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one can plot a single point in the
L
-dimensional Euclidean space corre-

sponding to each distinct
K
-landmark object. The collection of all such

points is called the
form space
corresponding to
K
-landmark objects.

Definition of the form space of K-landmark objects:
The form

space of
K
-landmark objects is a collection of points in

L

2 dimensional Euclidean space such that each point in

this collection corresponds to a form matrix of some
K
-landmark

object.

Every possible form with K-landmarks corresponds to one and only one

point in the identified form space. Similar to the three-landmark case,

this form space occupies only a subset of L-dimensional Euclidean

space (e.g., only that part of the space with positive numbers). When

the forms being studied include more than three landmarks (K > 3),

the additional constraints that specify the form space subset are com-

plicated. The exact mathematical description of these constraints is

provided in
Part 2
of this chapter.

K(K

1)

4.7.3 Studying the difference between two forms using the form space

Suppose we have two objects. Any two objects will do, but let's think

back to the red and green transparencies described previously. Given

our definition of form space, there is a single point in the form space

corresponding to the red transparency and all other three-landmark

objects with the same form. Similarly, there is another single point in

the form space corresponding to the green transparency and all other

objects with the same form. Since we are dealing with three landmark

objects, we can plot them in the form space as points R and G in
Figure

There are many different ways to describe the difference between

points R and G. We will present a few possibilities that have proved

useful in our own research. As long as this description depends solely

on the location of points R and G in the defined form space, and not on

any other extraneous information, the description will satisfy the

invariance requirement. Definitions of form difference should satisfy

the invariance requirement and be biologically interpretable.

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