Instead of finding three points for a triangle, build a triangle fan around a random point, and then group quads into polygons outside of that.
I'm going to fill this in with some map vertices. cx: 26, cy: 27
polygon(48% 45.4%, 37.08% 51.53%, 27.98% 50.55%, 23.55% 43.14%, 19.65% 39.12%, 18.87% 30.5%, 15.84% 22.98%, 18.28% 14.55%, 22.23% 13.45%, 24.87% 6.78%, 28.64% 10.47%, 32.24% 12.7%, 33.67% 21.26%, 36.43% 31.41%)
13 vertices
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