Biology Reference
In-Depth Information
solelyonthedistancebetweenthepointsandtherelativeheightsofthestalks,and
the total amplitude of bending. If we consider two different deformed plates, both
describingthesametotaloverallamountofchange(thesamesetofstalkheights)but
one with the stalks proportionately closer together, the one that is bent between the
more closely spaced points requires more energy than the one that is bent between
more widely spaced points.
The bending energy depends on the spacing of the stalks because it is a function of the
rate of change in the slope of the plate
i.e. whether the slope of the surface increases
rapidly or slowly. In these terms, more energy is required when the slope of the surface
changes at a higher rate (for the same net amplitude of change). Imagine a tall stalk sur-
rounded by short ones, which induces a steep slope in the curvature of the plate. The
steepness of that slope is proportional to the function being minimized
the rate of
change in slope of the surface
and, thus, the function being minimized is a function of
the second derivative (the slope of the surface is the first derivative) integrated over the
whole surface of the plate. It can also be termed the integral of the quadratic variation
over the plate.
To return from ideal plates to the analysis of a deformation, we now project the changes
that were visualized as if in the Z-direction back into the X, Y plane (the plane of our land-
mark data). The idea of bending that had a physical meaning when we were talking about
changes in the Z-direction is now reinterpreted as “spatially local information”. This inter-
pretation may not be intuitively obvious, but consider what a relatively rapid increase in
slope means
that there are contrasting displacements of closely spaced points. When
closely spaced points change in opposite directions it requires more energy to bend the
plate between them; so there is an inverse relationship between the spatial scale of the
change and its metaphorical bending energy. Minimization of bending energy is equiva-
lent to minimization of spatially localized information.
It is always possible to envision changes as highly local by assuming that the plate flat-
tens out immediately after rising, then rises again just at the next stalk, then flattens again,
then rises again, etc. The argument against doing so is that this would be the most unpar-
simonious interpretation possible. By minimizing bending energy, we obtain a more parsi-
monious description of the change. We do not assume highly localized change unless the
data demand doing so.
Uniform and Non-Uniform Components of a Deformation
Some transformations require no bending energy at all; these are equivalent to tilting or
rotating the plate. These are often called affine or uniform transformations, meaning that
they leave parallel lines parallel. The terms “affine” and “uniform” are both used to
describe the same component of a deformation; “affine” is favored by mathematicians, but
“uniform” appears more often in the geometric morphometric literature. Consequently,
we will use “uniform” for this component and “non-uniform” for its complement. In our
example (see
Figure 5.2
), if the entire fish simply elongates relative to its depth without
any disproportionate lengthening of one region relative to another, it is a uniform elonga-
tion. Uniform elongation is equivalent to uniform narrowing, as should be recalled from
