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shortening of the eye result solely from the generally lower cranial growth rates; rather,
there is an abrupt (and localized) deceleration of growth rates in the orbital region.
However, that does not, by itself, fully account for the apparent contraction of the grid in
the head, especially in S. gouldingi. Part of the relative shortening of the head, supraorbi-
tally, results from the displacement of the landmark at the epiphyseal bar (landmark 2)
towards the anterior landmark of the eye (landmark 14). Suborbitally, the apparent short-
ening of the head results from the displacement of the posterior jaw landmark (landmark
13) towards the posterior eye landmark (landmark 15), as well as from the more general
shortening of the snout and eye. These two species also differ in the ontogeny of posterior
body shape. In S. gouldingi, the caudal peduncle (the region bounded by landmarks 6, 7,
and 8) appears to contract, but no change appears to be localized there the posterior
body generally shortens (as does the head). Growth rates appear to decrease, moving pos-
teriorly from the mid-body to the tail. Because the caudal peduncle is the most posterior
part of the body, the growth rates are lowest there. In P. denticulata, growth rates decrease
more slowly, and most of the change in the posterior body seems to result from the poste-
rior displacement (and relative shortening) of the anal fin. That increases the distance
between the pelvic and anal fins (which expands the grid between them), but because that
is not a part of the general expansion of the mid-body (it is limited to the ventral region
between the fins), the change is ventrally localized. Due to the sparse sampling of land-
marks in the middle of the body, there is no abrupt contraction or expansion of the grid
such as we see in the head. Sparse sampling of that region makes it difficult to detect
localized changes because we cannot show what happens between landmarks when we
have not sampled them (quoting Gertrude Stein, “there is no there there”).
USING B ENDING ENERGY TO SUPERIMPOSE SEMILAN DMARKS
In the earlier discussion of semilandmarks, we discussed how semilandmarks could be
“slid” along curves to minimize the perpendicular distance between the specimens, and
thus the Procrustes distance between the specimens. This was a distance minimizing
approach, but it is also possible to use the thin-plate spline to slide landmarks to produce
an optimally smooth (non-localized) difference between semilandmarks on two specimens.
In this approach, developed by Green (1996) and Bookstein (1997) , the first step is a con-
ventional Procrustes superimposition (treating landmarks and semilandmarks as equiva-
lent) to compute a mean configuration and align the targets to it. This is followed by
moving the semilandmarks of each target to minimize the bending energy of the thin-plate
spline describing the deformation of the reference to that target. The semilandmarks
are not free to move in any direction; each is confined to “slide” along the line tangent to
the curve at that semilandmark ( Figure 5.8 ). The shape of the curve is not actually known,
so the tangent is estimated as the line parallel to the segment connecting adjacent land-
marks or semilandmarks ( Figure 5.9 ). After sliding, the superimposition is recomputed; if
the new mean configuration differs from the previous mean, the sliding and superimposi-
tion are reiterated until they converge on a solution. The justification for this sliding tech-
nique is that differences in relative positions of semilandmarks along the curve cannot be
informative because this spacing was defined arbitrarily (i.e. extrinsically). Thus, sliding to
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