In this paper, we study the contact geometry of second-order ordinary differential equations that are quadratic in the highest derivative (the so-called quadratic Abel equations). Namely, we realize each quadratic Abel equation as the kernel of some nonlinear differential operator. This operator is defined by a quadratic form on the Cartan distribution in the 1-jet space. This observation makes it possible to establish a one-to-one correspondence between quadratic Abel equations and quadratic forms on Cartan distribution. Using this realization, we construct a contact-invariant {e}-structure associated with a non-degenerate Abel equation (i.e., the basis of vector fields that is invariant under contact transformations). Finally, in terms of this {e}-structure we solve the problem of contact equivalence of nondegenerate Abel equations.
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