Hughes et al. 2002.1


2 3

The shape of the dental arches has held the attention 
of physical anthroplogists and dentists since the 
beginning of the last century. Many methods have been 
developed to describe dental arch morphology, ranging 
from simple geometric classifications (Hrdlička, 1916), 
through combinations of linear dimensions (Moorrees, 
1959) to various complex curve-fitting procedures 
(Lu, 1966; Jones and Richmond, 1989; Kasai et al., 
1995; Battagel, 1996). The application of fourth-order 
polynomials of the form:

y = a + bx + cx2 + dx3 +ex4
provides a number of advantages, the most significant 
being that the coefficients can be easily interpreted 
(Richards et al., 1990). The second (x2 or quadratic) 
and fourth (x4 or quartic) terms describe the arch shape 
while the first (x1 or linear) and third (x3 or cubic) terms 
describe asymmetry.

Lu (1966) drew attention to the inter-dependence 
of coefficients of simple polynomials. The sum of 
squares associated with the k coefficients cannot be 
partitioned into k parts, each attributable to a single 
degree of freedom. Consequently, it is not possible to 
assign accurate values to the relative contributions 
of symmetry and asymmetry to overall arch shape. 
The partition can be achieved, however, by using 
orthogonal polynomials (Kendall, 1959). Lu (1966) 
presented the first detailed account of fitting orthogonal 
polynomials to arch data. Unfortunately, although the 
theory for equally-spaced x-coordinates was sound, 

ABSTRACT There have been numerous attempts to 
quantify the shape of the dental arch mathematically, 
with orthogonal polynomial curves providing a robust 
and versatile method for quantifying variation in both 
shape and asymmetry. Lu (1966) first presented the 
theoretical basis for fitting orthogonal polynomials to 

Lu’s worked example contained some mathematical 
errors. Furthermore, the extrapolation to non-equally 
spaced data was flawed. Kendall (1959) provided the 
correct general parameterization for unequally-spaced 
data, and this was further simplified to a recursive 
method by Robson (1959). The aim of the current paper 
is to address the errors within Lu’s orginal paper, and to 
present a valid extrapolation of his work to unequally-
spaced arch data for use in quantitative assessments of 
arch form.

RESULTS AND DISCUSSION

The theoretical workings presented on pages 1058-
1062 of Lu’s original paper are substantially correct, 
with one small exception, and it would be inappropriate 
to reproduce this section in great detail. We urge 
those readers who are interested in Lu’s adaptation 
of orthogonal theory that produces a partition of arch 
shape variance for equally-spaced data to examine the 
original paper, noting that the calculation for the sums 
of squares for the intercept of the orthogonal regression 
on page 1059 reads:

SS =
Y

n

2

( )b0 0ξ
∑

when it should instead read:

SS =
Y

n
( )b0 0

2

ξ
∑( )

Theoretically, Lu’s concept cannot be faulted. 
However, the application of the theory was flawed, 
particularly in the use of published orthogonal 
polynomial tables for equally-spaced data (Fisher and 
Yates, 1957), leading to biased estimates of orthogonal 
coefficients. In the next section of this paper we reproduce 
verbatim the worked example from Lu’s original paper 

Form, Symmetry and Asymmetry of the Dental Arch: 
Orthogonal Analysis Revisited
Toby Hughes, Lindsay Richards, and Grant Townsend
Dental School, The University of Adelaide, South Australia, 5005.

Editor’s Note: Lu’s paper (1966) has been cited many 
times in the dental literature, but researchers have 
been confused by the nature of the analysis (orthogo-
nal regression is not the same as conventional regres-
sion analysis) and most biologists and clinicians have 
been unable to “break-through” Lu’s mathematics, 
particularly since there are several key errors in the 
paper. Toby Hughes and his colleagues were invited 
to submit this paper to facilitate the understanding 
and application of this useful analytic method.

Correspondence to: Toby Hughes, Level 6 Dental School, 
University of Adelaide University, Australia 5005
E-mail: toby.hughes@adelaide.edu.au

arch shape data. Whilst theoretically sound, Lu’s original 
paper contained several arithmetic errors and a number 
of incorrect assumptions. In this paper we present 
corrections for these errors and extrapolate the theory 
to unequally-spaced arch shape data using a simple 
recursive procedure first developed by Robson (1959).



