Statistical Methods for Comparison of Forms - An Invariant Approach to Statistical Analysis of Shapes - page 206

and,

Let

and define a new matrix,

E K ( K- 1)/2 there exists
a D -dimensional object with K landmarks, if and only if:
i) d belongs to the positive quadrant, that is, all the ele-
ments of this vector are non-negative, and,
ii) The matrix B d is a positive semi-definite matrix with
rank ( B d )

Theorem 1: Corresponding to a point d

D .

Proof: This is a standard result in multidimensional scaling litera-
ture. See Mardia et al. (1979, Chapter 14) for details.
Similar to the discussion about the form space corresponding to
three-landmark, two-dimensional objects in Part 1 , it can be noticed
that the set of points that satisfies the above condition constitute a
subset of the positive quadrant of the Euclidean space of dimension