What a great game!!
If you haven't already, you really ought to go check out planarity.net. John Tantalo has really made a great game for us. Here I share some strategy and some screen grabs from this game. Here are some from a level 25 run.
A great thing about planarity is that you just move the points around, and let your instinct tell you where to put them. Move them back and forth and you'll get a feel for it. After playing this game for a while, I realized that it doesn't really get harder at the higher levels - it only takes longer as there are more vertices.
But what is really going on? Can we quantify how the mind solves this game? The most straightforward approach is a nodal one. Pick one vertex at random, and move it to the top of a clear area. That will be your node 0. Put all vertices which it connects below it in a line. Now, for each of those nodes repeat the process, making sure that the edges above don't cross. This may require that you re-order already placed nodes. Here are a few pictures of games solved in this manner:
Another way to solve the puzzles in a strict algorithm that is a bit fast for humans is to make a bounding shape. To do so requires finding three or more vertices that can form a closed loop. This is easier than you might think, even in the highest levels. Once you have formed the closed loop, you can build a nodal solution (as described above) to the interior of the shape from each vertex. Here are a few pictures of games solved in this manner:
'Closed Loop Solutions
An adaptive closed loop is solution is more like what the brain does automatically.. it builds closed shapes and pursues a nodal solution, but when re-ordering the nodes becomes difficult a new bounding shape can be made in the exterior of the original one to speed up the process. Here are a couple of examples:
Adaptive Closed Loop Solutions
Finally, there is the artisitc approach. Here, we move the vertices as our aesthetic sense tells us in a pretty one. In general the puzzle forces a certain pattern on us, and we build our picture from there.
Good luck, and happy planaritying! I'd love to see your screengrabs - please send them to me and I'll post your picture and name - luke.saul[-\at-]gmail.com.