Methods for size reconstruction in distorted and damaged vertebrae of fossil tetrapods

Abstract

The methods for reconstruction of the initial dimensions of distorted and damaged vertebrae in fossil tetrapods are elaborated and tested, using the axial skeleton of horned dinosaurs from the Upper Cretaceous of Mongolia as an example. These methods allow not only more reliable resolution of the problems of vertebrae preservation for processing morphometric data, but also improvement of reconstructions of the vertebral column, as skeletons are mounted for museum exposition.

Full text

ISSN 00310301, Paleontological Journal, 2015, Vol. 49, No. 3, pp. 293–303. © Pleiades Publishing, Ltd., 2015.
Original Russian Text © V.S. Tereshchenko, 2015, published in Paleontologicheskii Zhurnal, 2015, No. 3, pp. 70–80.
293
INTRODUCTION
The study of changes in the patterns of the vertebral
structure along the vertebral column (serial variation)
is of great importance for the recognition in the axial
skeleton of taxonomically significant characters (Ter
eshchenko, 1991, 2001, 2004, 2007; Tereshchenko
and Alifanov, 2003; Tereshchenko and Sukhanov,
2009, 2014; Holmes and Ryan, 2013). In some cases,
it is necessary to reconstruct lost or damaged vertebrae
to obtain their true parameters. The previously devel
oped method (Tereshchenko, 1990) allows the recon
struction of angular characteristics (inclination of
diapophyses, facets of prezygapophyses and postzyga
pophyses, etc.) in distorted vertebrae and linear mea
surements of each second lost vertebra in a series of
wellpreserved vertebrae. The present study provides
the methods for reconstruction of linear measure
ments of satisfactory preserved vertebrae, which retain
general integrity of elements, but have damages and
(or) distortions. Although these methods are elabo
rated in the study of serial variation of the axial skele
ton of horned dinosaurs of Mongolia (protocerato
poids), they are applicable to all tetrapods.
MATERIAL AND METHODS
The preservation of specimens is illustrated in the
present study by examples of vertebrae of the following
horned dinosaurs, housed in the Borissiak Paleonto
logical Institute of the Russian Academy of Sciences,
Moscow (PIN): Bagaceratopidae indet., specimen
PIN, no. 614/29; Protoceratopidae indet., specimen
PIN, no. 614/35;
Protoceratops
sp., specimen PIN,
no. 3143/7;
Udanoceratops tschizhovi
Kurzanov, 1992,
specimen PIN, no. 3907/11;
?Udanoceratops
sp.,
specimen PIN, no. 4046/11. Collection numbers of
PIN correspond to the following localities: (614) Bayn
Dzak, (3143) Tugrikin Shire, (3907) Udyn Sayr,
(4046) Baga Tariach, which are dated Late Cretaceous
(Djadokhta Formation). Almost all the above proto
ceratopoids have wellpreserved vertebrae, although in
Protoceratops
sp. (specimen PIN, no. 3143/7), articu
lar facets of prelumbar vertebral centra are damaged
and cervical vertebrae are distorted.
The serial numbers of vertebrae in the vertebral col
umn and differences between two protoceratopoid
families were determined by the previously elaborated
key to the vertebrae (Tereshchenko, 2007); the paper
cited provides the terminology and system of vertebral
measurements (Fig. 1). Data processing is based on
the assumption that the dorsal and ventral lengths of
vertebral centra, cranial and caudal widths and heights
of vertebrae and their elements (centra and neurapo
physes) vary along the vertebral column independently
of each other. These variations can differ in members
of different sexes and taxa (Tereshchenko, 2001, 2004,
2007, 2008; Tereshchenko and Sukhanov, 2009; Ter
eshchenko and Singer, 2013). Therefore, attention is
paid not only to the vertebral measurements, but also
their mean values (parameters). The conditional mean
of angular and linear measurements of the right and
left sides of a vertebra is calculated to smooth the dif
ferences caused by preparation and gluing the frag
ments and also individual variation, if it is strongly
expressed. Individual variation in the vertebral column
is manifested in the asymmetry of shape, size, position
and development of paired vertebral structures
(Figs. 1e, 1g), abnormal fusion of elements, and relief
Methods for Size Reconstruction in Distorted
and Damaged Vertebrae of Fossil Tetrapods
V. S. Tereshchenko
Borissiak Paleontological Institute, Russian Academy of Sciences,
Profsoyuznaya ul. 123, Russia 123, Moscow, 117997 Russia
email: Tereshchenko@paleo.ru
Received November 12, 2013
Abstract
—The methods for reconstruction of the initial dimensions of distorted and damaged vertebrae in
fossil tetrapods are elaborated and tested, using the axial skeleton of horned dinosaurs from the Upper Cre
taceous of Mongolia as an example. These methods allow not only more reliable resolution of the problems
of vertebrae preservation for processing morphometric data, but also improvement of reconstructions of the
vertebral column, as skeletons are mounted for museum exposition.
