The metric and measure of the space of types of second order curves

The author has considered the space of types of second order curves, e.g. the set of classes of triples of real numbers $(a_{11},a_{12},a_{22})\ne(0,0,0)$ generated by translations of plane and multiplications by a non-zero constant. On this set classes an invariant Riemannian metric and measure is defined. The measure of elliptic and hyperbolic curve types has been evaluated.