A representation for the space of possible poses of a vertebrate neck is introduced and applied.  For a neck with K cervical vertebrae, and intervertebral joints with 2 degrees of freedom (mediolateral and dorsoventral flexion), the full configuration space has a computationally-unwieldy 2*K dimensions. But regarding the neck as permitting the head to reach to a specific location from a specific direction of approach, the configuration space reduces to a 6-space (3 positional, 3 directional). As the neck extends to reach increasingly far from the body, the range of approach directions (at the atlas) diminishes. In the limit, the “reachability envelope” (RE) is a surface defined by the points of greatest radial distance reachable without stepping. Vertebrate necks vary considerably in RE area, and the directional flexibility within the volume bounded by the RE (the swan, e.g., can preen points on its own neck from various directions). A computational model explores how neck flexibility (both RE and directional flexibility) is affected by factors including vertebral count and the distribution of centrum length and intervertebral flexibility along the neck. The flexibility observed in representative vertebrates is replicated, and implications for sauropod dinosaurs are derived.

The reconstruction of sauropod neck posture is critical to the interpretation of their feeding behavior. While the undeflected, osteologically neutral pose (ONP), has been proposed as the characteristic pose (CP) for the sauropod neck, the proposal that sauropods habitually held their necks in ONP has been questioned, since some extant vertebrates habitually hold the head higher than ONP. Rather than consider but one generic CP, we distinguish characteristic poses for locomotion (CLP), feeding (CFP), and vigilance (CVP), and examine these poses for a broad range of extant taxa (22 birds in 20 orders; 10 mammals; two non-avian archosaurs; and three other reptiles). CFP is generally below ONP, CLP is close to ONP, and CVP is close to, or above ONP. Exceptions where ONP is significantly lower than CLP and CVP (as noted in the ostrich) are consistent with other environmental requirements, not the least of which is the need for vigilance when not browsing at ground level. As the only pose that can also be configured with certainty in fossils, we conclude that it has high value for hypothesizing about sauropods, their having have no exact biomechanical analogue among extant tetrapods.

2009 SVP (Bristol)
Non-Parasagittal Yet Efficient: The Role of the Pectoral Girdles and Trunk in the Walk of Triceratops and Apatosaurus
Stevens, Kent A. & E.D. Wills

The robust forelimbs of Triceratops and Apatosaurus represent postural extremes of semi-sprawling versus nearly columnar, respectively. Reconstructions of their gaits remain controversial regarding forelimb, pectoral girdle, and trunk motions. Limbs are redundant kinematic systems with more degrees of freedom (DOF) than required for locomotion, resulting in an infinite space of possible walks, and requiring further constraint (e.g., from ground reaction forces or muscle moments). We introduce a purely kinematic constraint deriving from the phase in a walk cycle when both left and right hindlimbs (or forelimbs) are bearing weight (dual support). During dual support, the left and right kinematic chains (from ground to body) are constrained to propel the body on the same path. The kinematic analysis begins with a digital skeletal model, and for each joint involved in locomotion, the axis and range of motion are estimated for each DOF (e.g. flexion/extension). Next, a configuration space is created by computing the set of all combinations of deflections sampled at uniform intervals within each DOF. The dual support constraint then selects those poses consistent with dual support. For quadrupeds, further constraint is provided by coupling the fore and hind limbs through limited trunk flexion. Using a genetic search algorithm, walk cycles are then sought that maximize smoothness of body trajectory. An efficient gait is thus found without prior assumptions regarding osteological neutral pose, or the angular extremes of joint DOFs, or whether or not the limb’s movement is parasagittal. The method successfully replicates the sprawling gait of a reptile and the parasagittal gait of a cursorial mammal. In Triceratops, to achieve efficient locomotion, the semi-sprawling forelimb motion required compensation, e.g., by a combination of pronation/supination, pectoral girdle mobility and back flexion. For Apatosaurus, the forelimb movements, while not strictly parasagittal, were readily compensated for by moderate girdle and back mobility. This method allows exploration of the effect of pronation/supination, pectoral girdle, and trunk mobility in combination and in isolation.

