Besides the well know relationship in terms of the discriminant of a 2nd order polynomial, is there any else that relates in some sense the behavior of 2nd order PDEs (better, its solution) with the "geometrical shape" of the corresponding conic section?
I mean, given the well known geometrical symmetries of conic sections, is there any relationship with the underneath underneath symmetries of these PDEs solutions, e.g. the time reversal symmetry of hyperbolic PDEs solution (waves propagation) or the irreversibility of parabolic PDEs solution (diffusion process)? What about elliptic PDEs solution which does not depend upon time, at all?
I cannot withstand this is merely a fact of semantics/terminology. Isn't it?
If so, one might think about these particular kind of PDEs solutions as the "intersection" between two functional spaces? What would the "conic surface" functional space stand for? What about the "plane"?