Biology Reference
In-Depth Information
CHAPTER
3
Simple Size and Shape Variables:
Shape Coordinate
s
This chapter presents methods for obtaining shape variables. One is particularly
simple and easily understood, and we present it first because the method is so accessible.
This method is sometimes called the “two-point registration” and it produces coordinates
that are termed “Bookstein shape coordinates”, which can be used both for graphical
displays and formal statistical tests. The second method, the Procrustes superimposition
is perhaps less intuitive, but the method is the one most widely used, for reasons that will
become apparent in the next chapter. It is the one that we will use throughout the rest of
this topic so we introduce it now rather than laying its theoretical foundations in the next
chapter and deriving the superimposition method from theory. We focus on the simplest
possible application of the methods, the analysis of shapes with only three two-dimen-
sional landmarks (triangles). We also discuss how information about size can be restored
(because it is removed in the course of obtaining shape coordinates). We then extend the
analysis to three-dimensional landmarks and, in the case of the Procrustes superimposi-
tion, to semilandmarks, points along outlines or curves. As well as presenting the methods
for obtaining the coordinates, we also discuss the graphical description of results because,
to a large extent, it is the descriptive power of geometric morphometrics
the visualization
of shape change
that makes these methods so useful. The graphical results can differ
depending on the methods for obtaining the shape variables, so we show how apparent
inconsistencies can be reconciled.
SHAPE COORDINATES
In Chapter 1, we discussed the meanings of shape and size as they are defined in
geometric morphometrics. We defined shape in terms of operations that do
not
alter
shape
specifically, translation, rotation and rescaling. These operations can be applied to
a simple form, a triangle, allowing us to obtain a coordinate system. Because there is more
than one way to apply these operations, we can obtain different sets of coordinates from
