Projective Geometry – Book Suggestions

Question

I want to be acquainted with projective geometry, so I'm asking for a reference. I need some words to explain my specific background and motivation.

There are many things I learnt related to projective geometry. For example, the theory of Möbius transformations, i.e. fractional linear transformations. The relationship between complex plane, Riemann sphere and the complex projective line is obvious: $\mathbb C\subseteq S^2\cong\mathbb CP^1$ (but not mentioned in the textbook), and the concept of cross-ratio in complex analysis is just the counterpart of the same thing in projective geometry. It's known that a Möbius transformation is determined by three general points, but the proof is purely algebraic, and in fact, incomplete in some sense in our textbooks. The treatment of $\infty$ in the textbooks, in my opinion, is too arbitrary. And in fact, I need some geometric aspects.

I need to learn more on projective geometry. I don't want to go into the process of axiomatization (just like Hilbert's system of Euclidean geometry) if it's too tedious, but I want some rigorous solutions. In fact, we can do things without axiomatization, just as how we define manifolds, or the Euclidean plane as a metric space isometric to $\mathbb R^2$ with standard distance. Our textbook on analytic geometry covers some, but sometimes inrigorous so that I cannot accept. For example, they define (real) projective plane (translated into set-theoretic language) as follows:

The triple $(P,L,I\subseteq P\times L)$ is a projective plane (where $I$ is the incidence structure) if and only if it's isomorphic to the pencil $(P_1,L_1,I_1)$ of all lines and planes passing through $0$ in $\mathbb R^3$, where $P_1,L_1$ are sets of lines and planes, and $(l,\alpha)\in I_1\iff l\subseteq\alpha$. $(P_2,L_2,I_2)$ and $(P_3,L_3,I_3)$ are isomorphic if and only if there exist bijections $\phi_P\colon P_2\to P_3,\phi_L\colon L_2\to L_3$ such that $(p_2,l_2)\in I_2\iff(\phi_P(p_2),\phi_L(l_2))\in I_3$, i.e., bijections attain the incidence structure.

The preceding definition is rigorous. However, when they're defining cross-ratios and projective transformations, there's nothing rigorous. They only do them for specific models, for example, the extended Euclidean plane, and never check whether anything is well-defined, i.e, independent of choice of bijections of the isomorphic relation.

I need some introduction to projective geometry. It's better if the relationship between complex analysis or anything interesting and projective geometry is mentioned.

Any help? Thanks.

Answer 1

For a modern approach, see Richter-Gebert's

Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry.

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