Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 146

Figure 4.7. The subset of a three-dimensional Euclidean space that corresponds to the
form space of three-landmark objects. Every point in this subset corresponds to some
three-landmark object, and every three-landmark object corresponds to some point in
this subset. Points labeled R and G represent the location of the red and green trans-
parencies (three-landmark forms) in this space.

this space.
There are certain constraints on three-landmark objects that form
triangles that make this space only a subset of the entire three-dimen-
sional Euclidean space. For example, dimensions measured on an
object must be positive, and so any distance corresponding to the side
of a triangle on this object must be positive. Consequently, the location
of three-landmark objects in this form space will be restricted to the
positive quadrant of Euclidean space. In addition, a set of three posi-
tive numbers corresponds to a triangle if and only if the sum of any two
numbers exceeds the third. This further restricts the subset of space
corresponding to three-landmark objects that form triangles. There are
parts of the Euclidean space where this constraint is not met and
therefore cannot be part of the defined form space.
Generalization of these findings to K -landmarks is fairly straight-
forward. We have seen in Chapter 3 that the form of any K -landmark
object, whether two- or three-dimensional, can be represented by the
collection of all possible pair-wise distances. There are L 2

such distances, so that the dimension of this space is equal to L . Thus,

K(K 1)