4 5

Although the curve-fitting technique is well 
known in statistical circles, it is thought that the 
intended audience of this paper might not be as 
familiar, and for this reason the sample is explained in 
considerable detail.

The arch width is divided into 14 equidistant 
intervals defined by 15 points. We have the following 
observed data:

X (arch base): -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7

Y (arch height): 27.7, 20.1, 13.9, 9.4, 6.1, 3.4, 1.4, 0, 0.3, 
1.3, 3.2, 6.3, 11.0, 18.2, 29.0

The computations are illustrated in Table 1.

∑Yξ
1
 = 43.2 c ∑ξ

1
2 = 280

∑Yξ
2
 = 6932.2 a ∑ξ

2
2 = 37,128 a

∑Yξ
3
 = -684.4 a ∑ξ

3
2 = 39,780 a

∑Yξ
4
 = -105,958.6 ab ∑ξ

4
2 = 6,466,460 ab

AN ILLUSTRATIVE EXAMPLE

for equally-spaced arch shape data (boxed text, page 
4) from Lu’s page 1062 onwards, important errors are 
shown in bold, and footnotes (listed below) to the text 
contain explanations and appropriate corrections.

a ξi represents a polynomial of degree i in x (i.e. ξi 
= ϕ

i
(x

j
) ). The original table from Fisher and Yates 

(1957) specifies a series of pre-multipliers (λ
in
) for 

ξ
1
-ξ

4 
in footnotes beneath the table. Following Fisher 

(1921), these arbitrary constants are determined 
conveniently so that ξ

i
 is an integer for all j = 1, 2, 

..., n. They were not referenced in Lu’s paper, and 
his failure to apply them in subsequent calculations 
resulted in substantial errors throughout the 
remainder of the worked example.

b This value was incorrectly signed as negative (-) in 
Lu’s paper resulting in incorrect values for ∑Yξ

4
 and 

∑ξ
4

2.
c Whilst Lu’s use of sums and differences is 

arithmetically correct, we feel it adds unneccesary 
complexity to the calculations. Indeed, in the one 
series of calculations where the pre-multiplier was 1 
and in which Lu’s figures should have been correct, 
the sign of ∑Yξ

1
 was incorrectly reported as positive 

(+), presumably due to the incorrect summing of 
the cross-products of the differences. It is preferable 
to simply list the full table and obtain the cross-
products directly. Correct values for ∑Yξ

i
 and ∑ξ

i
2 

are as follows:
 ∑Yξ

1
 = -43.2 

 
 ∑ξ

1
2 = 280

 
 ∑Yξ

2
 = 2,310.7 

 
 ∑ξ

2
2 = 4,125

 
 ∑Yξ

3
 = 821.3 

 
 ∑ξ

3
2 = 57,283

 
 ∑Yξ

4
 = 2,590.2 

 
 ∑ξ

4
2 = 760,139

Due to the calculational errors noted above plus a 
number of subsequent arithmetic and typographical 
errors, the remainder of the worked example was 
substantially incorrect. The correct parameterization 
with the associated partition of variation (Table 2) is 
presented below:

 ∑Y2 = 2848.55 ∑Y =151.3 n = 15

 ∑Yξ
1
 = -43.2 ∑Yξ

2
 = 2,310.7

 ∑Yξ
3
 = 821.3 ∑Yξ

4
 = 2,590.2

Explanatory notes
From observed data, we compute ∑Y2 and ∑Y.
2. Column 1 is obtained by adding the Y values pair-

wised from the centre, e.g. 1.4 + 0.3 = 1.7; 3.4 + 1.3 
= 4.7, etc.c

3. Column 2 is obtained by subtracting the Y 
value corresponding to X from the Y value 
corresponding to –X, e.g. 1.4 – 0.3 = 1.1; 3.4 – 1.3 = 
2.1, etc.c

4. Columns 3, 4, 5 and 6 are obtained from the 
orthogonal polynomial tablesa (Fisher and Yates, 
1957) with n = 15. The ∑ξ

i
2 are also obtainable 

from this table. These values are only listed for 
the upper half of the entire polynomial.c For even-
powered ξ the omitted half are duplicates of the 
exhibited half; for odd-powered ξ the omitted half 
are numerically the same values as the exhibited 
half, except with the signs reversed.