Keywords
: Tetrapoda, vertebrae, methods, restoration
DOI: 10.1134/S0031030115030120

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PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
TERESHCHENKO
(a) (b) (c) (d)
(e) (f) (g) (h)
Ld
vpp
L
(
N
)
vpc
Lv
ldi L pd
lpa
lpr
ipdf
lpt Hn
L ps
Ln
vpc
H
(
C
)
caud
H
(
N
)
caud
H
caud
isp
B sp
H sp
HV
H
caud
H
(
N
)
caud
H
(
C
)
caud
B
(
C
)
caud
B
(
N
)
di
B prz
B npr
iprg
bn
S pa
B
(
C
)
cr
H
nc
S pa
hn
S di
B
(
N
)
di
S sp H di
idiv
lpa
A pr
B
(
N
)
pa
H
(
C
)
cr
H
(
N
)
cr
1
1
B di
ciz
idif
B prz
cpd
B fpr
L tr
itr
BV
H tr
H
cr
B ptz

PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
METHODS FOR SIZE RECONSTRUCTION 295
features of the articular surfaces of the centra, which,
however, do not influence characteristic structural fea
tures of vertebrae in particular taxa of different rank
(Kovalenko, 1983, 1986, 1992; Tereshchenko, 1990;
Filler, 2007; Holmes and Ryan, 2013). If measure
ments on the right and left sides only slightly differ, the
conditional mean can be neglected, taking either mea
surement for the parameter. The same is applicable,
for example, for the determination of the neurapophy
sis length (Figs. 1a,
L
(
N
), 2b, 2c) or the space between
the parapophysis and anterior edge of the vertebral
centrum (Figs. 1b, 1f,
lpa
). However, in some verte
brae of Bagaceratopidae indet. (specimen PIN,
no. 614/29), these parameters were established based
on the conditional mean. In particular, in the sixth
cervical vertebra, imperfect gluing the neurapophysis
roof (tectum neurale) on the arch pedicel resulted in
essential difference in length between the right and left
sides (Fig. 3c,
L
(
N
)
lf
,
L
(
N
)
rt
). In addition, in the first
thoracic vertebra of this specimen, as in the ninth cer
vical vertebra of
?
Udanoceratops
sp., specimen PIN,
no. 4046/11, the facet centers of the right and left
parapophyses are located at different levels (Fig. 1e);
this is probably connected with individual variation.
A measurement and parameter coincide if they are
determined by one measurement, as, for example, in
the case of the total vertebral width (measured at the
ends of the costal processes), which in the presacral
vertebrae corresponds to the width at the vertebral
diapophyses, (Figs. 1f, 1h,
BV
,
B
(
N
)
di
). It is possible
to determine the length of the vertebral centrum by
two methods, either based on the midline connecting
the centers of the anterior and posterior surfaces, or on
the parametric mean of the dorsal and ventral lengths
1
of the centrum (Fig. 1a,
Lv
,
Ld
). The parameters of
the centrum width and height coincide with the mean
of measurements of the cranial and caudal sides
(Figs. 1c–1f,
B
(
C
)
cr
,
B
(
C
)
caud
,
H
(
C
)
cr
,
H
(
C
)
caud
). The
neurapophysis width corresponds to the mean value of
the vertebral width measured at the ends of prezygapo
physes and postzygapophyses (Fig. 1g,
B prz
,
B ptz
).
The height of the neural arch pedicels is measured
posteriorly from the centroneurapophyseal suture to
the neural spine base, which usually coincides with the
dorsal surface of the postzygapophyses (Fig. 1b,
Hn
).
It is frequently important to determine not only this
parameter, but also the
conditional neurapophysis
height
, the halfsum of conditional and parametric
mean values of which includes the measurements from
the dorsal surface of the vertebral centrum (bottom of
the spinal canal) to the lateral edges of the prezygapo
physes and postzygapophyses on the cranial, caudal,
right, and left sides of a vertebra (Figs. 1c–1f,
H
(
N
)
cr
,
H
(
N
)
caud
). The
total vertebral height
is measured verti
cally on the posterior side from the lower edge of the
centrum to the neural spine apex (Fig. 1c,
HV
). Note
that the neural spine length (
L sp
) is considered here as
the distance from the base to apex irrespective of the
inclination extent; the height (
H sp
) is measured from
the apex along the perpendicular to the base level; the
width (
B sp
) is a craniocaudal (longitudinal) measure
ment; and the thickness (
S sp
) is transverse to the last
(Tereshchenko and Singer, 2013). The same measure
ments are applicable to other processes (costal, diapo
physeal, etc.: Figs. 1c, 1e–1h).
