Plant-pollinator specialization: Origin and measurement of curvature

A feature of biodiversity is the abundance of curves displayed by organs
and organisms. Curvature is a widespread, convergent trait that has
important ecological and evolutionary implications. In pollination
ecology, the curvature of flowers and pollinator mouthparts (e.g.
hummingbird bills) along the dorsiventral plane has been associated with
specialization, competition, and species co-existence. Six methods have
historically been used to measure curvature in pollination systems; we
provide a solution to this inconsistency by defining curvature using
well-established concepts from differential geometry. Intuitively,
curvature is the degree to which a line is not straight, but more
formally, it is the rate at which the tangent of a curve changes direction
with respect to arc length. Here, we establish a protocol wherein a line
is fitted against landmarks placed on an image of a curved organ or
organism, then curvature is computed at many points along the fitted line
and the sum taken. The protocol is demonstrated by studying the
development of nectar spur curvature in the flowering plant genus
Epimedium (Berberidaceae). By clarifying the definition of curvature, our
aim is to make the language of comparative morphology more precise and
broadly applicable to capture other curved structures in nature.