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translation, and reflection change the values of the coordinate loca-

tions of landmarks that define the triangle, these operations have no

effect on the vector of distances between landmark pairs. A vector of all

possible inter-landmark distances representing the relative location of

the landmarks is invariant with respect to rotation, reflection, and

translation of the triangle.

In fact, a stronger property holds. There is a one to one correspon-

dence between a given triangle and a vector of three distances such

that given this vector of distances, one can draw (or reconstruct) the

original triangle. For example, in simple Euclidean geometry terminol-

ogy, two triangles have the same form if they are congruent (Fishback,

1969). When a triangle is drawn from a vector of inter-landmark dis-

tances, it will be a triangle
congruent
with the original triangle from

which the distances were calculated. Given one type of data, we can

construct the other
uniquely
(up to translation, rotation, or reflection).

Such a one-to-one, invariant transformation of the data is called a

“
maximal invariant.
”

At this point, the following statement bears repeating: the land-

mark coordinate matrix is not invariant to translation, rotation, or

reflection, but the corresponding vector of all possible pair-wise dis-

tances
is
invariant to these operations. Our example has focused on

triangles but these findings are true for objects defined by more than

three landmarks in two- or three-dimensional space. Given a landmark

coordinate matrix, there is a corresponding and unique vector of all

possible pair-wise distances. Conversely, given a vector of all possible

distances, one can construct the original landmark coordinate matrix

up to translation, rotation, and reflection. We have neither gained nor

lost information, whether the form is recorded as a vector of distances

or as a landmark coordinate matrix. All the relevant “geometric” infor-

mation is identical in both representations, and the transformation

from one to the other is exact. The main difference between these two

representations of the form is that the vector of distances is indepen-

dent of the choice of an arbitrary orientation, whereas the landmark

coordinate matrix is not. The vector of distances is a
coordinate system-

free
representation of form. We will show that a vector of all possible

distances is biologically interpretable as well.

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