We introduce a new method to construct a large family of Lagrangian surfaces
in complex Euclidean plane by means of two planar curves making use of their
usual product as complex functions and integrating the Hermitian product of
their position and tangent vectors.
Among this family, we characterize minimal, constant mean curvature,
Hamiltonian stationary, solitons for mean curvature flow and Willmore surfaces
in terms of simple properties of the curvatures of the generating curves. As an
application, we provide explicitly conformal parametrizations of known and new
examples of these classes of Lagrangians in complex Euclidean plane.Comment: 15 pages, 5 figure