I am an urban planning graduate with experience in related software. I found two topics regarding cities' plans interesting:
- Delaunay triangulation
- Voronoi diagram
Imagine! I can use Voronoi diagram in order to create zones based on the centers (these centers can be Hospital, Fire station, school, etc).
I am looking for more topics regarding geometry which are possible to apply in the 2D plane. In this case, I am not sure which topics, books, or resources I should research more. I would be appreciated if you give me some reference to read or name a few diagrams or geometry topics which it is possible to use in 2D plans.
When you refer to zoning, I assume you mean this. Unless you specify what you're trying to optimize for and in what context you intend to apply the knowledge you seek, I cannot give you any better recommendations. In fact, this is a question that is better suited for the Engineering StackExchange, where you will find civil engineers and perhaps other urban planners such as yourself that might have experience dealing with issues such as this and thus could give you guidance that is a bit more focused than you will find here. If you were to ask your question there and then come back here with more specific requests, I'm sure we would be able to better assist you (for example, if you wanted to optimize for traffic flow, I could point you to traffic flow theory).
Having said that, graph theory, as @justabit has mentioned, and in particular the four color theorem could be of interest to you. Start with the wikipedia articles and, if you feel like there might be something to it, there are a variety of different books on the topic (Springer's Graduate Texts in Mathematics collection offers you a few good picks); fair warning, most of them will be quite advanced. Graph theory is a part of discrete mathematics, about which a book has already been recommended to you (the book justabit recommended includes a section on Voronoi diagrams and mentions the Delaunay triangulation, in case you haven't checked it out yet).
I have also found this article, which touches very lightly on a few ideas that you might enjoy. Check out the bibliography at the end of the page and see if the writings/authors have anything worthwhile to offer you (in particular Nikos Salingaros).
Lastly, a quick web search of "mathematics of urban planning" will produce a variety of results, some of which you may find worth looking into.
I don't know if these recommendations will be helpful to you, but I hope this will at least give you some idea about where to start looking. There's a lot to learn on nodes and the connections between them through graph theory, for example. Keep in mind that as someone whose background is in mathematics only I have no idea how you might (or if you could, even) apply these. I will again recommend that you pose this question to the civil engineers on the Engineering SE.