  We consider how the problem of determining normal forms for a specific class
of nonholonomic systems leads to various interesting and concrete bridges
between two apparently unrelated themes. Various ideas that traditionally
pertain to the field of algebraic geometry emerge here organically in an
attempt to elucidate the geometric structures underlying a large class of
nonholonomic distributions known as Goursat constraints. Among our new results
is a regularization theorem for curves stated and proved using tools
exclusively from nonholonomic geometry, and a computation of topological
invariants that answer a question on the global topology of our classifying
space. Last but not least we present for the first time some experimental
results connecting the discrete invariants of nonholonomic plane fields such as
the RVT code and the Milnor number of complex plane algebraic curves.