In the previous studies, the parameter Q was used
for construction of formulas of intervertebral mobility
in tetrapods (Kuznetsov and Tereshchenko, 2010); it
Fig. 1.
Schemes of vertebral measurements in protoceratopoids: (a, d, g) Protoceratopidae indet., specimen PIN, no. 614/35,
seventh cervical vertebra; (b, e, f)
Udanoceratops
sp., specimen PIN, no. 4046/11: (b, f) first thoracic vertebra, (e) ninth cervical
vertebra; (c)
Udanoceratops tschizhovi
, specimen PIN, no. 3907/11, sixth caudal vertebra; (h)
Protoceratops
sp., specimen PIN,
no. 3143/7, third caudal vertebra; (a) right, (b, c) left, (d) posterior, (e, f) anterior, and (h) dorsal views. Designations: (
A pr
) angle
of divergence of prezygapophyses, (
B di
) diapophyseal width, (
B
(
C
)
caud
) caudal width of vertebral centrum, (
B
(
C
)
cr
) cranial width
of vertebral centrum, (
B fpr
) width of prezygapophyseal facet, (bn) spinal canal width, (
B
(
N
)
di
) vertebral width at diapophyses
ends, (
B
(
N
)
pa
) vertebral width at parapophyses ends, (
B npr
) width of interprezygapophyseal space, (
B prz
) vertebral width at
prezygapophyses (cranial width of neurapophysis), (
B ptz
) vertebral width at postzygapophyses (caudal width of neurapophysis),
(
B sp
) neural spine width, (
BV
) total vertebral width, (ciz) interzygapophyseal crest (crista interzygapophysalis), (cpd) postzyga
diapophyseal crest (cr. postzygadiapophysalis), (
H
caud
) conditional vertebral height posteriorly, (
H
(
C
)
caud
) caudal height of ver
tebral centrum, (
H
cr
) conditional vertebral height anteriorly, (
H
(
C
)
cr
) cranial height of vertebral centrum, (
H di
) diapophyseal
height, (
Hn
) pedicel height of neural arches, (
hn
) spinal canal height, (
H nc
) height of connection of neural arch pedicels with
vertebral centrum, (
H
(
N
)
caud
) caudal conditional height of neurapophysis, (
H
(
N
)
cr
) cranial conditional height of neurapophysis,
(
H sp
) neural spine height, (
H tr
) costal process height, (
HV
) total vertebral height, (idif) angle between diapophysis and longitu
dinal vertebral axis in frontal plane, (
idiv
) angle between diapophysis and vertical vertebral axis in transverse plane, (
ipdf
) angle of
inclination of vertebrocostal articulation to longitudinal vertebral axis in sagittal plane, (
iprg
) angle of inclination of prezygapo
physeal facet to transverse vertebral axis in transverse plane, (
isp
) angle of between neural spine and longitudinal vertebral axis in
sagittal plane, (
itr
) angle between costal process and longitudinal vertebral axis in frontal plane, (
Ld
) dorsal length of vertebral
centrum, (
ldi
) distance between diapophysis and anterior edge of vertebral centrum, (
L(N)
) vertebral length at prezygapophyses
and postzygapophyses (neurapophyseal length), (
Ln
) pedicel length of neural arches, (
lpa
) distance between parapophysis and
anterior edge of vertebral centrum, (
L pd
) length of axis of vertebrocostal joint, (
lpr
) length of prezygapophysis projection beyond
level of anterior edge of vertebral centrum, (
lpt
) length of postzygapophysis projection beyond level of posterior edge of vertebral
centrum, (
L sp
) neural spine length, (
L tr
) costal process length, (
Lv
) ventral length of vertebral centrum, (
S pa
) parapophysis
thickness, (
S di
) diapophysis thickness, (
S sp
) neural spine thickness, (
vpc
) peripheral ridge of cranial articular surface of vertebral
centrum, (
vpp
) peripheral ridge of caudal articular surface of vertebral centrum; in (d, g) dashed line designates reconstructed sites
of neurapophysis, (
1–1
) horizontal in transverse or frontal plane of undistorted vertebra is parallel to its transverse axis; for other
designations, see the text. Scale bar, 20 mm.

