AMER. ZOOL., 20:627-641 (1980)
When One Form is Between Two Others:
An Application of Biorthogonal Analysis 1 ' 2
FRED L. BOOKSTEIN
Center for Human Growth and Development, University of Michigan,
Ann Arbor, Michigan 48109
SYNOPSIS. This essay presents a method for measuring the degree to which one biological
outline form lies in between two others. The procedure does not measure forms separately, but rather compares pairs of tensors expressing D'Arcy Thompson's "Cartesian
transformations" according to the biorthogonal formalism of Bookstein. In analogy with
conventional methods, betweenness is computed as a similarity score, the cosine of a nonEuclidean angle between the tensors. The new quantities, size-betweenness and shapebetweenness, enable comparisons of form series against a priori orderings intra- and interspecifically.
uously and smoothly correspond a single
of each other form. Because this subpoint
In the quantitative appreciation of organic evolution one of the great neglected ject-matter is one of form purely, the data
themes is orthogenesis, evolution through for analysis are purely geometric: outline
grades monotone in form. Operational- forms of two-dimensional sections or proized, orthogenesis is nothing but the extent jections. As for the homologies, data supto which generations of a lineage are sys- ply these mainly at landmarks, special
tematically in between their ancestors and points whose equivalence between forms
their descendants; then the fundamental is granted a priori. Interpolations bridge
metric for this theme is the extent to which the landmarks to impute the homology
one form lies between two others. In other function all the way around the boundcontexts too, an appreciation of between- aries of a pair of outlines and throughout
ness is crucial in determining whether a their interiors; landmarks calibrate this
series of forms is concordant with some function but bear only part of its inforexternal ordering. The study of correla- mation content. I have discussed these
tion between morphology and cline, for in- matters in some detail elsewhere (Bookstance, often reduces to a consideration of stein, 1978).
forms presumed ordered in a geographic
arc. Classification of an extinct form often Size and shape measures
hinges on its suitability for inclusion in a
Whether y is in between x and z is oblineage of which two other forms are vious when x, y, z are ordinary decimal
known, as in the search for "missing links." numbers. Such simplicity may be imitated
When a collection of forms shows topo- in the study of forms if the crucial paramlogical incompatibilities—parts connected eter behaves like a strictly ordered contininconsistently from specimen to speci- uum; the commonest underlying dimenmen—we are blocked from imagining any sion behaving so is size. The particular
easy transformation between them. In this number we choose as measure of size
article I will treat only series which can be ought to accord with homology, of
formalized as smooth distortions from course—which means only that coordigrade to grade. The distortions are to ex- nates of homologous points should be
press a differentiable point homology: To subjected to identical algebraic manipevery point of every form must unambig- ulations—but otherwise may be a nondecreasing function of any relevant dis1
From the Symposium on Analysis of Form pre- tances across the inside of the form.
sented at the Annual Meeting of the American So- This definition guarantees that, for increciety of Zoologists, 27-30 December 1979, at Tampa, mental growth, all size measures are
Florida.
monotonic with respect to each other and
2
This essay is dedicated to the late Kenneth B.
all
betweenness estimates consistent among
Leisenring, Professor of Geometry at the Univerthemselves.
sity of Michigan.
INTRODUCTION
627
628
A
FRED L. BOOKSTEIN
B
C
FIG. 1. Three rectangles A, B, C of sizes 3 x 3, 4 x
8, 9 x 9 units. B is in between A and C on all size
measures and on no shape measures.
Nevertheless it would be unwise to define betweenness of forms in terms of size
measures only. Consider, for instance, the
three forms in Figure 1: rectangles A, B,
C measuring 3 x 3 , 4 x 8 , and 9 x 9 , with
the corners presumed homologues as
drawn. According to any size measure B
is in between A and C. Yet clearly the distortion wreaked in passing from A to B
was all but undone between B and C,
masked only by the general increase in
size. It is unhelpful to declare B "between"
A and C when A and C agree in all respects
but one and B disagrees with both; better
to separate out size-betweenness, since it
can be defined so easily, and deal with
form-betweenness as a separate issue.
Shape measures, such as angles and ratios, are functions according with homology that take on the same values for exactly similar forms. In Figure 1, form B is
not in between A and C as determined by
shape measures, since on all shape mea-
FIG. 2. Three vectors a, b, c in series define two
translations t(a,b), t(b,c). When b is exactly in between
a and c the translations have the same direction and
sense.