2007 SVPCA (Glasgow)
Kinematic Constraints on the Reconstruction of Dinosaur Gaits
Stevens, K.A. & E.D. Wills

Reconstructions of dinosaur locomotion may draw on multiple sources of evidence, including estimates of limb posture, joint ranges of motion, muscle lines of action and moments, inertial properties of body parts, and estimates of metabolic cost. Presuming that the skeletal and muscular systems work in concert to achieve kinematically and dynamically efficient locomotion, to what extent might the kinematics of the appendicular skeleton in isolation reveal an efficient gait? Towards that end, dimensionally and morphologically accurate skeletal models were created of dinosaurs (from a variety of sources including digitization) and joint range of motion estimated and represented. Genetic Algorithms (GA) that ‘reward’ particular observables (e.g., forward travel) and penalize others (e.g., metabolic cost) have been used successfully to synthesize gaits [Sellers, et al., 2004]. These applications of GA rely on p hysical simulation (to model muscle moments and inertial and gravitational forces). In distinction to these approaches, we u se GA to evolve purely kinematic solutions, which are then compared with gaits observed with extant animals and as derived by other methods.
Sellers, WI, L.A. Dennis, W.J. Wang, and R.H. Crompton. Evaluating alternative gait strategies using evolutionary robotics. Journal of Anatomy 204, 343–351, 2004.

2007 SVPCA (Glasgow)
Isolating Functional Degrees of Freedom in Limbs During Locomotion
Wills, E.D. & K.A. Stevens

The pattern of limb movements characteristic of a given vertebrate during locomotion is, at least in part, reflected in its osteology. The limbs, working in concert, produce an efficient (hence smooth) forward movement of the body. Candidate movements for a given imb can be formalized as trajectories within a configuration space defined by all possible combinations of joint deflection angles, to within some given angular resolution. This space can then be explored using Genetic Algorithms (GA) techniques [Holland, 1992], here a candidate movement is evaluated on how well it transports the body while minimizing angular and translational deviations during the forward path. This application focuses on the pectoral girdle and forelimbs of the sauropod Apatosaurus, to determine ow the limb might have functioned during locomotion. The wrist, elbow, shoulder, and pectoral girdle are provided appropriate functional degrees of freedom (FDOFs) per joint (e.g., flexion/extension at the elbow; flexion/extension plus abduction/adduction and medial- and lateral-rotation at the shoulder). By selectively adding and removing specific FDOFs it possible to isolate the elative contribution of each towards the locomotion task. This is particularly important in examining the potential mobility of the pectoral girdles, for which no strong constraint on their position and range of motion is provided by their osteology.
Holland, John H. 1992. Adaptation in Natural and Artificial Systems, Cambridge, MA: MIT Press, 228 pp.

Download PDF of poster here.

Allometry, the differential growth of body parts in relation to an entire organism, is usually analyzed and presented graphically, wherein a measure of relative growth is expressed by a power law, for either an intraspecific (ontogenetic) or interspecific (phylogenetic) range of individuals. In contrast, dynamic visualization of a three-dimensional skeletal model that undergoes allometric change provides a more immediate and inclusive grasp of that same progression. To visualize allometry, two or more skeletal models are created paramerically in the DinoMorph software. The modeled individuals that comprise the sequence are homeomorphic, e.g., they have identical phalangeal and vertebral formulae. The models differ geometrically, not only in the major dimensions of corresponding bones, but for each bone, the geometric model capturing its specific morphology is also homeomorphic across the individuals in the sequence. This permits visualization of orphological change for individual bones in addition to appreciating their proportional changes during the allometric progression of the overall skeleton. While conventional allometric analyses frequently use femur length as a proxy for overall body size, the current method permits allometry with respect to reconstructed body size directly, and indeed to test the often-presumed sometry of femur length as a function of overall body size. For animals in which the hindlimbs are themselves involved in significant functional change, such as tyrannosaurid allometry, the ability to select other, functionally independent, bases is particularly advantageous. DinoMorph further provides a representation of joint flexibility that quantifies the full six degrees of freedom of elative orientation and position of each articulated pair of bones. As a consequence, changes in posture and flexibility can also be interpolated and appreciated with relation to a given stage of allometric growth. Body movements, expressed relative to the individual achievable range of motion, can then be correlated with allometric change to the entire skeleton.

