Graphs are mathematical models that contain information about objects (vertices) and relations between those objects (edges). A drawing, also called an embedding, of a graph is made by representing the vertices as points in R² and the edges as curves between their endpoints. When these curves only intersect in their endpoints, the embedding is planar. Graphs that have such an embedding are planar graphs. There is no fixed set of rules when it comes to drawing a graph. So how to decide how a graph should be drawn?}

One method of drawing a graph is called the rubber band representation, where some vertices are initially fixed as a strictly convex polygon on the Euclidean plane and all remaining vertices are placed in the barycenter of their neighbours. For the rubber band representation of 3-connected planar graphs, further referred to as the Tutte embedding, Tutte's theorem states that all connected regions that are bounded by edges (called faces) are strictly convex, these faces do not contain vertices or edges.

This research adapted the optimization problem corresponding to the Tutte embedding by changing the objective function, while making sure the embedding remained compliant with Tutte's theorem. In doing so, new methods for graph drawing can be examined and used in various areas, depending on the individual context and application of the drawing.

Two types of objective functions were analysed. The first type, based on existing research, minimizes the difference between the length of the edges and their desired length. The second type was proposed by this research and minimizes the differences between the surface area of all faces within the embedding.

The first type of objective functions yielded embeddings with a similar structure as the Tutte embedding, maintaining the symmetries of the graph, but with different proportions. The embeddings that were yielded by the second type of objective function were different. Some of which do not maintain the symmetric characteristics of the graph. A notable feature is that for some graphs, there exist several local minima with corresponding embeddings that are significantly different from each other.

This report contains the analysis of the different objective functions and their results. The results show that there are multiple ways to draw 3-connected planar graphs compliant with Tutte's theorem. Moreover, it provides options for further analysis of the objective functions and paves the way for further research possibilities.