To characterize the diversity of planar shapes in
such instances as insect wings and plant leaves, we
present a method for the generation of a smooth
morphometric mapping between two planar domains
which matches a number of homologous points. Our
approach tries to balance the competing requirements
of a descriptive theory which may not reflect
mechanism and a multi-parameter predictive theory
that may not be well constrained by experimental
data. Specifically, we focus on aspects of shape as
characterized by local rotation and shear, quantified
using quasi-conformal maps that are defined precisely
in terms of these fields. To make our choice optimal,
we impose the condition that the maps vary as
slowly as possible across the domain, minimizing
their integrated squared-gradient. We implement this
algorithm numerically using a variational principle
that optimizes the coefficients of the quasi-conformal
map between the two regions and show results for
the recreation of a sample historical grid deformation
mapping of D’Arcy Thompson. We also deploy
our method to compare a variety of Drosophila
wing shapes and show that our approach allows
us to recover aspects of phylogeny as marked
by morphology