Monoplots - Understanding Biplots - page 437

Thus, it turns out that it only makes sense to seek approximations in an even number of
dimensions, sometimes referred to as bimensions but we prefer hedron . Previously we
have often chosen to give two-dimensional maps, but now we are obliged to, and if
we want better approximations we must use 4, 6, ... dimensions.
We next consider a single hedron and to simplify notation now drop the hedron
identifier i and the dimension identifiers. Thus, we write u i 1 u i 2 − u i 2 u i 1 as uv − vu .
As usual, we may plot the rows of ( u , v )asasetof p points, but what does it mean?
If P i ( u i , v i ) and P j ( u j , v j ) are two of these points, then the area of the triangle OP i P j
is

1
2

( u i v j − v i u j ) . Thus the interpretation of this kind of monoplot is in terms of area.
Figure 10.9 shows the geometry. Skew-symmetry is reflected in the angular sense in
which the area is measured. We take the anticlockwise sense to be positive, so that
OP i P j =− OP j P i . Distance is not an appropriate measure, as although OP i P j is zero
when P i and P j coincide, so is it zero when O, P i and P j are collinear, however distant
P i may be from P j . In conventional Euclidean plots all points equidistant from any point
P i lie on a circle; now all points P with equal skew-symmetry n ij with P i lie on a
line parallel to OP i as in Figure 10.9. There is another line parallel to OP i which has
skew-symmetry − n ij , indicated by P in the diagram.
Axes are shown in Figures 10.9 and 10.10 although, as usual, they are merely
scaffolding for plotting the points. Consequently, in Figure 10.10 the axes have been
de-emphasized. From the figures, we can see that the area P i P j P k = n ij + n jk + n ki .
Clearly, this result is independent of the position of the origin. However, apart from this

P j

P i

P ′

O

P ′′

Figure 10.9 The geometry of skew-symmetry. Positive skew-symmetry n ij is indicated
by area measured in the anticlockwise sense OP i P j , negative skew-symmetry in the clock-
wise sense OP j P i . The line through P (P ) is the locus of all points with skew-symmetry
n ij ( − n ij ) .