Fic. 3. An appropriate continuously varying measure of the betweenness of b with respect to a and c,
btw(b;a,c), is the cosine of the angle between t(a,b)
and t(b,c) at b.
sures A and C have the same score. Having
so easily shown B to be not in between A
and C, we might be tempted to essay the
obverse as a definition—that B is between
A and C to the extent that B's score is consistently between A's and C's on arbitrary
shape measures.
Unfortunately, owing to the great generality of the class of shape measures, this
definition proves nugatory—the case of
Figure 1 can always be made to apply. In
the Appendix is the proof of the rather
dismaying Shape Nonmonotonicity Theorem: for any triple A, B, C of homologous
outlines, there exist indefinitely many
shape measures which accord with homology and for which A and C have exactly
the same score, so that B, whatever it looks
like, cannot be in between. Whereas determining whether B is in between A and
C using just size measures led to loss of
information, however consistent the decisions, deciding the matter using all possible shape variables is invariably inconsistent.
Such an impasse hints at entrapment by
FIG. 4. Btw(b;a,c) is a function of the angle at b, not
the distances from b to a or c. We have btw(b;a,c) =
btw(b';a',c') = btw(b";a",c").
WHEN ONE FORM IS 'BETWEEN TWO OTHERS
0
1 2
3
4
5
629
6
\
H*
^—
- --^
71
I
1
0
1
2
Skull of chimpanzee.
3
4
Human skull.
5
Skull of baboon.
FIG. 5. D'Arcy Thompson's Cartesian transformation grids from Diodon to Orthagonscus and from "human
skull" through chimpanzee to baboon. (See text.)
some assumption not yet made explicit. To
extricate ourselves from it and get on with
the work of comparative morphology we
might restrict the domain of measures in
some severe fashion, or average somehow
over the conclusions drawn according to
all possible variables, or otherwise evade
the issue. But the dilemma can be resolved
properly only through a radical redefinition of the task. I have let you assume, as
630
FRED L. BOOKSTEIN
A
B
C
Fie. 7. The betweenness of form B is embodied in
the relation of the two independent crosses superimposed upon it.
k
lations t(a,b) from a to b, t(a,c) from a to
c, and t(b,c) from b to c all have the same
direction and sense.
Even when b is not on the line ac, as in
Figure 3, we may examine the translations
t(a,b), t(b,c) to see how nearly they are
aligned. For this purpose we would like a
continuously varying measure, btw(b;a,c),
which equals +1 whenever b is exactly in
between a and c, and - 1 when one of a,
c is in between the other and b. A suitable
quantity is the cosine of the angle made at
b between the two translation vectors, one
into point b, one out. This cosine is also
the proportional component of t(b,c) along
t(a,b), likewise of t(a,b) along t(b,c). When
FIG. 6. A series of three parallelograms A, B, C is t(b,c) is perpendicular to t(a,b), its value is
described by two strains, A —» B and B —* C. Each zero. In this case the additional translation
strain is defined by a cross of principal axes bearing from b to c is irrelevant to the direction of
dilatations, superimposed on either form.
I did when I began this research, that the
estimation of betweenness begins with
measures of three separate forms. In fact,
it cannot. Betweenness is rather a measure
of pairs of transformations.
EUCLIDEAN BETWEENNESS AS
CONCORDANCE OF TRANSLATIONS
Suppose that to the usual three outlines
A, B, C are associated three "measurement
vectors," points a, b, c in an N-dimensional
space, as in Figure 2. Here, as for real
numbers, point b is between points a and
c when it lies within the finite line-segment
drawn to connect them. In terms of the
translation vectors which connect the
points one to the next, point b is exactly in
between a and c if and only if the trans-
FIG. 8. Two examples of perfect betweenness of
form B—exact alignment of the crosses and log-proportionality of the dilatations. Here we have log d'AB/
log d'ac = log d2AB/log d2Br = 1.
631
WHEN ONE FORM IS BETWEEN TWO OTHERS
translation from a to b, adumbrating neither a positive nor a negative multiple.
The angle we are extracting is a function
of the directions of t(a,b), t(b,c), not their
magnitudes. Except for measurement
errors induced by noise, it is immaterial
for betweenness how much closer b is to
one of a, c than to the other. For instance,
we may vary t(b,c) by iteration (multiplication of the vector by a positive constant),
and thereby move c while leaving btw(b;a,c)
unchanged, as in Figure 4.
The vectors t of this little example can
be computed only by measuring the forms
A, B, C separately and then subtracting.
Our generalization will require for forms
not embedded in a parameter space this
same possibility of computing the similarity of a pair of transformations. If we could
measure each transformation directly, we
would neatly escape the paradox of size
and shape measures by avoiding measurement of forms, "points" undergoing
"translations," completely.