296
PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
TERESHCHENKO
corresponds to the mean of four measurements taken
from the ventral edge of the centrum to the dorsolat
eral edges of prezygapophyses and postzygapophyses
on the cranial and caudal sides of a vertebra, which is
named in this work the
conditional vertebral height
(Figs. 1c, 1d, 1f,
H
cr
,
H
caud
). It should also be noted
that, in morphological studies based on individual ver
tebrae lays, the parameter coincides with the mean of
respective measurements of a vertebra in question. In
functional studies, the measurements at junctions
between the vertebrae (articular facets of centra, joints
of zygapophyses, etc.) are of particular interest.
Therefore, the parameter is equal to the mean of mea
surements on the anterior and posterior sides of adja
cent vertebrae.
The position of particular vertebral structures rela
tive to the sagittal, transversal, and frontal morpholog
ical projections (planes) is determined as follows. The
height of the vertebral centra and neurapophyses is
determined relative to the vertical axis, which belongs
to the sagittal and transverse (segmental after Akae
vskii, 1975) planes; the width is determined relative to
the transverse axis (after Dubrovsky and Fedorova,
2008), which belongs to the transverse and frontal
(horizontal after Sumida, 1990) plains; and the length
is determined relative to the longitudinal axis (after
Dubrovsky and Fedorova, 2008), which belongs to the
sagittal and frontal planes (Fig. 2a). The measure
ments of unpaired processes (spinous, haemal), con
nected with the length and height, are measured in the
sagittal plane, and paired processes (parapophyses,
diapophyses, costal processes) are measured in the
transverse and frontal projections (Figs. 1c, 1e, 1f, 1h).
The
thickness
of the first is determined relative to the
transverse axis; in the second, relative to the vertical
axis; the
width
of all processes is connected with the
longitudinal vertebral axis (Figs. 1c, 1e–1g). The
inclination of these processes in the sagittal and frontal
planes is determined relative to the longitudinal verte
bral axis and in the transverse plane, relative to the ver
tical or transverse plane (Figs. 1c, 1f–1h,
idif
,
idiv
,
isp
,
itr
). The inclination of articular facets of zygapophyses
is determined in the transverse projection relative to
the sagittal plane by both the angle to the vertical axis
(Sukhanov and Manzii, 1986) and the angle of conver
gence of the right and left facets of prezygapophyses
(Fig. 1f, angle
A
pr
; Sumida, 1990, textfig. 31A).
Since it is impossible to reconstruct the last parameter
in distorted vertebrae and inclination to the sagittal
plane is difficult to measure, in the study of serial vari
ation, the angle of inclination of prezygapophyseal
facets to the horizontal was measured (Fig. 1e,
iprg
).
In this case, the transverse axis coinciding with equal
levels of lateral edges of prezygapophyseal or postzyg
apophyseal facets of contralateral sides is taken for the
horizontal (Figs. 1g, 2c, 1e, 3a, 3c, straight line
1–1
).
Restoration of Measurements of the Vertebrae Differing
in the Extent of Preservation
Since fossil vertebrae are always more or less dam
aged and distorted, it is necessary to reconstruct the
initial parameters of their elements. The measure
ments of a lost vertebra or its elements were deter
mined based on the mean of measurements of the pre
ceding and succeeding vertebrae (Tereshchenko,
1990). In some cases, it is possible to reconstruct cer
tain measurements on one side of vertebra based on
the structures preserved on the opposite side (Figs. 1d,
1g
, dashed line). If the angles of inclination of paired
vertebral structures (facets of prezygapophyses, diapo
physes, etc.) are different in distorted vertebrae in the
transverse or frontal planes, it is possible to reconstruct
them by calculation of the conditional mean (Fig. 2d,
idiv rt
,
idiv lf
).
Vertebrae may be distorted in all direction, fre
quently resulting in asymmetry and changes in angular
and linear parameters. As the initial vertebral mea
surements are reconstructed, it is only possible to con
sider deformation in the transverse or frontal planes,
sometimes neglecting transformation in the sagittal
projection (Fig. 3). In the case of transverse and fron
tal distortion, one half of a vertebra is displaced rela
tive to the counterpart, resulting in a decrease in width
(Figs. 2d, 3b, 3d,
b
(
C
)
cr
,
b prz
,
b ptz
). In addition, in
the case of distortion in the transverse plane, it seems
that the vertebra increases in conditional height
(Figs. 2d, 3b, 4a,
H
max
,
H
cr max
,
H
(
C
)
cr max
,
Hcr rt
).