2005 "100 Years of Tyrannosaurus rex" Symposium
"Rex, sit. Modeling Tyrannosaurid Postures"
Stevens, K.A., E.D. Wills, P.L. Larson, and A. Andersen

A three-dimensional digital model of an adult tyrannosaurid is created to visualize and explore a variety of hypothesized standing and resting postures. The skeleton of Tyrannosaurus rex specimen “Stan” (BHI-3033) at the Black Hills Institute of Geologic Research is modeled in DinoMorph™, with major elements of the cranium, appendicular skeleton based on digitized scan data, and axial elements represented with dimensionally accurate but more simplified morphology including a ribcage based on the specimen. The distribution of body mass is estimated by associating approximations to the volume associated with each element, ith estimates of net density that take into account pneumatization and respiratory structures. Estimates of the range of motion of major elements of the hindlimb are incorporated into the model, such that when the skeleton is animated, the joint movements are onstrained to within the overall range of achievable postures. During the model’s movements the distribution of mass is tracked and the instantaneous center of mass visualized to show deviations from a condition of static equilibrium. A range of transitions from standing posture to various repose and sitting postures are animated and analyzed in terms of balance and stability.

2005 German Sauropod Group (Aathal)
The Neck Position of Sauropod Dinosaurs: Virtual Reconstruction within DinoMorph
Stevens, K.A. & J.M. Parrish

To visualize ontogenetic changes in the tyrannosaurid skeleton, articulated three-dimensional digital skeletal models were created in DinoMorph™. Black Hills Institute of Geologic Research specimen ‘Stan’ (BHIGR 3033) was modeled as a representative adult. The head, limbs, and girdles of ‘Stan’ were digitized; the remaining (primarily axial) elements were represented in dimensionally-accurate but simplified form. This adult tyrannosaurid model was then used as the basis for a parametric adaptation to two other tyrannosaurids: the Burpee Museum tyrannosaurid specimen ‘Jane’ and the Field Museum of Natural History specimen ‘Sue’ (FMNH PR2081). The parametric representation of dimensional information throughout the entire skeleton allows for the visualization of continuous ‘morphing’ between two given models. For each of the hundreds of parameters in each model, a weighted-sum interpolate was computed, as controlled by a factor that varied between 0 and 1. The interpolated model permitted dynamic visualization of the differences etween the two models, e.g. between ‘Stan’ and ‘Sue’, presumably representing male and female adult morphs, and between ‘Jane’ and the adult forms. By normalizing the two models with respect to overall body length, the interpolation between ‘Jane’ and the adult form is marked by a dramatic increase in the robustness of the thorax, head and presacral vertebral column, accompanied by relative shortening of the neck, and a remarkable reduction in the relative length and proportions of the hindlimbs.

The feeding range of sauropod dinosaurs was constrained by the height of the base of the neck, which was itself constrained by the nature of the articulation between the scapulocoracoid and the trunk. Sauropod scapulocoracoid orientation and mobility remain controversial because no tight, bony articulations were present between these bones and the vertebral column, and few osteological markers are available to constrain shoulder orientation. Previous hypotheses of scapulocoracoid position in sauropods were inferred from death poses or through simple goodness-of-fit criteria, but were not developed within a phylogenetic context. We examined the scapulocoracoids, ribcage, and sterna of several neosauropods, basal sauropodomorphs, extant archosaurs, and lepidosaurs in order to: 1) assess the soft tissue contributions to scapulocoracoid orientation and mobility; and 2) determine osteological correlates associated with shoulder position. Archosaurian scapulae vary in position from nearly parallel to the vertebral column (birds) to nearly vertical (crocodilians) but in all cases the scapular blade is nearly parallel to the top of the neural spines along its distal extent, suggesting a similar orientation in dinosaurs. Positions and homologies of the musculature supporting the scapulocoracoid were conservative within taxa. Scapulocoracoid movements against the sternum, affected by the Mm. serratus, levator scapulae, and sternocoracoideus groups, become more restricted in archosaurs, particularly in birds. Flattened areas on the external surfaces of the dorsal ribs ("facets") are present in birds, sauropods, and other dinosaurs but are absent in crocodilians. Dissection and CT-scan data show that the scapular blade bows away from the dorsal ribs in Alligator whereas it lies in close contact with the dorsal ribs in birds, which may explain the lack of "facets" in crocodilians. Rib facets correlate with the neutral orientation of the scapular blade in birds. We suggest the presence of rib "facets" and the more restricted movements of the scapulocoracoid in diapsid outgroups support a constrained, sub-vertical orientation of the pectoral girdle in sauropods.

