Embedding a general planar graph into a grid

I have here a little problem with my homework, and would appreciate some direction.

I am attempting for some time now to show that every planar graph is embeddable into a grid (As large as needs be).

I tried to make this argument inductively in many ways, but I am afraid that I don't see any good way to do that so far. My main problem is that while the subgraph may be embeddable in a grid, it might not still be possible to connect the relevant nodes to the nodes added in the inductive stage after the reorganization.

While writing this down, I thought of another possible solution which might solve it - observing the embedding to the plain as an embedding in the Cartesian plane, and moving every node to a "very close" point whose both coordinates are rational. Taking the greatest common denominator of all the coordinates, I believe that I'll find myself in a (huge) grid as required.

There is a really neat result which answer's this question: Schnyder's Theorem. Another nice result is that of de Fraysseix, Pach and Pollack.

Here is a reference for both algorithms. The "Realizer Method" corresponds to Schnyder's Theorem and "Canonical Orderings" corresponds to de Fraysseix et al.'s approach.

These algorithms can embed any planar triangulation in an $O(n)\times O(n)$ grid. You can embed a planar graph $G$ by adding edges until you obtain a maximal planar graph $T$. Then you run any of these algorithms with input $T$. The output will be a drawing of $T$ in an $O(n)\times O(n)$ grid, which is also a drawing of $G$ in that same grid.

StackExchange Computer Science Q#26330, answer score: 4