THE APPEARANCE OF PERFECT
BETWEENNESS FOR PARALLELOGRAMS
The method I propose is described most
easily for the special sort of distortions that
are affine transformations, homogeneous
strains which leave parallel lines parallel.
Let A, B, C be three parallelograms with
corresponding edges homologous linearly
along their lengths, so that the transformation T(A,B) by which B derives from A
is a homogeneous strain, and T(B,C) likewise.
How is the relation T(A,B) between two
such forms to be visualized, in order that
it might be compared to T(B,C)? D'Arcy
Thompson (1917), in his famous method
of "Cartesian transformation," suggested
the use of a grid aligned with some feature
of the form, as in the famous examples of
Figure 5. I suggested (Bookstein, 1978)
aligning instead with the shape change itself, making use of the principal axes of the
strain, the two directions which are at 90°
both before the transformation and after
(hence the neologism, "biorthogonal analysis").
These directions may be characterized
instead by the dilatations d'AB, d2AB, the
CM
O
log d 1
FIG. 9. Plot of a transformation as a point on a line
on a scatter of log d1 by log d2.
factors of multiplication of length, along
them. For one principal axis the dilatation
is maximal, and for the other, minimal,
among all directional rates of length-multiplication from one form to the other. Figure 6 displays each transformation, A —»
B or B —» C, by a cross upon either form
aligned with the principal axes and with
each arm of length proportional to the dilatation, d'AB or d2AB, upon it. The strain
from B to A bears the same principal axes
as that from A to B and dilatations d'BA,
d2BA which are the reciprocals of d2AB,
We may draw one cross upon form B,
the middle form, for each of the distortions T(A,B), T(B,C). Toward the measurement of btw(B;A,C), the series of
three parallelograms is reduced to the
comparison of two crosses on the single
parallelogram B, as in Figure 7. One cross
describes the passage into stage B, the other the passage out of it.
Just as we examined the concordance of
the two translations t(a,b), t(b,c) in Figure
3 by reference to the vectors bearing them,
so we need to examine the agreement of
these two transformations T(A,B), T(B,C)
632
FRED L. BOOKSTEIN
btw
btw,
size
shape
b<0
FIG. 10. The non-Euclidean angle measures of betweenness. (A) The major diagonal is perpendicular to all
lines (transformations) with respect to shape-betweenness. (B) The minor diagonal is perpendicular to all
lines with respect to size-betweenness. The inequalities in various sectors of the diagrams bound the betweenness score, size or shape, with respect to the (arbitrary) transformation plotted.
by reference to the crosses bearing them.
Recall that a point b is in between two
points a and c when the translation t(a,b)
is an exact proper fraction of t(a,c)—when
by iterating the motion t(a,b) some (in general nonintegral) number of times we arrive at point c. For iteration leaves the direction of t unaltered, and the geometry
of vectors is such that if the directions of
t(a,b), t(a,c) are the same then the existence
of the appropriate iterate is guaranteed.
A partially analogous statement is true
for the strains T transforming these parallelograms. The principal axes of a transform remain the same under iteration, so
that B can be exactly in between A and C
only if the principal axes (in B) of T(A,B)
are exactly aligned with those of T(B,C).
But even then, in this tensor geometry, the
existence of a suitable iterate is not guaranteed, but must be explicitly demanded
instead. For B to lie exactly in between A
and C there must exist a single exponent
which raises the dilatations of T(A,B) simultaneously to their counterparts for
T(A,C). Such an exponent is the ratio of
logarithms log d'AC/log d'AB if its two values, for i = 1 and i = 2, are equal and positive. In that case T(A,C) is an exact iterate
of T(A,B), so that parallelogram B may be
said to be exactly in between A and C, just
as for the vectors of Figure 3—but the
forms A, B, C have not been separately
measured.
Figure 8 shows two examples of exact
betweenness for parallelograms. In both
cases the middle form B is halfway in between the other two.
WHEN ONE FORM IS BETWEEN TWO OTHERS
633
THE ESTIMATION OF IMPERFECT
BETWEENNESS FOR PARALLELOGRAMS
We assign the value +1 to btw(B;A,C),
and the value - 1 to btw(A;B,C) and
btw(C;A,B), in both the cases of Figure 8.
Now a general series of parallelograms will
fail to fit this unambiguous construction in
two ways: The ratios of log dilatations, that
is, the apparent exponents of iteration
along the two principal axes, may not be
the same; or the axes of T(A,B), T(B,C)
may not be exactly aligned. I will deal with
these difficulties in turn.
