# Parabolic, Hyperbolic, Elliptic

The terms "parabolic," "hyperbolic" and "elliptic" are used to classify certain differential equations. The terms "hyperbolic" and "elliptic" are also used to describe certain geometries. Is there a connection between these usages, and, if so, what is it?

• Related – Giuseppe Negro Oct 19 '17 at 13:44
• It has to do with the characteristic curves of the equation, as an intuitive example, the wave equation has two families of nonparallel lines (degenerated hyperbola) and the heat equation has one family of lines (degenerated parabola). But I would like to read a more developed answer as well. – Koto Oct 19 '17 at 13:51
• For “elliptic” in particular I had asked a similar question in the past, albeit a bit broader by taking other uses into account as well. – MvG Oct 28 '17 at 7:15
• In my mind the elliptic/parabolic/hyperbolic language all stems from conic sections - the conic $a^{ij}x_i x_j + b^i x_i = 0$ and the PDE $a^{ij} \partial_i \partial_j u + b^i \partial_i u =0$ are classified near identically. Elliptic/hyperbolic geometries have models built from circles/hyperbolae, and volume growths given by circular/hyperbolic trigonometric functions. – Anthony Carapetis Oct 28 '17 at 7:22