  This paper illustrates a computational approach to Culler-Morgan-Shalen
theory using ideal triangulations, spun-normal surfaces and tropical geometry.
Certain affine algebraic sets associated to the Whitehead link complement as
well as their logarithmic limit sets are computed. The projective solution
space of spun-normal surface theory is related to the space of incompressible
surfaces and to the unit ball of the Thurston norm. It is shown that all
boundary curves of the Whitehead link complement are strongly detected by its
character variety. The specific results obtained can be used to study the
geometry and topology of the Whitehead link complement and its Dehn surgeries.
The methods can be applied to any cusped hyperbolic 3--manifold of finite
volume.