Axes aligned but dilatations
disproportionate
Whenever the principal axes of the
transformations T(A,B), T(B,C) are
aligned, the relation between the transforms is embodied in the two pairs of dilatations (d' AB , d2AB), (d'ec, d2BC)- On a diagram whose axes are the logarithms of the
dilatations, as in Figure 9, each pair, and
so each strain, appears as a single point.
Invariance of betweenness under iteration
of transformations away from B (just like
iterations of translation in Figure 4) implies that its value may depend only on the
half-lines from the origin through the observed log-dilatation Cartesian pairs, as
drawn, and not on position upon those
lines. As a function of a pair of half-lines
it is clearly a sort of angle measure.
The diagonal lines of this plot serve a
special function. For instance, transformations which are plotted on the diagonal
from upper right to lower left, the major
diagonal log d1 = log d2, are pure changes
of scale: they multiply distances in all directions by the same factor, thereby leaving shape unchanged. Then insofar as we
are measuring betweenness of shape these
pure changes of scale must be orthogonal
to all other transformations. In geometric
terms, the major diagonal must be perpendicular to all other lines in the diagram.
Betweenness for shape, then, must behave as sketched in Figure 10a. It corresponds exactly to a nearly forgotten variant of non-Euclidean geometry, Galilean
geometry, familiar only to old-fashioned
mathematicians; there the measure we
seek is called parabolic angle measure. T h e
FIG. 11. Affine distortion of the major and minor
diagonals so that slope-differences for size-betweenness and shape-betweennness can be weighted for
misalignment of the two crosses of principal axes.
formulas which will appear below as dei ex
machina are derived, in the classical treatises, as necessary consequences of the properties required of the measure. The reader
who wishes to appreciate and understand
these matters is referred to the excellent
elementary review by Yaglom (1979).
The quantity we seek takes its simplest
embodiment when the diagram is rotated
by 45° counterclockwise, so that the major
diagonal lies vertical. There is a quantity
kshape which is the difference between the
slopes of the two half-lines in this coordi-
634
FRED L. BOOKSTEIN
pair of strain-tensors is reduced to a pair
of angle measures, one for size and one
for shape: two coefficients which together
measure betweenness in series of three
parallelograms. The existence of these two
independent betweennesses is a funda- log(d>ABd*AI0/log(d»AB/d*AB).
mental difference between analysis of paThis k ranges from — oo to +°° and is pre- rameters, via vectors, and analysis of transcisely zero for perfect betweenness (slope formations via tensors.
Of course, in the perfect series of Figdifference zero). It behaves, then, like the
tangent of the angle we seek. A more con- ure 8, both cossize and cosshaPe were idenvenient version of this same ordination is tically +1, so we could not yet see the
the "cosine" ± 1/^/(1 + k2), which ranges need for two distinct quantities. But in Figbetween — 1 and +1 just like the cosine of ure 1 one can compute kslze = 1.24, kshape =
Figure 3. For half-lines on the same side 3.19, corresponding to "cosines" of 0.627
of the diagonal, the + sign is used, other- and —0.299, respectively, and "angles" of
wise the — sign. As either strain approach- 51° and 107°. The middle form is more
es a pure change of scale, k becomes ar- than twice as far out of shape-betweenness
bitrarily large and the computed "cosine" as it is out of size-betweenness with respect
tends toward zero, implying orthogonality to the two squares.
of the two transformations, as we reAxes misaligned
quired.
The preceding development assumed a
The major diagonal, representing pure
changes of scale, is orthogonal to all other well-defined pairing between the dilatachanges only when we are talking about tions of the two transformations. To the
shape purely. The other diagonal, log d1 = extent that the axes are out of alignment,
— log d2, plays a similar role in a different points and slopes representing the two
sort of measurement. Along this diagonal. strains become less directly comparable.
are found transformations which leave The ambiguity is worst when the axes of
area unchanged (since the product of the either cross squarely bisect the angles of
dilatations is unity). These are therefore the other: when they are just 45° out of
pure shape-changes which must be for- alignment. (Rotated beyond 45°, one cross
mally orthogonal to all other transforma- will tend toward alignment with the other
tions under another angle measure, for again, though with dilatations oppositely
"size-betweenness," sketched in Figure paired.) In the 45° case, either strain has
but a single effect upon the two principal
10b.
axes of the other. The transformations reThis measure is also based in a differ- late to each other as pure size changes,
ence of slopes, now after a rotation of Fig- decimal numbers without any remaining
ure 9 clockwise by 45° so that the minor discriminanda of orientation.
