  Culler and Shalen, and later Yoshida, give ways to construct incompressible
surfaces in 3-manifolds from ideal points of the character and deformation
varieties, respectively. We work in the case of hyperbolic punctured torus
bundles, for which the incompressible surfaces were classified by Floyd and
Hatcher. We convert non fiber incompressible surfaces from their form to the
form output by Yoshida's construction, and run his construction backwards to
give (for non semi-fibers, which we identify) the data needed to construct
ideal points of the deformation variety corresponding to those surfaces via
Yoshida's construction. We use a result of Tillmann to show that the same
incompressible surfaces can be obtained from an ideal point of the character
variety via the Culler-Shalen construction. In particular this shows that all
boundary slopes of non fiber and non semi-fiber incompressible surfaces in
hyperbolic punctured torus bundles are strongly detected.
