 Characteristic direction

For a scalar second-order partial differential equation in two variables, a tangent direction along which Cauchy data fail to determine the highest transverse derivative through the equation. For principal part $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}$, a tangent $(dx,dy)$ is characteristic exactly when $A(dy{)}^2-2Bdxdy+C(dx{)}^2=0$. Equivalently its conormal annihilates the principal symbol. A hyperbolic partial differential equation has two distinct real characteristic directions at each point.