Consider a polynomial f∊R[x; y] whose Hessian curve is compact and the unbounded connected component of its complement is hyperbolic. We study the fields of asymptotic directions on this component. Thus, we determine an index formula for the field of asymptotic directions involving the number of connected components of the Hessian curve constituting the boundary of this component, and the number of the corresponding Gaussian cusps. As an application of this study we show an example of a polynomial of degree 4 with 10 Gaussian cusps. Moreover, we determine the parity of the Gaussian cusps on the boundary of the unbounded region of a graph of a polynomial.
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