5. To obtain ∑Yξ
1
 and ∑Yξ

3
 we obtain the sum of 

cross products of the differences (Column 2) with 
Column 3 and Column 5 respectively.c

6. To obtain ∑Yξ
2
 and ∑ξ

4
, we obtain the sum 

of cross products of the sum (Column 1) with 
Column 4 and Column 6, respectively.c 

ORTHOGONAL ANALYSIS REVISITED



4 5

b =
Y

n
=

151.3
15

=10.08670
∑

b =
Y

=
-43.2
280

=-0.15431
1

1
2

ξ

ξ
∑
∑

b =
Y

=
2310.7
4125

=0.56022
2
2

ξ

ξ
2∑

∑

b =
Y

=
821.3
57283

=0.1433
3
2

ξ

ξ
3∑

∑

b =
Y

=
2590.2
760139

=0.00344
4
2

ξ

ξ
4∑

∑

SS (b
1
ξ

1
) = b

1
∑Yξ

1
 = (-0.1543)(-43.2)

 = 6.67

SS (b
2
ξ

2
) = b

2
∑Yξ

2
 = (0.5602)(2310.7)

 = 1294.45

SS (b
3
ξ

3
) = b

3
∑Yξ

3
 = (0.0143)(821.3)

 = 11.74

SS (b
4
ξ

4
) = b

4
∑Yξ

4
 = (0.0034)(2590.2)

 = 8.81

SS (b
0
ξ

0
) = b

0
∑Y = (10.0867)(151.3)

  = 1526.12

SS (total) = ∑Y2 – SS (b
0
) = 2848.55 –

 1526.12 = 1322.43

Total SS explainable by regression:

R =
1322.43-0.76

1322.43
=0.99972

Index of total symmetry:

A=
V +V

V +V +V +V
x1002 4

1 2 3 4

=
1303.26
132167

 x 100 = 98.31%

Index of total asymmetry:

B = 100 - A = 1.69%

Index of taperedness:

A =
V

V +V
x 1002

2

2 4

 
=

1294.45
1303.26

 x 100 = 99.32%

Index of squaredness:

A
4
 = 100 - A

2
 = 0.68%

Index of lopsidedness:

ORTHOGONAL ANALYSIS REVISITED

 Sum Difference ξ
1
a ξ

2
a ξ

3
a ξ

4
a

 
 0.0 0.0 0 -56 0 756
 1.7 1.1 1 -53 -27 621
 4.7 2.1 2 -44 -49 251
 9.3 2.9 3 -29 -61 -249
 15.7 3.1 4 -8 -58 -704
 24.9 2.9 5 19 -35 -869
 38.3 1.9 6 52 13 -429
 56.7 -1.3 7 91 91 -1001b

TABLE 1. Computational table for fitting a fourth order orthodongal polynomial to 15 points

 Source d.f. Sum of Squares

 Total  14 1322.43

 Symmetry (b
2
 + b

4
) 2 1303.26

  Quadratic (b
2
) 1 1294.45 . . . V

2

  Quartic (b
4
) 1    8.81 . . . V

4

 Asymmetry (b
1
 + b

3
) 2   18.41

  Linear (b
1
) 1    6.67 . . . V

1

  Cubic (b
3
) 1   11.74 . . . V

3

 Remainder 10    0.76

TABLE 2. Partition of variation



6 7

B =
V

V +V
x 100 = 

6.67
18.41

 = 36.26%1
1

1 3

Index of tiltedness:

