind of indentation and for the rest it is square. The underlying multi-polygons are better seen if the pink one is removed:

So this is clear: there are two multi-polygons here, complete with holes and self-tangencies. These multi-polygons have overlap in some places, no overlap in other places. They nearly fill the area, there are only three "boxes" free. The blue and green colors nicely mix into blue-green. The colors and transparencies are selected for that. SVG definitions are:

Green: style="fill-opacity:0.5;fill:rgb(153,204,0);stroke:rgb(153,204,0);stroke-width:3"

Blue: style="fill-opacity:0.3;fill:rgb(51,51,153);stroke:rgb(51,51,153);stroke-width:3"

I like this combination.

And with the pink style on top of it...

Pink: style="fill-opacity:0.2;stroke-opacity:0.4;fill:rgb(255,0,0);stroke:rgb(255,0,255);stroke-width:8"

...all colors change, however, they are still clearly distinct. Yellow-orange like for green+pink; lila for blue+pink; and brown for the blue+green+pink. So we have seven unique distinct colors now, which happens to be the ideal number of colors for a map.

By the way, this image is created using Boost.Geometry. Boost.Geometry can easily create SVG images. And Boost.Geometry can overlay multi-polygons (already possible for a long time) having all kinds of self-tangencies (new!). A self-tangency in a multi-polygon is a place where two polygon-members touch each other (orange dots in the map below), or where a polygon touches itself (the red dot in the map).

Actually... this could also be a polygon, having five interior rings (holes) which might be mutually tangent and tangent to they exterior ring.

OK, one more image...

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