Equivalent Straight Line Embedding of a Planar Graph Drawing on a Grid
"Schnyder's algorithm" computes an embedding on an $O(n)\times O(n)$ grid. See here for a reference, and see here for a more general treatment that is not behind a paywall. Note that if your initial (topological) embedding is not unique, you can just add edges (while preserving planarity) until the graph becomes 3-connected (or even a triangulation) and then apply some drawing algorithm. Once the layout is computed you simply remove the augmentation.

I want to point out that there is also a different drawing algorithm for the $O(n)\times O(n)$ grid by De Fraysseix, Pach and Pollack. It is based on the canonical order of the planar graph. See here for the reference if the graph is a triangulation, and here for 3-connected planar graphs.enter link description here. These articles are behind paywalls, but the algorithms are standard and you will find lecture notes if you google the keywords. You might also try to check out the handbook of graph drawing for more information.