B
3
 = 100 - B

1
 = 63.77%

The procedure outlined above is suitable for data 
obtained at equidistant increments of X. However, on most 
occasions, investigators wish to define the dental arch in 
terms of specific anatomical landmarks. In such cases, the 
width distances of the arch may increase unequally and 
the use of tabulated orthogonal coefficients is invalid. 
Lu’s (1966) analytical extension to unequally-spaced data 
was flawed, irrespective of the numerous typographical 
errors that were present in the derivation. Lu noted that in 
computing the following simple polynomial regressions:

Y = a + b
1
x

Y = a’ + b’
1
x + b

2
x2

Y = a’’ + b’’
1
x + b’

2
x2 + b

3
x3

Y = a’’’ + b’’’
1
x + b’’

2
x2 + b’

3
x3 + b

4
x4

it can be shown that:
Y = a - b

1
ϕ + b

2
ϕ2 + b

3
ϕ3 + b

4
ϕ4

Fig. 1. Graphical representation of cusp tip spacings 
used to define arch shape. 

X (arch base): -19.12, -17.68, -14.81, -11.63, -8.03, -2.20, 2.22, 9.41, 14.46, 17.38, 20.19, 21.54

Y (arch height): 35.18, 31.78, 24.41, 18.28, 13.61, 11.80, 11.79, 13.77, 18.42, 24.43, 31.84, 35.75

Fig. 2. Arch shape described by an orthogonal fourth or-
der polynomial, with antimeric points joined to illustrate 
the degree of asymmetry. 

where ϕ
1, 
ϕ

2
, ϕ

3
, and ϕ

4
 are orthogonal polynomials and 

their coefficients are the last unprimed coefficients of each 
of the four equations respectively. However, this is only 
true in the case of equally-spaced data, a fact overlooked 
in the original paper. Even were it appropriate for use 
on unequally-spaced data, the subsequent partition of 
variance that was presented (cited from Ostle, 1958) 
was also incorrect, a fact which can be easily verified by 
application to the equally-spaced data from the same 
paper.

Kendall (1959) presented the analysis of equally-
spaced x-values as a special case of the more general 
usage of orthogonal polynomials for all data-types. 
Robson (1959) extended the analysis of non-equally 
spaced x-values by presenting a simple recursive 
procedure to estimate appropriate orthogonal polynomial 
equations. An alternative construction procedure, also 
recursive but requiring the solution of r linear equations 
for the construction of ∫

r
(x

i
) was described by Grandage 

(1958). Robson’s (1959) methodology is robust and 
efficient and remains the method of choice for both 
equally- and unequally-spaced arch data. For the full 
methodology, a detailed examination of the original paper 
is recommended.

A simplified protocol appropriate for fitting a fourth-

TABLE 4. X and Y values for the dental arch used in the illustrative example

ORTHOGONAL ANALYSIS REVISITED



6 7

 Source d.f. Sum of Squares

Total 11 935.87

Symmetry (b
2
+b

4
) 2 932.19

 Quadratic (b
2
) 1 918.21 ...V

2

 Quartic (b
4
) 1 13.98 ...V

4

Asymmetry (b
1
+b

3
) 2 1.00

 Linear (b
1
) 1 1.00 ...V

1

 Cubic (b
3
) 1 0.00 ...V

3

Residual 7 2.68

TABLE 3. Partition of variation

order orthogonal polynomial to arch shape data, and the 
subsequent partition of variance (Table 3) and derivation 
of shape-indices is presented below. The worked example 
uses data from a single arch (Hughes et al., 2001) for 
illustrative purposes (Fig. 1).  The data are listed in Table 
4.

ξ0  = 
1

n

ξ
ξ ξ

ξ ξ
1

0

0 0
2

 = 
x- 0 x

x x

∑
∑∑ −( )