Despite changes in vertebral proportions, our experi
ence in morphometry suggests that, in the case of
frontal distortion, it is allowable to measure structures
directly in the vertebra. In this case, it is desirable to
measure separately the length of the vertebral centra
and neurapophyses on the right and left sides to take
the mean value if measurements differ. In so doing, the
maximum values on the cranial and caudal sides are
taken for the centrum and neurapophysis widths
(Figs. 3b, 3d,
B prz
and
B ptz
). Similarly, it is possible
to measure the conditional height of the neurapophy
sis and vertebra distorted in the transverse plane
(Figs. 2d, 3a, 3b, 4a,
H
cr
lf
,
H
cr
rt
,
H
(
N
)
cr
lf
,
H
(
N
)
cr
rt
).
However, actually, the cranial and caudal measure
ments of the neurapophysis width can be directly used
for determination of the parametrical mean, whereas
the parameters of the conditional vertebral and
neurapophyseal heights are only used after averaging
the cranial and caudal measurements on the right and
left sides. Note that searching for the maximum (ini
tial) measurements of the width and conditional
heights of the neurapophysis and vertebra requires
specific measurements and presence of reference
points, which are in this case the bottom center of the
spinal canal or the most remote sites of the ventral sur
face of the centrum and lateral edges of zygapophyses.
In the absence of these reference points, the width and
height of the vertebral centrum of a distorted vertebra

PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
METHODS FOR SIZE RECONSTRUCTION 297
(a)
(b) (c)
(d) (e)
Transverse plane
Sagittal plane
Frontal plane
Transverse axis
Longitudinal axis
Vertical axis
L
(
N
)
Lv
idiv rt
H
cr
rt
H
(
C
)
cr
max
b
(
C
)
cr
idiv lf H
cr
if
1
2
3
4
H
(
N
)
caud
H
(
C
)
caud
H
caud
1
11
1
1
2
4
L
(
N
)
5
Fig. 2.
Vertebrae of
Protoceratops
: (a) diagram of vertebra position (Protoceratopidae indet., specimen PIN, no. 614/35, fifth thoracic vertebra) relative to the sagittal, transverse,
and frontal planes; (b–e)
Protoceratops
sp., specimen PIN, no. 3143/7, fifth cervical vertebra: (a) right, anterior, and dorsal views, (b) left, (c) dorsal, (d) anterior, and (e) pos
terior views. Designations: (b(
C
)
cr
) cranial centrum width of distorted vertebra, (
H
(
C
)
cr
max
) greatest height of vertebral centrum, (
Hcr lf
) conditional cranial height of vertebra
in left view, (
Hcr rt
) conditional cranial height of vertebra in right view, (
idiv lf
) angle between left diapophysis and vertical vertebral axis in transverse plane, (
idiv rt
) angle between
right diapophysis and vertical vertebral axis in transverse plane, (
1–5
) horizontal in transverse plane of distorted vertebra, (
2–3
) midline of vertebra, (
2–4
) vertical vertebral axis;
for other designations, see Fig. 1. In (b) dashed line designates reconstructed site of vertebral centrum; for other explanations, see the text. Scale bar, 20 mm.

298
PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
TERESHCHENKO
would be estimated incorrectly, differing from the ini
tial values before deformation (Figs. 2d, 2e, 3a, 3b).
It is important that reconstruction of the condi
tional vertebral height in the case of transverse distor
tion with the help of a computer (option
distortion
) is
impossible, because the initial vertebral width and
angle of displacement of one side relative to the other
are unknown, which were set when a normal vertebra
was transformed (Fig. 3a, 3b). As a result of such a
“reconstruction,” it seems that the angles and condi
tional height on the displaced side decrease, but incli
nation of structures on the opposite side, along with
decreased width, are retained (Figs. 4a, 4b). However,
it is possible to apply a calculation method for recon
struction of the width and height of the vertebrae dis
torted in the transverse plane, which is exemplified
below by cervical vertebrae of
Protoceratops
sp., speci
men PIN, no. 3143/7 (Figs. 2b–2e, 4a).
Initially, it is necessary to determine the angle of
displacement of one vertebral side relative to the other.
In so doing, we should pay attention to the position of
the transverse vertebral axis (Figs. 1g, 2c, 2e, 3a,
straight line
1–1
) rather than the change in inclination
of its midline from the sagittal plane, since its initial
state is not known (Fig. 2e, angle
3–2–4
). Superpos
ing the midline of the distorted vertebra (Fig. 2e,
line
2–3
) on the sagittal plane (Figs. 2d, 2e, straight
line
2–4
), two lines are drawn from the point at the lat
(a) (b)
(c) (d)
1b ptz
H
(
N
)
H
(
C
)
cr
B
(
C
)
cr
H
max
H
(
N
)
rt
H
(
N
)
lf
B ptz
L
(
N
)
lf L
(
N
)
rt
34
1
B ptz
b
(
C
)
cr
L
(
N
)
rt
3
11
11
22
B ptz B ptz
b ptz
H
Fig. 3.
