Theory of shape

This chapter covers the basic theory of shape, beginning with the definition of shape and proceeding through the characterization of several theoretical spaces. Some of the mathematics may look a bit difficult, but it is important to grasp the basic ideas, which is presented verbally as well as mathematically. Interestingly, many of the techniques used in geometric morphometrics were developed independently of this theory even though they are justified by it. As the theory matured, it became possible to synthesize a large body of techniques that had been developed independently of each other and to explicate their interrelationships. Perhaps most importantly, this theory also allows one to judge whether or not methods are valid. The theory provides the underlying justifications for all the techniques involved, thereby allowing researchers to make inferences about shape without worrying that those inferences are somehow based on arbitrary or mathematically faulty choices made in the course of several analyses. Freed of such concerns, one can concentrate on the biological meaning of the results.