section_path: Role and characterizations in mathematics/Projective geometry
section_name: Projective geometry
paragraph_text: Projective geometry
  
  Let V be the set of all twice differentiable real functions f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } that satisfy the ordinary differential equation f ″ (x) + f (x) = 0 {\displaystyle f''(x)+f(x)=0}. Then V is a two-dimensional real vector space, with two parameters corresponding to a pair of initial conditions for the differential equation. For any t ∈ R {\displaystyle t\in \mathbb {R} }, let e t: V → R {\displaystyle e_{t}:V\to \mathbb {R} } be the evaluation functional, which associates to each f ∈ V {\displaystyle f\in V} the value e t (f) = f (t) {\displaystyle e_{t}(f)=f(t)} of the function f at the real point t. Then, for each t, the kernel of e t {\displaystyle e_{t}} is a one-dimensional linear subspace of V. Hence t ↦ ker e t {\displaystyle t\mapsto \ker e_{t}} defines a function from R → P (V) {\displaystyle \mathbb {R} \to \mathbb {P} (V)} from the real line to the real projective line. This function is periodic, and the quantity π can be characterized as the period of this map. This is notable in that the constant π, rather than 2 π, appears naturally in this context.