Changes in the proportions and morphometric parameters of the sixth cervical vertebra of Bagaceratopidae indet. (speci
men PIN, no. 614/29), artificial distortion (using computer) in (a, b) transverse and (c, d) frontal planes. Designations:
(
bprz
) cranial neurapophysis width of distorted vertebra, (
b ptz
) caudal neurapophysis width of distorted vertebra, (
H
) parameter
of conditional height of undistorted vertebra, (
H
max
) increase in conditional height of displaced vertebral side, (
H
(
N
)) parameter
of conditional neurapophysis height, (
H
(
N
)
lf
) conditional neurapophysis height in left view, (
H
(
N
)
rt
) conditional neurapophysis
height in right view, (
L
(
N
)
lf)
neurapophysis length in left view, (
L
(
N
)
rt
) neurapophysis length in right view; for other explana
tions, see the text and Figs. 1 and 2. Scale bar, 20 mm.

PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
METHODS FOR SIZE RECONSTRUCTION 299
eral edge (in this case, the left prezygapophysis); the
first is perpendicular to the sagittal plane (line
1–1
before distortion) and the second extends to the lateral
edge of the right prezygapophysis (line
1–1
after dis
tortion). It is possible to measure the angle of displace
ment obtained not only by a protractor in a photo
graph, but also by a goniometer directly in the vertebra
(Figs. 2d, 4c, angle
1–1–5
). This angle should not be
greater than
45
°
–50
°
, otherwise, compressive defor
mation of one side and stretching of the other would
have changed the vertebral shape to a hardly restorable
state.
Let us consider the opportunity to determine true
height and width of distorted vertebrae, using an
example of the centrum of the fourth cervical vertebra
(specimen PIN, no. 3143/7), with the help of the
scheme constructed on its sides (Fig. 4c). In the
scheme,
CD
corresponds to the side of the vertebral
centrum, relative to which the opposite side
EF
, along
with a part of its width (
AE
), is displaced. The rectan
gle
ABCD
is comparable to the position of the centrum
before distortion and the parallelogram
EFCD
shows the
state after distortion. The angle between the parallelo
gram side
FC
and its height
CB
is equal to the angle of dis
placement determined above (Fig. 4c, angles
1–1–5
,
BCF
, and
ADE
are equal to
α
). Thus, the parallelo
gram sides
CD
and
EF
correspond to the desired cra
nial height of the centrum (
H
(
C
)
cr
) and the sides
ED
and
FC
are its width (
B
(
C
)
cr
), which seems to be possi
ble to measure in distorted vertebrae. However, in
practice, this is difficult not only because of the
absence of a reference point for measuring the centrum
width in distorted vertebrae (Fig. 2e, midline
2–3
or
vertical axis
2–4
), but also because the point of measur
ing the maximum centrum height may fall beyond the
midline (Figs. 2d, 2e, 4c, lines
2–3
and
2–4
passing
near the measurement of the maximum centrum
height). Therefore, the centrum height and width of
distorted vertebrae should be measured traditionally,
but several times, selecting the greatest values
(Figs. 2d, 4c,
H
(
C
)
cr max
,
b
(
C
)
cr
). Then, we should turn
to the calculation method for determination of mea
surements, taking into account the fact that
Н
(
С
)
cr max
=
Н
(
С
)
cr
+
Х
,(1)
where
X
is increment of the centrum width (
AE
) (1).
The solution of any rectangular triangle (Fig. 4c,
AED
=
BFC
), the sharp angle and adjacent cathetus of
which are known (measurement of the maximum cen
trum width and angle of displacement), allows the cal
culation of the hypotenuse and opposite cathetus
lengths (increment and desired centrum width):
X
=
b
(
С
)
cr
tan
α
and
В
(
С
)
cr
=
b
(
С
)
cr
/cos
α
. (2)
Transforming formula (1) and substituting the
value of
X
, we get the required centrum height:
Н
(
С
)
cr
=
Н
(
С
)
cr max
–
b
(
С
)
cr
tan
α
. (3)
Since the above scheme of the vertebral centrum,
which explains the geometrical sense of the formulas,
is applicable to the vertebrae distorted in the transverse
and frontal planes (Figs. 4c, 4e), they allow the recon
struction not only of the initial cranial and caudal con
ditional heights of the neurapophysis or vertebra, but
also the width and length of its elements. Thus, for
mula (2) gets the following general view:
B
=
b
/cos
α
and formula (3),
H
=
Н
max
–
b
tan
α
(in the case of
transverse distortion) and
L
=
L
max
–
b
tan
α
(frontal
distortion), where b is the greatest cranial or caudal
width of the centrum of the distorted vertebra or
neurapophysis,
L
max
is the greatest length of the struc
ture in question. For example, the last measurement of
the neurapophysis is made from the caudal edge of the
left postzygapophysis to the cranial end of the right
prezygapophysis (Fig. 4e,
L
(
N
)
max
). It is more conve
nient to make this measurement after reorientation of
the distorted vertebra, so that its midline coincides
with the sagittal plane (Figs. 3c, 3d, 4e, lines
2–3
and
2–4
). Taking this into account, formula (3) for calcu
lation of the conditional cranial vertebral height and
neurapophysis length is represented as follows:
Н
cr
=
Н
cr max
–
b
(
С
)
cr
tan
α
, at
b
ptz
≤
b
(
С
)
cr
and
L
(
N
) =
L
(
N
)
max
–
b
(
N
)
caud
tan
α
, at
bprz
≤
b ptz
.
