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Procrustes Problems$

John C Gower and Garmt B Dijksterhuis

Print publication date: 2004

Print ISBN-13: 9780198510581

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198510581.001.0001

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(p.205) Appendix D Oblique axes

(p.205) Appendix D Oblique axes

Source:
Procrustes Problems
Publisher:
Oxford University Press

Let x be a row-vector giving the coordinates of a point relative to a set of P orthogonal Cartesian axes. Suppose now, we wish to refer this same point to a set of oblique axes whose directions relative to the Cartesian axes are given by the columns of a matrix C. Thus the kth column c k of C gives the direction cosines of the kth oblique axis. There are at least two ways of representing coordinates relative to oblique axes. These are shown in Figure D.1, below.

In the first version (a) one projects orthogonally onto the oblique axes, so that the oblique coordinates y are given by (C.3) as:

(D.1)
y = x C .
In Figure D.1(a), y 1 = 3 and y 2 = 3.

In the second version (b), the parallel axes system, x is the vector-sum of the coordinates (z 1, z 2, z 3, …, z p). Thus x = z 1 c 1 + z 2 c 2 + z 3 c 3 + … + z p c p so that x k = z 1 c 1k + z 2 c 2k + z 3 c 3k +…+ z p c pk and x = zC′ giving:

(D.2)
z = x ( C ) 1 .
In Figure D.1(b), z 1 = 2 and z 2 = 1.

It follows from (D.1) and (D.2) that:

(D.3)
z = y ( C C ) 1 and y = z C C
Appendix D Oblique axes

Fig. D.1. Oblique axes coordinate systems. In (a) the coordinates are given by orthogonal projection onto the oblique axes. In (b) the projections are parallel to the oblique axes.

(p.206) give the transformations between the two oblique representations. The cosines of the angles between the oblique axes are the elements of C′C, which has a unit diagonal. When the oblique axes are themselves orthogonal then the cosines are zero so that C′C = I, and C is orthogonal. In this case y and z coincide, as is obvious from the geometry of Figure D.1.

Using (D.1) we can evaluate the distance between two points given by their projection coordinates y 1 and y 2. From (D.1), their Cartesian coordinates are y 1 C −1, y 2 C −1 so that their squared distance is given by:

(D.4)
( y 1 y 2 ) ( C C ) 1 ( y 1 y 2 ) .
Similarly, from (D.2) the squared distance between two points given by their parallel axes coordinates z 1 and z 2 is:
(D.5)
( z 1 z 2 ) ( C C ) ( z 1 z 2 ) .