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marks. This transformation eliminates the nuisance parameters of

translation, rotation, and reflection. One important consequence of any

such transformation adopted for the purpose of eliminating nuisance

parameters is that the original parameters of interest, namely, the

mean form and the perturbation structure, are transformed as well. As

a result, only the transformed parameters are estimable and not the

original parameters. But are these
transformed
parameters useful for

scientific interpretation? Our experience is that in the case of land-

mark coordinate data used in biological analysis, the transformed

parameters described are meaningful and interpretable.

In the next section, we explain the necessary data transformation and

corresponding parameter transformation that eliminate the nuisance

parameters of translation, rotation, and reflection. We also demonstrate

that the transformed parameters are biologically interpretable.

3.5 A definition of form

To demonstrate that the suggested transformation does not affect the

study of form, we must begin with a precise definition of the concept of

form
of an object.

Definition: The
form
of an object is the characteristic that remains

invariant under any translation, rotation or reflection of the object.

To clarify this definition, consider the simple situation of a triangle,

defined by the location of three landmarks. Suppose we rotate or trans-

late or reflect the triangle by an arbitrary amount. Any such movement

of the triangle results in changes in the coordinate locations of the

three vertices. Although no changes have been made regarding the rel-

ative location of the landmarks, a new set of coordinates is required to

define the new location of the three landmarks once the triangle has

been translated, rotated, or reflected. This means that the landmark

coordinate matrix changes upon reflection, translation, or rotation and

that the landmark coordinate matrix is not invariant with respect to

translation, rotation, and/or reflection.

Now, consider characterizing the form of a triangle as a vector of

distances between all possible pairs of landmarks (a vector of three dis-

tances in the case of a triangle). This vector of inter-landmark

distances is equivalent to the definition of the triangle by landmark

locations with one subtle but important difference: a coordinate system

is not required to record the inter-landmark distances. While rotation,

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