  We describe a new method for constructing a spectrahedral representation of
the hyperbolicity region of a hyperbolic curve in the real projective plane. As
a consequence, we show that if the curve is smooth and defined over the
rational numbers, then there is a spectrahedral representation with rational
matrices. This generalizes a classical construction for determinantal
representations of plane curves due to Dixon and relies on the special
properties of real hyperbolic curves that interlace the given curve.
