Simple Size and Shape Variables: Shape Coordinates - Geometric Morphometrics for Biologists - page 52

moment, pick any two points as the endpoints of the baseline (A and B); we will discuss
how to choose them later.

1.) 54 306

2.) 223 447

3.) 632 300

Now take the next three coordinate pairs, and draw that triangle:

1.) 11 342

2.) 251 520

3.) 769 318

Now draw both triangles using the same baseline (with point A and B superimposed),
and draw the vector extending between the one free landmark (C xy ) on both triangles.
That vector is the shape variable describing the difference between the triangles.

Comparing Many Individual Triangles
Of course, we rarely (if ever) compare only two specimens (or triangles). We now con-
sider how to compare many individual triangles (below we discuss comparing forms
more complex than triangles). The same procedure (and formulae) applies no matter how
many triangles or individuals are examined. For example, given a collection of triangles
( Figure 3.3A ), we assign points A and B the coordinates (0, 0) and (1, 0) and compare all
these triangles as whole triangles ( Figure 3.3B ), or as scatter-plots of the one free point
( Figure 3.3C ).
The scatter-plot is useful for checking the repeatability of your landmarks, as well as
for studying the variability of shape or differences in shape. For all these purposes, it is
important that the axes of the scatter-plot be sized so that a square shape is shown as a
square

that is, the length of the interval from 0 to 1 on the X -axis should be the same as
the length of the interval from 0 to 1 on the Y -axis. Many programs do not do this scaling
of axes automatically, so you may have to scale the axes yourself. Often this can be done
by first calculating the maximum and minimum values for the X - and Y -coordinates;
the difference between those values, i.e. the range of values, should be set equal for both
coordinates. For example, if the X-coordinate ranges from 0.030 to 0.060 and the Y -coordinate
ranges from 0.020 to 0.060, both axes should be 0.040 units long (the Y -coordinate has the
slightly larger range). In this case, the minimum on the X -axis could be set to 0.025 and the
maximum on the X -axis to 0.065. This distributes the extra length equally above and below
the observed values, and should enforce a 1:1 aspect ratio for the graph.
Whentheaxesareonthesamescale,anapproximately circular scatter of points
indicates that there is a reasonably equal amount of variation in all directions. Random
digitizing error should be circular; systematic errors, in contrast, will look elliptical.