diagonal is vertical. In terms of the original
Then the more misaligned are two crossdilatations,
es of principal axes, up to 45°, the less can
kslze = log(d1BC/d2Bc)/log(d1Bcd2BC)
we measure shape-betweenness at all, and
the more does size-betweenness reduce to
-log(d1AB/d2AB)/log(dIABd2AB),
the alternatives +1, —1 as for decimal
and "cosinesize" =
+ k size 2 ).
numbers. In terms of the half-lines of FigThis formal parallelism of cosinesize and ure 10, misalignment of the crosses incosine shapc , size-betweenness and shape- creases all shape-related slope differences
betweenness, is quite pleasing. As the rel- (raising kshape and lowering cosineshape) and
ative orientation of a pair of translation- decreases all size-related slope differences
cosinesize). In
vectors was reduced to a single angle (lowering ksize and raising
1
2
measure, so the relative orientation of a effect, the size axis log d = log d has been
nate system. Each slope is a linear fractional transform (x -I- y)/(x — y) of the original
Cartesian coordinates (x,y) upon the diagram; then
WHEN ONE FORM IS BETWEEN TWO OTHERS
635
PflN.M
PfiN.M
H.SHP.M
PflPIO.H
FIG. 12. Principal axes for the transformation of an adult male chimp to an adult male human and to an
adult male hamadryas baboon. The data are seven-landmark abstractions from midsagittal sections, as described in the text. The crosses on each of the lower figures are in correct relative proportion.
expanded, and the shape axis compressed,
COMPUTATION FOR THE
by the same factor; the affine geometry is
GENERAL TRANSFORMATION
as in Figure 11.
In sufficiently small regions, smooth
A convenient quantity for this dependence of the betweennesses on misalign- transforms are indistinguishable from homent is cos 2$, where <>/ is the smallest an- mogeneous strains. Then to analyze begle between any arm of one cross and any tweenness for arbitrary curving forms A,
arm of the other. When </> = 0° the factor B, C it suffices to execute the preceding
is unity, leaving size-betweenness and analysis separately neighborhood by
shape-betweenness commensurate; when neighborhood of the outlines. There result
<f> = 45° the factor is zero, reducing thetwo betweenness measures at each point of
analysis to a binary measure (+1, - 1 ) of form B, whose values collectively form new
spatial fields. We can locate regions of high
size alone.
636
CIZL
FRED L. BOOKSTEIN
fcETWfcTINNESS
UF P A N . M
WITH KIT iJPI/Cl
fO H . S A P . M
0.99
i.oo
AND PAP 1 0 . HA
o . W 0.99
o . y y o . v y 0 . 9 9 o . v v 0 . 9 a 0 . 9 is
I. . 0 0 1 . 0 0 0 . 9 9 0 . 7 9 0 . 9 9 0 . 9 ' - ' 0,7l>> 0 . V 9
0,V9
1-00
1 . 0 0 0 . 9 V 0 •)'> 0 . 9 9 0 . 9 9 0 . 9 9 1 . 0 0
0.90
O.vR O . v y 1,00 0 . V 9 0 . 9 9 0 . 9 9 0 . 9 9 1,00 1.00
0.96
0 . V / 0 . 9 9 1.00 0.V9 0 . 9 8 0 . 9 9 1 ,00 0.99 1 .00
0-93
0 . 9 4 0 . 9 8 0 . 9 9 0 . 9 9 0 . 9 9 0 . 9 , ' O.(>>2 '),«•)</ 0 . 9 1 .
0,9/.
O.Sn
0 . 9 8 0 . 9 V 1 . 0 0 O . V j Qt/2
0.79
O.! ; i:v
0 , 9 9 1 . 0 0 1. . 0 0 0 > M 0 . , . ! 6 ~ 0 , 2 0
0.64
O . H';:, :l , 0 0 0 . 9 9 0 , 4!) 0 . 0i'f - 0 . 21 •••• 0 • A1
0,40
0,ci/>
O.iii'J O . ' i l 0 . 4 " >
0 . 6 1 0 . . ' ' . 0 - 0 . 0 A - 0 . ' 3 0 - 0 . !-"vl
O.U*
0 . 2 6 0 . 2 7 0.')8-0,•.;:!.-0.4.1. - 0 . 5 4
0 . 1. I
0 . 0 9 - 0 . 0 8 - 0 , 65 - 0 . 5 S - 0 . 61.
• 0 . 4 7 0 . 4 5 - 0 . I J 6 - 0 , ?0- -0,0'-.'-O . 9 1
- o . 4 9 - o . :-j8••••<>. 9::;- o . 9 0 - o . 9 2 - 0 . 9 3
•- 0 . 4 4 •••• 0 . 4 9 - •• 0 . 6 V - 1 . 0 0 - 0 . 9 9
•0.41.
