“Motivic” Integral Geometry

I am trying to understand a beautiful work of Pierre Schapira [5]. Notations and Conventions • For any set S we denote by IS the identity map S → S. For any subset A ⊂ S we denote by 1A the characteristic function of A. • In the sequel all manifolds will be assumed real analytic. A morphism of manifolds will be a real analytic map between two manifolds. Introduction Suppose we are given a simple closed smooth curve C ↪→ R2. Suppose that for every affine line L ⊂ R2 we know the number nC(L) of intersection points of C with L. How much information about C can we extract from this information? Intuitively, this knowledge ought to differentiate between different curves. Clearly, we would like to be more specific than this, and better yet, we would like to put numbers behind such statements. We will sketch a classical approach to this problem following the beautiful presentation in [3, I.§2]. We begin by giving a more useful description of the counting function nC(L). Denote by G̃ the set of affine lines in the plane. A line L is determined by its normal coordinates (θ, c). More precisely, the line L(θ, c) is given by the equation x cos θ + y sin θ = c. We deduce G̃ ∼= { (θ, c) ∈ R } /(θ, c) ∼ (θ + π,−c). ∗Notes for myself and whoever else is reading this footnote.