Biology Reference
In-Depth Information
avoid interpreting the changes in regions between those landmarks, except to say (per-
haps) that the long bony structures are relatively elongated or reoriented more laterally.
However, this can be a problem when multiple landmarks are located at tips of long
structures and no other landmarks serve to pin down what is happening to the regions
between them. It is possible to analyze the changes in relative position and length of
those tips using shape coordinates, but it may not be wise to draw a grid interpolating
changes at those tips to regions between them
there is no organismal tissue there.
If we do not have one of the special cases described above
that is, if we do not have
evidence that some landmarks are largely independent of the others
then we can apply
an interpolation function to understand changes between landmarks. Because the interpo-
lation function is continuously differentiable, relative displacements of landmarks can be
used to calculate the displacement of any location on the organism. These inferred displa-
cements between landmarks can be illustrated using a variety of graphical styles;
Figure 5.2
demonstrates the one most often used, a deformed grid in the style of D'Arcy
Thompson (1992)
.
THE PHYSICAL METAPHOR
The mathematical basis for drawing the picture of the deformed grid is a metaphor
the bending of an idealized steel plate (
Bookstein, 1989
). According to this metaphor, dis-
placements of landmarks in the X, Y plane (the plane in which we have drawn them in
Figure 5.1
) are visualized as if they were transferred to the Z-coordinate of an infinite, uni-
form and infinitely thin steel plate. That is, instead of depicting a landmark as displaced
in some direction within the plane of this page, it is visualized as if it were displaced in
the third dimension (out of this page).
The metal plate is constrained by little stalks that weld the landmarks in one shape to
the landmarks in the other. This is difficult to draw because the imagery is inherently
three-dimensional, so imagine two plates and place a configuration of landmarks on each.
Now, put one plate above the other, and construct little stalks that attach a landmark on
one plate to its homologue on the other plate. If a landmark in one shape is displaced a
long distance relative to the other landmarks, construct a long stalk. Thus, when the land-
mark is displaced a long distance in one direction (such as far anteriorly), the stalk is long;
conversely, when displaced only a short distance, the stalk is short. Therefore the stalks
are of uneven lengths, and that unevenness means that one plate cannot be flat. The con-
formation that plate takes is determined by the relative heights of the stalks, and by the
distances between them on the plate.
In some cases, the plate simply tilts or rotates (it does not actually bend); in other
cases the plate must actually bend, such as when a point in the middle is elevated
higher than four surrounding points. That bending may be gentle or quite sharp. For
real steel plates, the conformation of the plate tends to minimize the magnitude of
bending over the whole plate (as well as the physical energy required to produce
thatbending).Hereweusetheexpressiontends to minimize the magnitude and
energy of bending, because real steel plates may have flaws, and the situation is not a
pure case of work against elasticity. In the ideal case, the bending energy depends