ξ
ξ ξ ξ ξ

ξ ξ ξ ξ
2

2
0

2
1

0
2

0 1
2

1

2
 = 

x- 0 1x x

x x x

−

− −( )
∑∑

∑∑∑

ξ
ξ ξ ξ ξ ξ ξ

ξ ξ ξ
3

3
0

3
1 2

3
2

0
3

0 1
3

 = 
x- 0 1x x x

x x x

− −

− −

∑∑∑
ξξ ξ ξ1 2

3
2

2
−( )∑∑∑∑ x

ξ
ξ ξ ξ ξ ξ ξ ξ ξ

ξ
4

4
0

4
1 2

4
2 3

4
4

0

 = 
x- 0 1x x x x

x x

− − −

−

∑∑∑∑
44

0 1
4

1 2
4

2 3
4

4

2
ξ ξ ξ ξ ξ ξ ξ− − −( )∑ ∑∑∑∑ x x x

b
1
 = ∑yξ

1
 = 1.00 SS

1
 = b

1
2

b
2
 = ∑yξ

2
 = 30.30 SS

2
 = b

1
2

b
3
 = ∑yξ

3
 = 0.04 SS

3
 = b

3
2

b
4
 = ∑yξ

4
 = 3.74 SS

4
 = b

4
2

Total SS = 
y - y

n
= 935.87

2 ∑∑ ( )2

Total SS explainable by regression:

R  = 
935.87 - 2.68

935.87
 = 1.002

Shape indices can be calculated as outlined earlier. Total 
symmetry = 99.89% is composed of taperedness (98.50%) 
plus squaredness (1.5%). Total asymmetry = 0.11% is 
composed of lopsidedness (100.00%) plus tiltedness 
(0.00%). The relative magnitudes of these indices are 
illustrated in Figure 2, which shows the fitted curve with 
connected antimeres.

CONCLUSION

Lu’s original 1966 paper remains of value for illustrating 
the utility of orthogonal polynomials in the analysis 
of arch shape data, and clearly the original theoretical 
considerations were of merit. Unfortunately, the numerous 
mathematical errors contained within the paper make its 
application to real-world data misleading and inaccurate. 
The corrections outlined in the present paper should now 
enable researchers to carry out more accurate and reliable 

ORTHOGONAL ANALYSIS REVISITED



8 9

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quantitative assessments of dental arch form using the 
orthogonal polynomial approach.

ACKNOWLEDGEMENTS

This study was supported by a grant from the National 
Health and Medical Research Council (NH & MRC) of 
Australia.

LITERATURE CITED

Battagel JM. 1996. Individualized catenary curves: their 
relationship to arch form and perimeter. Br J Orthod 
23:21-28.

Fisher RA. 1921. Studies in crop variation. I. An 
examination of the yield of dressed grain from 
Broadbalk. J Agric Sci 11:107.

Fisher RA, Yates F. 1957. Statistical tables for biological, 
agricultural and medical research.  New York: Hafner.

Grandage A. 1958. Orthogonal coefficients for unequal 
intervals. Biometrics 14:287-289.

Hrdlička A. 1916. Contribution to the anthropology of 
Central and Smith Sound Eskimos. Ann Pap Am Nat 
Hist 5:177-285.

Hughes TE, Richards LC, Townsend GC. 2001. Dental 
arch forms in young Australian twins. In: Brook A, 
editor. Dental Morphology 2001. Sheffield: Sheffield 
Academic Press Ltd, p 309-320.

Jones ML, Richmond S. 1989. An assesment of the fit of 
a parabolic curve to pre- and post-treatment dental 
arches. Br J Orthod 16:85-93.

Kasai K, Richards LC, Townsend GC, Kanazawa E, 
Tadamasa I. 1995. Fourier analysis of dental arch form 
in South Australian twins. Anthropol Sci 103:39-48.

Kendall MG. 1959. The advanced theory of statistics, 3rd 
ed.  Vol. II. New York: Hafner.

Lu KH. 1966. An orthogonal analysis of the form, 
symmetry and asymmetry of the dental arch. Arch 
Oral Biol 11:1057-1069.

Moorrees CFA. 1959. The dentition of the growing child. 
Cambridge: Harvard University Press.

Ostle B. 1958. Statistics in research. Iowa: Iowa State 
University Press.

Richards LC, Townsend GC, Brown T, Burgess VB. 1990. 
Dental arch morphology in South Australian twins. 
Arch Oral Biol 35:983-989.

Robson DS. 1959. A simple method for constructing 
orthogonal polynomials when the independent 
variable is unequally spaced. Biometrics 15:187-191.

ORTHOGONAL ANALYSIS REVISITED

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