0.46
FIG. 13. Size-between ness and shape-betweenness fields for Pan with respect to Homo and Papio, from the
tensors of Figure 12. Each printed number appropriately scores the pair of crosses in the same relative
position in the Pan outlines (Fig. 12, top).
or low betweenness, size- or shape-, and
estimate, wherever either betweenness is
high, how far B has come along the path
from A to C. Extended regions of divergent betweenness may correspond to different forms of functional regulation of
size and shape or different regimes of selective pressure.
For the computation of betweenness in
this wise one needs the principal axes of
the distortions T(A,B), T(B,C) at every
point of B. A computer algorithm published previously (Bookstein, 1978) supplies those fields of crosses, sampled at the
points of a grid, in three steps.
1. Outlines are recorded as homologous
landmarks joined by homologous boundary curves. The landmarks are simply
points coded by their Cartesian coordinates; boundary arcs are input as linear or
quadratic arcs between landmark pairs,
There may also be landmarks interior to
the boundary and not lying upon arcs.
2. For any pair of forms, the landmarks
and arcs together sample the distortion
function of the one whole form onto the
other. We extend this correspondence
throughout the interiors of the forms by
interpolation from the boundary data. A
result suitably smooth borrows the com-
WHEN ONE FORM IS BETWEEN TWO OTHERS
SHrtriT BETUEENNEttS OF PAN.M
637
WITH RESPECT TO H.SAP.M
AND PAPIO.HA
0.07 0.09 0.10
0.05 0.06
0.08
0.09 0.10 0.12
0.15
0.0:1 0.05 0.07 0.08 0.08 0.09 0 . 1 1
0.19
- 0 . 0 3 •0.01 0.05 0.07 0,08 0.08 0.08 0.10
0.15
••O.o4-0. 0 4 - 0 - 0 2 0.05 0.08 0.09 0.08
0.09 0.14
- 0 . 0 6 - 0 . 0 5 ~ 0 . 0 3 0.07 0.10 0.09 0.09
0.08-0.03-0.02
-0.09- 0,07-0.04
0.10 0.17
0.25
0.14-0.06-0.11-0.13-0.11
-0 . 1 4 - 0 . 1 0- 0 . 02 0 . 2.2 0 . 87-0 . 1.0-0 - 23- 0 . 33-0 . 3 4 - 0 . 3 7
0.00-0.25-0.44-0.6 4
•0.1 8 0 . . I 3 («,1« 0.09
-0.29-0,15
0.75
0.17-0.21-0.39-0.57-0,74
- 0 , 5 0 - 0 . 3 3 -0.25-0.29
0,40-0.53-0.65
-0. H .5-0 . 68-0 . 5 3 - 0 . 4 9 - 0 . 52-0 . 5 8 - 0 . 64
- O. i i :> - 0. 6 8 - 0 . 6 .'=. - 0 . 6 7 - 0. 7 0 - 0 . 7.1
-0.54-0,62-0.72-0.80-0.84-0.85
0. 33 0 • ?/- 0.4.1. - 0 . 75- 0 . 91 - 0 . 95
0,40 0,38
0.24-0.80-0.99
0.4 9 0-4/5
FIG. 13. Continued.
puter code of certain algorithms used in
computational physics. This extended homology function is sketched by its effect
on a square grid superimposed over one
form, as in Thompson's original Cartesian
transformation drawings.
3. At every point of either form, the
principal axes of the strain which approximates the transformation locally can be
computed by simple algebraic manipulations of the grid. A sample of crosses is
then drawn out upon both forms.
From two samples of these crosses, for
maps of a single form B onto two others,
A and C, the betweenness fields of B with
respect to A and C follow directly. One can
display these numbers in their appropriate
geometric ordering, inspect them, and average them over homogeneous regions of
B and over the whole,
By way of example I will analyze some
higher primate forms. The data are taken
from a larger exploratory study, reported
in Bookstein (1978), of the cranium in
midsagittal section. From images of sections variously published I abstracted sets
of landmarks and smooth curves connecting them. In the diagrams which follow,
landmarks are indicated by an X on the
outline. Clockwise from lower left, they
are: prosthion, the frontmost point of the
alveolar margin of the maxilla; rhinion,
the bottom of the internasal suture; nasion, the top of that suture; inion, center
638
FRED L. BOOKSTEIN
RLOURT.F
RLOUHT.I
FIG. 14. Principal axes for the transformation of an adult female Alouatta to an infant and to an adult male.
of the bump where the nuchal torus crosses the midplane; basion, frontmost point
of the foramen magnum; and an unnamed
point where the last molar erupts from the
alveolar bone. In addition there is one interior landmark, sella, drawn as a dot near
the lower central margin of all the forms.
