Projective Geometry Question.

Question

I want to learn projective gemetry. I have the coxeter book and found some youtube videos. I have just started. I have tried answering this Challege question from the video, and so far. I have made many lines on paper, but haven't gotten anywhere near having anything that is worthwhile.

So I am asking for help, because projective geometry is strange for me, but I like it and would like to really understand it. Thus, hints, video, flash games, and/or solutions would be warmly welcomed.

How can you tell if an arbitary 5-gon is the image of a regular 5-gon under projectiviy?

I am thinking that I will need 10 lines, 3 for the line and there are five sides which each need a line, but trying to draw this, is beyond my ability. Edit: I have done more research, and now believe that The Theorem of Pappus is what I need, now just trying to draw it.

Answer 1

You sound like you want to get stuck into the subject by doing it rather than theorising about it.

I most strongly recommend you the book on Projective Geometry by Lawrence Edwards. My Second (revised) Edition was published by Floris in 2003, I do not know if there might be a later one. Edwards takes you through with short, almost equation-free discussions followed by plenty of drawing exercises. Pappus' theorem first appears on Page 30, in a drawing exercise, naturally - just the way Pappus did it!

In your 5-gon problem, we begin by noting that the before and after images of any quadrilateral define a projective transformation. When you add a fifth corner point to make a 5-gon, its image is determined by that transformation. All you need to do is to see whether that coincides with the fifth point of the other 5-gon. Since one of the pentagons is regular, you may find that use of cross-ratios will simplify things, since these are preserved under projective transformation. This may be solved analytically or, if you enjoy Edwards' treatment, on the drawing board.