For instance, pick a geometry we have studied which you find interesting. Is there some aspect of it which was discussed briefly in class but which we didn't pursue? Is there some way of changing the rules which intrigues you?
If you're not having much luck, browse through the textbooks. Look at the books on reserve at the library. Talk to me.
Once you have tentatively chosen a topic, write a few sentences explaining it. If you are creating your own model, describe exactly what it is. If there's something missing from a proof, or from the coverage of a topic in one of the books, or whatever, describe what's missing.
Turn in your choice of topic, together with the brief explanation.
Now that you have chosen the topic, you should know at least in principle what geometry model(s) you will be working with. The next step is to decide what questions to ask about it. So make up a list of questions about your model. Does it need a distance function? Do you plan to determine what corresponds to circles?
Select several of these questions (1 is too few; 10 is too many) which you hope to answer while writing your essay. Divide them into appropriate categories. Now you're ready for the outline: Start with an introduction, end with a summary/discussion/conclusion, and put the various (categories of) problems in the middle. Briefly describe each part.
Turn in your outline.
Solve the problems. This is the fun part!
Write up what you did. You need to include enough detail so that people can understand it. Most calculations should be given explicitly. Lots of figures (with suitable captions/descriptions) are a big help. But you also need to include enough words so that people can understand it; theorems and proofs may be appropriate, but are certainly not sufficient.
Turn in your rough draft.
Be a perfectionist. Fix your math mistakes. Fix your grammar mistakes. Fix your spelling mistakes. Make sure your logic is sound. Make sure your reader will know at each stage what you're doing. Perhaps some reminders are needed: ``Now we will solve the Dray conjecture'' or ``We therefore see that the Dray conjecture is false''.
A discussion of the history of non-Euclidean geometry is not appropriate. A comparison of different (historical) versions of neutral geometry might be.
This does not necessarily mean that you must do something nobody's ever thought of before, although you'll certainly get brownie points if that is the case. You do need to work through the math yourself, and present the results in your own words.
You may use whatever references you can find which might be appropriate. But you must give appropriate credit. A direct quote, for instance the statement of a postulate or a theorem, should be clearly labeled as such. A figure which appears elsewhere must be so labeled. It is not appropriate to make minor changes in text, or to redraw a figure, without giving a proper reference; this is plagiarism. By all means paraphrase an argument you find elsewhere. But give credit to the author. And don't fill up the entire essay this way; that's a book report.
Your references should appear separately at the end of your essay, with a section heading such as References or Bibliography. Full publication data must be given, including title, author(s), publisher, and year. Page numbers may be given if appropriate.
Your essay should be easy to read. Ask a friend to read it. Tell them not to worry about the details. Is the argument clear? They should be able to read the introduction and conclusion and tell you what your essay is about. Can they?
Your essay should be easy to read in another sense: Type it (or use a word processor)! Get that new ribbon/cartridge you've been thinking about! Use section headings. Indent your paragraphs. Don't run lengthy equations into the text - display them neatly on separate lines. (You may hand-write equations if you can not type special symbols.)
By all means include lots of figures! These can appear in the text or on separate pages at the end, and may be hand-drawn. Each one should have a label such as Figure 1 as well as a caption. You must describe each figure in the text in enough detail so the reader can figure out why it's there.
Your essay should be 5-7 pages long not counting figures and lengthy equations. Somewhat longer is OK; shorter is not.
It's a good idea to get the math right!