# What knowledge should I have before learning projective geometry? (I'm an 11th grade student)

I wanted to learn projective geometry. I don't know much about it. I came across projective geometry in a book called 'Euclidean geometry in mathematical olympiads' and I was very interested in it. So I tried the book projective geometry by HSM Coxeter but so far it has no mention of cross ratios which seemed quite important.

I looked at a previous thread about what book to read on it Book suggestions on projective geometry but the book mentioned 'perspectives on projective geometry' is out of my skill level (I am just starting my 11th grade).

So is there any book within an 11th grade skill range and if not, what do I need to learn before I am ready for projective geometry?

If you want book recommendations, I can recommend Dover Publications books highly. They are inexpensive and often are more easily understood than books from academic publishers. The Dover book Intoduction to Projective Geometry by C. R. Wylie, Jr. seems to be the kind of book you are looking for. As for prerequisites, a feeling for geometry is probably all that you need for a start. Don't expect to understand everything the first time through, unlike a novel. You have to play with the ideas and objects yourself and construct your own understanding. It will grow on you.

There are several approaches to projective geometry. In the metric/analytic approach, the use of cross-ratio is very important. In the combinatorial/incidence approach, the dual indicidence relations between lines and points is important including finite projective planes. In linear algebra approach, the use of homogeneous coordinates in connection with affine and projective spaces is important. In the abstract algebra approach, the algebraic structures that some projective planes posses are important. For example, ternary operations or binary operations that may not be commutative or associative. There are probably other approaches and subfields you may encounter.

If you already know some Euclidean geometry, it will give you a wider perspective. For example, all conic sections are projectively equivalent, except for degenerate cases like two intersecting lines. The existence of a "line at infinity" where parallel lines intersect removes exceptions because in projective planes, all lines intersect each other. You may also be interested in the unified projective generation of conics. You may find Poncelet's Porism of great interest because it is a purely projective result discovered by the geometer who essentially founded the subject.

As you mention a book on Euclidean geometry, I'd like to emphasize that knowing a lot of theorems in plane Euclidean geometry would provide good motivation (at least to be able to see the contrast not analogies!)

This is more than the usual theorems covered in 11th grade. Remember in Projective geometry length and angles don't play a role,: it is concurrency of certain lines and collinearity of certain points. So knowing a lot of theorems in Euclidean geometry of this kind would be helpful. (medians of a triangle are concurrent etc) And a lot about conics (which means some co-ordinate geometry). would be a great advantage.

As a student currently in a Projective Geometry course at a university, I would say the highlights of the class this semester have been affine planes, projective planes, configurations, quadrangles/quadrilaterals, and multiple theorems (i.e. Desargue's theorem and Pappus' theorem). Before coming into this class, I had Euclidean geometry. However, projective geometry is very abstract and very different from high school classes. So, it's best to keep an open mind and try your best to not rely completely on the knowledge you already have about geometry. That being said, you will still use some information. For example, it's important to have the slope and point-slope formulas memorized. You also need to have a strong background in algebra, linear algebra knowledge, and a proofs class. I would say that 90% of my upper classes rely heavily on writing proofs. So, it is very important that you understand the formatting and specifics of proofs.