Editing Morphometrics (section)

===Outline analysis===
[[File:Thelodont morphometrics.svg|thumb|The results of [[principal component analysis]] performed on an outline analysis of some [[thelodont]] denticles.]]
[[Outline analysis]] is another approach to analyzing shape. What distinguishes outline analysis is that coefficients of mathematical functions are fitted to points sampled along the outline. There are a number of ways of quantifying an outline.  Older techniques such as the "fit to a polynomial curve"<ref>
{{cite journal
 | author = Rogers, Margaret
 | year = 1982
 | title = A description of the generating curve of bivalves with straight hingess
 | journal = Palaeontology
 | volume = 25
 | pages = 109–117
}}</ref>
and Principal components quantitative analysis<ref>{{cite journal
 | author = Glassburn, T.A.
 | year = 1995
 | title = A new palaeontological technique describing temporal shape variation in Miocene bivalves
 | journal = Palaeontology
 | volume = 38
 | pages = 133–151
}}</ref> have been superseded by the two main modern approaches: [[eigenshape analysis]],<ref

 name=Lohmann1983>{{cite journal
 | author = Lohmann, G.P.
 | year = 1983
 | title = Eigenshape analysis of microfossils: A general morphometric procedure for describing changes in shape
 | journal = Mathematical Geology
 | volume = 15
 | issue = 6
 | pages = 659–672
 | doi = 10.1007/BF01033230| s2cid = 120295975
 }}</ref> and [[ellipse|elliptic]] [[Fourier analysis]] (EFA),<ref name=Ferson1985>{{cite journal
 | author = Ferson, S. |author2=Rohlf, F.J. |author3=Koehn, R.K.
 | year = 1985
 | title = Measuring Shape Variation of Two-Dimensional Outlines
 | journal = Systematic Zoology
 | volume = 34
 | issue = 1
 | pages = 59–68 
 | doi = 10.2307/2413345
 | jstor = 2413345}}</ref> using hand- or computer-traced outlines.  The former involves fitting a preset number of semilandmarks at equal intervals around the outline of a shape, recording the deviation of each step from semilandmark to semilandmark from what the angle of that step would be were the object a simple circle.<ref name=MacLeod1993>For an example "in use", see {{cite journal
 | doi = 10.2475/ajs.293.A.300
 | author = MacLeod, N.
 |author2=Rose, K.D.
 | date =  January 1, 1993 | title = Inferring locomotor behavior in Paleogene mammals via eigenshape analysis
 | journal = American Journal of Science
 | volume = 293
 | issue = A
 | pages = 300–355
 | bibcode = 1993AmJS..293..300M
 }}</ref> The latter defines the outline as the sum of the minimum number of ellipses required to mimic the shape.<ref name=Schmittbuhl2007>e.g. {{cite journal
 | author = Schmittbuhl, M.
 |author2=Rieger, J. |author3=Le Minor, J.M. |author4=Schaaf, A. |author5=Guy, F.
 | year = 2007
 | title = Variations of the mandibular shape in extant hominoids: Generic, specific, and subspecific quantification using elliptical fourier analysis in lateral view
 | journal = American Journal of Physical Anthropology
 | volume = 132
 | issue = 1
 | pages = 119–31
 | doi = 10.1002/ajpa.20476
 | pmid = 17063462
}}</ref>

Both methods have their weaknesses; the most dangerous (and easily overcome) is their susceptibility to noise in the outline.<ref name=Haines2000>{{cite journal
 | author = Haines, A.J.
 |author2=Crampton, J.S.
 | year = 2000
 | title = Improvements To The Method Of Fourier Shape Analysis As Applied In Morphometric Studies
 | journal = Palaeontology
 | volume = 43
 | issue = 4
 | pages = 765–783
 | doi = 10.1111/1475-4983.00148
|bibcode=2000Palgy..43..765H
 |s2cid=129091685
 }}</ref> Likewise, neither compares homologous points, and global change is always given more weight than local variation (which may have large biological consequences).
Eigenshape analysis requires an equivalent starting point to be set for each specimen, which can be a source of error
EFA also suffers from redundancy in that not all variables are independent.<ref name=Haines2000/>  On the other hand, it is possible to apply them to complex curves without having to define a centroid; this makes removing the effect of location, size and rotation much simpler.<ref name=Haines2000/>
The perceived failings of outline morphometrics are that it does not compare points of a homologous origin, and that it oversimplifies complex shapes by restricting itself to considering the outline and not internal changes.  Also, since it works by approximating the outline by a series of ellipses, it deals poorly with pointed shapes.<ref>{{cite book|last=Zelditch|first=M.L|author2=Swiderski, D.L. |author3=Sheets, H.D. |author4=Fink, W.L. |title=Geometric Morphometrics for Biologists: A Primer|year=2004|publisher=Elsevier Academic Press|location=San Diego}}</ref>

One criticism of outline-based methods is that they disregard homology – a famous example of this disregard being the ability of outline-based methods to compare a [[scapula]] to a potato chip.<ref>{{cite journal|last=Zelditch|first=M.|author2=Fink, W. L |author3=Swiderski, D. L |title=Morphometrics, homology, and phylogenetics - Quantified characters as synapomorphies|journal=Systematic Biology|year=1995|volume=44|issue=2|pages=179–189|doi=10.1093/sysbio/44.2.179 }}</ref> Such a comparison which would not be possible if the data were restricted to biologically homologous points. An argument against that critique is that, if landmark approaches to morphometrics can be used to test biological hypotheses in the absence of homology data, it is inappropriate to fault outline-based approaches for enabling the same types of studies.<ref name=MacLeod1999>{{cite journal
  |title = Generalizing and Extending the Eigenshape Method of Shape Space Visualization and Analysis
  |last = MacLeod
  |first = Norman
  |journal = [[Paleobiology (journal)|Paleobiology]]
  |issn = 1938-5331
  |volume = 25
  |issue = 1
  |year = 1999
  |pages = 107–38
  |jstor = 2665995
}}</ref>