Biology Reference
In-Depth Information
mandible is homologous to a landmark located in the middle of the mandibular foramen
of another individual. Conversely, a landmark located in the middle of the mental foramen
of the mandible of one individual is not homologous with a landmark located in the
middle of the mandibular foramen of another individual. The mental and mandibular
foramina do not correspond to each other, as may be obvious from the fact that they have
different names (it is even more obvious anatomically because one is located on the exter-
nal/lateral side of the mandible, the other on the internal/medial side; one is located ante-
riorly and the other posteriorly. That these two points do not correspond is obvious,
perhaps so obvious as to go without saying. But there are many examples of traditional
morphometric measurements that disregard homology, and it is in contrast to them
that the criterion of homology is so important for ensuring comparability of shapes.
For example, measurements such as “greatest skull breadth” or “least interorbital width”
may be taken between different pairs of endpoints in different specimens because they are
measured where the skull is widest, wherever that is, or where the interorbital region is
narrowest, wherever that is. The endpoints of those measurements need not correspond
one-to-one from one specimen to another; it is only that the length is greatest or least that
makes the measurements comparable at all. These measurements are not made between
homologous points and, in this context, the converse of “homology” is “not the same point”.
Calling landmarks “homologous” may seem to be a curious usage of the term for two
reasons. First, biologists often use homology for similarities due to common ancestry,
implying more than that the points correspond. Second, the term is usually applied to
structures rather than to points. If using “homology” for corresponding points bothers
you, you can always call them “corresponding” rather than “homologous” points.
In geometric morphometrics, homology has been stressed above all other criteria for
selecting landmarks for both mathematical and biological reasons. The mathematical
reasons are important to understand because semilandmarks, which are not usually
argued to be homologous biologically, are nonetheless treated as if they are homologous
mathematically. The mathematical issues are discussed in more depth in the next two
chapters, but you will likely select your landmarks before you read them so you need an
intuitive feel for the mathematical issues before choosing them. The primary mathematical
issue is that the coordinates of points (whether landmarks or semilandmarks) are treated
as if they correspond one-to-one when computing the difference between shapes. That is,
the coordinates are averaged, and deviations between individuals are quantified by sum-
ming the squares of the differences between the coordinates of the points. If the points in
one specimen do not correspond to the points in another, averaging them is like averaging
apples and oranges. The calculations make sense when each point on one specimen corre-
sponds to the same point on another; more specifically, when landmark 1 in one organism
corresponds to landmark 1 in another, as do landmarks 2 and so forth. That assumption
of correspondence is built into the methods whether they are applied to landmarks or
semilandmarks.
Semilandmarks are usually not regarded as homologous points. It is the curve that they
sample, not the semilandmarks individually, that are viewed as if corresponding one-
to-one. The positions of the semilandmarks along a curve are not viewed as informative
about shape and so information about their position along the curve will be removed
when removing other kinds of non-shape information from the coordinates. This is the
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