  This is an introduction to noncommutative geometry, from an affine viewpoint,
that is, by using coordinates. The spaces $\mathbb R^N,\mathbb C^N$ have no
free analogues in the operator algebra sense, but the corresponding unit
spheres $S^{N-1}_\mathbb R,S^{N-1}_\mathbb C$ do have free analogues
$S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$. There are many examples of real
algebraic submanifolds $X\subset S^{N-1}_{\mathbb R,+},S^{N-1}_{\mathbb C,+}$,
some of which are of Riemannian flavor, coming with a Haar integration
functional $\int:C(X)\to\mathbb C$, that we will study here. We will mostly
focus on free geometry, but we will discuss as well some related geometries,
called easy, completing the picture formed by the 4 main geometries, namely
real/complex, classical/free.