Each diagram depicts one transformation
by its strain-tensor field drawn out over the
two forms. (The arms of the crosses, furthermore, are to correct relative scale
upon the bottom forms.) The numeric
printouts of betweennesses display values
upon the grid of locations of crosses for
the top form.
D'Arcy Thompson used the "series"
from human, through chimpanzee, to baboon (Fig. 5) to exemplify the role of the
Cartesian method in studies of betweenness. He concluded, rather nonquantitatively:
. . . It is obvious that the transformation [from
man to baboon] is of precisely the same order
[as the transformation from man to chimp], and
differs only in an increased intensity or degree
of deformation. . . . It becomes at once manifest
that the modifications of jaws, brain-case, and
the regions between are all portions of one continuous and integral process.
A general review of Thompson's approach
to transformation analysis may be found
in Bookstein (1978, ch. v). For re-analysis
of this matter I replaced Thompson's specious figures by appropriate seven-landmark abstractions. Figure 12 displays the
two tensor fields of strain for the transformations from Pan to Homo and to Papio,
and Figure 13 the two betweenness fields.
We notice that although Pan is clearly in
between the other two forms in size over
the calvarium, the principal axes of the
transformations are nearly as misaligned
as they can be, so that shape-betweenness
is generally very low. Around the maxilla,
size-betweenness and shape-betweenness
are both substantial and negative, indicat-
WHEN ONE FORM IS BETWEEN TWO OTHERS
639
SIZE BETWEENNESG OF ALOUAT.F WITH RESPECT TO ALOUAT.I AND ALOUAT.M
0.03
-0.16 0.72 0.40 0.49 0.56 0.53 0.57
0.17
-0.13 0.69 1.00 0.99 0.97 0.96 0.96 1.00
0.83
-0.24 0.67 0.96 1.00 0.97 0.95 0.99 0.91 0.86
0.89
-0.84 0.74 0.94 0.98 1.00 0.99 0.92 0.84 0.81 0.U4
0.92
-0,69 0.95 0.97 0.97 0.93 0.99 1.00 0.96 0.81 0.84 0.90
0.91
0.78 0.89 1.00 0.98 0.98 0.98 0.98 0.96 0.94 0.99 0.94
0.97
0.84 0.87 0.93 3.00 0.98 0.96 0.95 0.94 0.89 0.85
1.00 1.00 0.98 0.99 0.99 0.97 0.95 0.93 0.90 0.88
3.00 0.99 0.98 0.99 0.99 0.96 0.93 0.91
0.95 3.00 1.00
0.97
0.87
0.88
1.00 0.93 0.95 0.97 0.86
0.9V 1.00 1.00 0.98 0.94 0.89
0.90
1.00
SHAPE Br rWLKNNh !.'S OK ALOUAI'.I- U]1H KESPFCT TO ALOUAT. 1 ANTi ALOUAt.M
0.89
0.81 0.99 0.9/ 0.96 0.93 0.84 0.46-0.62
0.61' 0.91
1.00 0.99 0.96 0.91 0.70 0.08-0.35
0.37 0.6S 0.93 3.00 0.88 0.68 0.48-0.3.0-0.13-0.10
0.06 0.35 0.58 0.82 0.99 0.77 0.13-0.14-0.17-0.13-0.07
-0.38-0.OH O.?7 0.45 0.60 0.75 0.94-0.07-0.16-0.16-0.3 2-0.10
-0.1.9-0.11 0.14 0.31 0.42 0.51 0.53 0.27 0.15 0.16 0.38
0.84
-0.16-0.12-0.08 0.07 0.23 0.35 0.45 0.53 0.53 0.48 0.71
-0.02-0.02-0.03-0.02 0.10 0.20 0.32 0.45 O.b3 0.64
0.72 0.28 0.12 0.08 0.08 0.16 0.32 0.60
0.04 0.44 O.SS 0.16-0.04-0.05 0.00
0.62
0.68
0.28
-0.04 0.28 0.54-0.05-0.08-0.10-0.03
0.33
FIG. 15.
Size- and shape-betweenness fields for the adult female with respect to the infant and the male.
ing alignment of the axes but a seriation
opposite to what Thompson implied. The
transformations of the chimp's upper jaw
to the other two forms are more nearly
iterates than inverses of each other. Of
course, we no longer have any reason to
believe that these forms fall in a series,
Another example, rather more reasonable, examines the sexual dimorphism of
the howler monkey Alotiatta for evidence
640
FRED L. BOOKSTEIN
of a "series" from infant through female
to male. The biorthogonal tensor fields are
in Figure 14, the betweenness scores on the
same grid in Figure 15. Here shape-betweenness is large and positive throughout
large regions of the cranium, such as the
sector from hyoid to forehead, while sizebetweenness is consistently very large
throughout. There is clearly suggested a
single system of form-control throughout
the skulls, in terms of which the male represents a hypermorphosis of the female.
