  We show that the shape of a complex cubic field lies on the geodesic of the
modular surface defined by the field's trace-zero form. We also prove a general
such statement for all orders in \'etale Q-algebras. Applying a method of
Manjul Bhargava and Piper H to results of Bhargava and Ariel Shnidman, we prove
that the shapes lying on a fixed geodesic become equidistributed with respect
to the hyperbolic measure as the discriminant of the complex cubic field goes
to infinity. We also show that the shape of a complex cubic field is a complete
invariant (within the family of all cubic fields).