The use of this method for reconstruction of the
centrum height and width of cervical vertebrae 3–9 of
Protoceratops
sp., specimen PIN, no. 3143/7, dis
torted in the transverse plane, allowed reconstruction
of the initial shape of their cranial and caudal surfaces
(Figs. 4a, 4d; Table 1). Note that, in this specimen,
distortion decreases from the third cervical to first tho
racic vertebrae from
35
°
to
0
°
, with the change in the
angle of displacement
≈
5
°
in each succeeding vertebra.
Based on these angles and centrum width and height
of distorted vertebrae, formulas (2, 3) allow the esti
mates of reconstructed measurements (Table 1). As a
result, it turned out that the centra of these vertebrae
were low, wider than high (Figs. 4b, 4d), whereas
before reconstruction, they were regarded as moder
ately high (relatively round).
In a search for the patterns of changes in the struc
ture of vertebrae and their elements along the vertebral
column, we use measurements and parameters that are
either connected with the animal’s size (linear param
eters) or independent of it (angular parameters). In the
first case, the registration of the serial variation (tables,
diagrams, etc.) is performed for each particular speci
men. For angular parameters, it is allowable to use
additionally the mean values of the parameter in ques
tion in closely related taxa, which provide information
on the change in inclination of particular vertebral ele
ments along vertebral column throughout the group.
When generalizing the linear parameters, it is possible
to use the ratios of two measurements or parameters
calculated in each vertebra of all specimens included
in the study. Using the limits of variation of ratios, it is

300
PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
TERESHCHENKO
(a) (b)
(c)
(d)
(e)
1
H
(
C
)
cr
max
b
(
C
)
cr
H
(
N
)
cr
rt
H
(
N
)
cr
lf
4
2
b ptz b ptz
1
H
(
C
)
cr
“
H
cr
”
b
(
C
)
cr
b
(
C
)
cr
H
(
N
)
cr
rt
H
cr
max
H
(
C
)
cr
max
L
(
N
)
max
b ptz
A
B
C
D
E
α
X
F
A
BC
D
E
α
F
4
2
1
1
5
X

PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
METHODS FOR SIZE RECONSTRUCTION 301
possible to calculate the mean values, which provide
the data on changes in particular structures along the
vertebral column within the entire group. In addition,
the patterns of changes in these ratios can be used for
reconstruction of damaged and lost vertebrae in every
specimen irrespective of the animal’s size.
Paying attention to the articular facets on the cen
tra of prelumbar vertebrae of
Protoceratops
sp., speci
men PIN, no. 3143/7, it should be noted that, on their
anterior and posterior surfaces, peripheral ridges which
are normally projecting edges of their articular surfaces,
are disrupted (Figs. 1a, 1c,
vpc
,
vpp
, 2b, 2d, 2e). These
ridges are hardly discernible on the anterior and poste
rior surfaces of the centra in the area of the spinal canal
bottom, where they are usually subject to minimal dis
ruption. Therefore, as the ventral centrum length of
these vertebrae is reconstructed, it is possible to be
guided by the measurements of their dorsal length. For
example, when reconstructing the ventral centrum
length of the cervical vertebrae, true measurements of
which are undoubtedly underestimated in a certain
specimen (Fig. 2b), it is important to be convinced
that the measurements of their dorsal length are rather
accurate (Table 2). In our case, among reconstructed
measurements of the dorsal centrum length, true mea
surements of the sixth and eighth vertebrae, the dorso
caudal surface of which is somewhat damaged, are
questionable. Taking into account the patterns of the
change in the dorsal length of the vertebral centra and
ratios
Lv
/
Ld
along the vertebral column of protocer
atopoids, it is possible to reconstruct in specimen
PIN, no. 3143/7 both the dorsal centrum length of the
sixth and eighth cervical vertebrae and ventral length
of all vertebrae based on the reconstructed dorsal mea
surements (Table 2).
