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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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