With the ordination supported by the scalars, one may return one's eyes to the tensor diagrams and extract distances and angles which would most precisely capture
the morphological ordination. We locate
two nearly perpendicular craniometric distances—one from mid-hyoid to acrion, one
from nasion to basion—which are aligned
with the principal axes in the region showing the most consistent betweennesses of
the series. The ratio of these two distances
is therefore a very likely candidate for dependent variable in analysis of intra- and
interspecific variation of this aberrant primate form.
FIG. Al. Diagram for the proof of the Shape Nonmonotonicity Theorem.
The principle of the demonstration is
clear from any specific example. For any
point P, on outline A, for instance, there
is a point P2 on A which is halfway around
the outline from P, (see Fig. Al). Conversely, P, is halfway around the outline
from P2. Call the segment P,P2 the bisecting
line through P,; it is likewise the bisecting
line through P2. Let Qi, Q2 be the homologues of P,, P2 on outline B. In general
the line QiQ2 will not be a bisecting line of
B. Assume, without loss of generality, that
the bisecting line of B through Q,, the line
C^Qa' in the figure, falls to the left of
Q,Q2.
Now consider a point P moving around
outline A. To this point P corresponds a
ACKNOWLEDGMENTS
moving point P' such that PP' is always a
The research reported here was support- bisecting line of A. Let Q, Q' be the homed by N.S.F. grant SOC 77-21102 to F. L. ologues of P, P' on outline B, and define
Bookstein and N.I.H. grant DE-03610 to the defect of P to be the excess of B's perimeter between Q and Q' on the left.
R. E. Moyers.
Note that although P moves on outline A,
the defect function is measured on outline
REFERENCES
B in accordance with the homology beBookstein, F. L. 1978. The measurement of biological tween the outlines. Whenever the homolshape and shape change. Lecture notes in biomath- ogy between A and B is smooth, the defect
ematics, v. 24. Springer-Verlag, Berlin.
is a continuous function of the position of
Thompson, D'Arcy W. 1917 (1942, 1961). On growth P on A.
and form. Cambridge University Press, CamFor P = P,, the homologue QiQ2 of the
bridge.
Yaglom (IAglom), I. M. 1979. A simple non-Euclidean bisecting line PiP2 through P leaves too
geometry and its physical basis. Springer-Verlag, much perimeter on the left, so that the
New York.
defect at P, is positive. For P = P 2 , the
homologue of the bisecting line is the same
line Q2Qi as before, but viewed, now, from
APPENDIX
the other end. In this position it leaves too
T H E SHAPE NONMONOTONICITY THEOREM
much perimeter of B on the right, so that
Here I show that for any two homolo- the defect at P2 is negative.
gous smooth outlines A, B, there are alThe defect has changed sign in passing
ways measures on which they exactly from P = P, to P = P2. Since defect is a
agree.
continuous function of the position of P,
WHEN ONE FORM IS BETWEEN TWO OTHERS
somewhere between P, and P2 it must be
zero. In other words, there exist points P 3 ,
P3' on A such that P3P3' and its homologue
Q3Q3' each bisect their respective outlines.
"The ratio of perimeters in which the
segment homologous to P3P3' cuts the
forms of a lineage" is a perfectly valid measure of these forms; and by construction
of P3, forms A and B have the same value
on this shape measure, namely, unity.
This sturdy demonstration can be easily
generalized. Define an involution to be a
function from an outline onto itself which
is its own inverse, such as the assignment
P,«-> P2 in the example. Let an involutory
pair be a pair of points P,, P2 which correspond under an involution. Then if A
and B are outlines linked by a smooth ho-
641
mology, for any continuous involutions on
A and B without fixed point there exist
an involutory pair on A and another on B
which correspond under the homology,
and therefore a measure on which A and
B have the same score. The proof is by
forcing the existence of a point of zero
"defect" exactly as in the special case.
Hence there exist indefinitely many paradoxical shape measures, one for each involution function. For example, the map
which sends any point P, to the point P2
such that P ^ j bisects the area inside the
outline yields another point P 3 and
another measure on which outlines A and
B agree; for convex blobs, the map which
sends P! into the point P2 whose tangent
is parallel to P^s yields another; and so on.