Table 1.
Changes in the width (B, b) and height (H) of the vertebral centra (mm) along the cervical region of the vertebral
column of
Protoceratops
sp., specimen PIN, no. 3143/7 before (distorted vertebrae) and after reconstruction (restored
vertebrae). Vertebral measurements reconstructed based on the mean of measurements of the preceding and succeeding
vertebrae are shown in parentheses (Tereshchenko, 1990); for designations, see the text and captions to Figs. 2d, 2e, 4a,
4c, and 4d
Vertebra no.
Distorted vertebrae
Angle of displacement
Reconstructed vertebrae
measurements parameters measurements parameters
b(C)
cr
b(C)
caud
H(C)
cr max
H(C)
caud max
b(C)
max
H(C)
max
B(C)
cr
B(C)
caud
H(C)
cr
H(C)
caud
B(C) H(C)
3 22.0 27.0 28.0 27.5 35
°
26.9 11.6 12.6 12.1
4 26.7 25.4 28.5 31.3 26.1 29.9 30
°
30.8 29.3 13.1 14.5 30.1 13.8
5 28.3 27.0 28.0 30.0 27.5 29.0 25
°
30.9 29.8 14.9 16.4 30.4 15.7
6 29.5 28.3 27.5 29.0 28.9 28.3 20
°
31.4 30.1 16.8 18.7 30.8 17.8
730.0 28.0 15
°
31.1 (28.3) 20.0 (20.1) 29.7 20.1
8 27.0 26.0 26.0 26.0 26.5 26.0 10
°
28.4 26.4 21.2 21.4 27.4 21.3
9 26.5 25.0 25.8 25.0 25.8 25.4 5
°
26.6 25.1 23.5 22.8 25.9 23.2
Fig. 4.
Reconstruction of measurements in distorted vertebrae in (a–d) transverse and (e) frontal planes: (a–d)
Protoceratops
sp.,
specimen PIN, no. 3143/7, fourth cervical vertebra, anterior view; (e) Bagaceratopidae indet., specimen PIN, no. 614/29, sixth
cervical vertebra, dorsal view; (a) parameters of distorted vertebra, (b) the same vertebra after computer transformation; (c, d, e)
reconstruction of vertebra by calculation method: (c, e) parameters used for calculation during reconstruction of vertebra, (d) ver 
tebra after reconstruction. Designations: (
α
) angle of displacement, (“
H
cr
”) imaginary decrease in conditional height of displaced
vertebral side, (
H
(
N
)
cr
lf
) conditional cranial height of neurapophysis in left view, (
H
cr max
) greatest cranial measurement of con
ditional vertebral height, (
H
(
N
)
cr
rt
) conditional cranial height of neurapophysis in right view, (
L
(
N
)
max
) greatest neurapophyseal
length, (
X
) increment of initial height of vertebral centrum (in the case of transverse distortion) and neurapophysis length (frontal
distortion); for other designations, see Figs. 1, 2, and 3. For explanation, see the text. Scale bar, 20 mm.

302
PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
TERESHCHENKO
CONCLUSIONS
Summarizing, it should be noted that the methods
proposed for restoration of the initial measurements of
vertebrae and their elements damaged and distorted in
the transverse and frontal planes provide improved
data on serial variation of the axial skeleton of extinct
tetrapods. In so doing, the calculation method allows
the estimation of distortion in vertebrae changing
along the vertebral column. In addition, these meth
ods improve the reconstruction of lost vertebrae in the
vertebral column of skeletons mounted for exposition.
ACKNOWLEDGMENTS
I am grateful to V.B. Sukhanov for taking photo
graphs, perusal of the manuscript, and valuable
remarks.
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Table 2.
Changes in the vertebral centrum length (L) in mm and ratio Lv/Ld in the series of cervical vertebrae of
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atops
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range mean
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5 23.0 20.5 22.3 1.07 1.20–1.25 1.23 25.2 20.5 22.9
6 22.1 18.8 20.5 1.17 1.25–1.30 1.28 25.0 (19.5) 22.3
7 18.8 20.2 1.17 1.30–1.40 1.35 (25.4) 18.8 22.1
8 25.5 17.9 21.7 1.42 1.20–1.30 1.25 25.9 (20.7) 23.3
9 23.7 22.7 23.2 1.04 1.10–1.20 1.15 26.1 22.7 24.4

PALEONTOLOGICAL JOURNAL Vol. 49 No. 3 2015
METHODS FOR SIZE RECONSTRUCTION 303
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Translated by G. Rautian
SPELL: 1. ok