Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 145

tor of form change utilizes only this unambiguous information and
describes the difference in orbits that the forms occupy. We now use the
principles of EDMA as presented in Chapter 3 to create descriptors of
form change that are invariant and useful to the investigator who is
striving to propose scientific inference from observations of form.

4.7.2 The form space

We continue with a simple situation in which we have a form repre-
sented by three landmarks. Recall that the form of an object
represented by three landmarks can be expressed as a vector of three
distances. We have shown that the vector of three distances is an
invariant representation of the form. That is, no matter how the object
is rotated, reflected, or translated, the vector of three distances
remains unchanged.
Let us consider the object represented by the landmark coordinate

matrix .

We have seen in the last chapter that the form matrix corresponding
to this object is given by

, and that the form matrix can be written FM(M)

as a vector of all entries above the diagonal, .

Now consider all distinct three-landmark objects that form a trian-
gle . There is a vector of three distances corresponding to each of these
objects. Each vector, although consisting of linear distances, corre-
sponds to a point in a three-dimensional Euclidean space. The
collection of points corresponding to all three-landmark objects is
called the form space of three landmark objects. Figure 4.7 shows the
form space of all three-landmark objects that form triangles with vec-
tors corresponding to the measure of each linear distance of the
triangle. The vector of three distances is recorded as a single point in