What is the standard definition of Planar 3-SAT? I have seen a number of different definitions. What was the original paper that defined it and proved it to be NP-complete?
There's a nice compilation of definitions of related NP-complete planar satisfiability problems at http://courses.csail.mit.edu/6.890/fall14/scribe/lec7.pdf
One of them, planar monotone 3-sat, allows you to split each terminal into positive and negative, with the terminals placed along a line with the positive part on one side of the line and the negative part on the other side of the line. The clauses have only positive or only negative terminals and are placed on the positive or negative side of the line respectively.