This was meant to be a comment but it was a bit too long, sorry!
There is a well known algorithm to draw a planar graph, namely Tutte's drawing algorithm. The input graph is assumed to be 3-connected and planar. The idea of the algorithm is to fix the position of vertices of a face in convex position and from those coordinates deduce the positions for the rest of the vertices. The resulting drawing being a planar drawing of the input graph. Perhaps you can find more about this at .
With respect to grids, there are a couple of well known algorithms for drawing planar 3-connected graphs. One is Schnyder's algorithm (through the a Schnyder wood decomposition of the edges), and the canonical ordering algorithm (through a partition of the vertices of the graph). Both of these algorithms run in $O(n)$ time and produce drawings in an $O(n)\times O(n)$grid, where $n$ is the number of vertices of the input graph. At  you may find an overview of these two algorithms for the case of planar triangulations. You could even go to references  and  cited there for the original papers. It is worth noting that these algorithms have been generalized to work on 3-connected planar graphs as stated above.
Are you thinking about drawings subject to more specific constraints?