2002 SVP (Norman)
Mass-based Biomechanical Computations on Sauropod Dinosaurs
Stevens, K.A. & J.M. Parrish

Sauropod mass estimates are generally provided by measuring the volume of a model or by inference based on a two-dimensional life reconstruction. Such techniques do not lend themselves to systematic study in which alternative reconstructions vary ribcage volume, abdomen volume or the distribution of muscle mass along the axial and appendicular skeletons. It has therefore been difficult to substantiate or reject some proposals regarding the muscular abilities of sauropods, such as the popular suggestion that some sauropods were capable of elevating themselves into a bipedal or tripodal (hindlimbs plus tail) posture. The DinoMorph^tm software provides a parametric framework with which to study the distribution of mass throughout a skeleton, and through the computational modeling of muscles, an ability to test the feasibility of proposals such as sauropod bipedalism. Dimensionally accurate three-dimensional digital reconstructions have been made of the axial skeleton, ribcage, and appendicular skeleton of <i>Camarasaurus</i> plus the two diplodocids <i>Apatosaurus</i> and <i>Diplodocus</i>. For a given reconstruction of the body plan, volumetric primitives are then associated with individual bones, then parametrically adjusted to reflect a range of interpretations regarding the distribution of muscle mass and viscera. The center of mass is then computed for these sauropods and compared with estimates using other techniques. Major muscle groups are introduced as relevant to specific tasks, such as those that might elevate the sauropod by pivoting the axial skeleton about the acetabulum, with the presacral mass partly counterbalanced by that of the tail. The muscular tension to achieve and maintain the elevated posture can then be computed and compared for alternative taxa and for alternative estimates of their mass, assuming that dynamics does not play the dominant role in achieving the bipedal posture. Alternative models for the hindlimb posture and the contribution of postural adjustments along the axial skeleton and forelimbs are incorporated in this study.

2002 SVP (Norman)
Rib Angulation, Scapular Position, and Body Profiles in Sauropod Dinosaurs
Parrish, J.M. & K.A. Stevens

Ribs are one of the most ignored parts of the skeletons of fossil vertebrates.  In dinosaurs generally and sauropods in particular, the neutral position of the dorsal ribs has a significant impact on calculations of body mass and of the positioning of the shoulder girdle.  Rib articulation is reconstructed by analysis of the joint articular surfaces in sauropods, and of the normal neutral orientation and range of mobility of the ribs in extant archosaurs.  A rib is raked when the distal end of the rib is posterior to the proximal head as viewed laterally relative to the long axis of the centrum.  In dorsal view the rib may also be swept, such that the plane containing the rib's curvature is not perpendicular to the long axis of the centrum. The relative rake and sweep of the ribs of Apatosaurus and Camarasaurus are compared, and used as the basis for their ribcage reconstructions. A digital model of the sauropod ribcage is animated to examine the role of rake and sweep in the volumetric changes associated with lung ventilation.  A flattened area on the lateral surface of dorsal ribs 2-5 in Apatosaurus indicates the likely position of attachment of the scapula to the trunk. If these contours indeed reflect the position of the scapula to the trunk, this would favor a subhorizontal orientation of the shoulder girdle, with the glenoid opening almost directly ventrally. The more nearly continuous curvature of the ribs in Camarasaurus precludes the use of similar landmarks to position the scapulae, but similarities in shape and medial curvature of the scapula suggests a shoulder girdle orientation similar to that in Apatosaurus.  A reconstruction of the curvature of the dorsal column along with the dimensions of the ribcage and the position of the scapulocoracoid permits a geometrically constrained skeleton which can then form the basis of an estimation of body profile and derivative volume and mass estimates.

The DinoMorph parametric modeling software was used to create digital models of three diplodocid sauropods (Apatosaurus, Diplodocus, Dicraeosaurus), plus Brachiosaurus and Camarasaurus. For each sauropod the major elements of the appendicular and axial skeletons were modeled with dimensional accuracy. The intrinsic curvature of the cervical vertebral column was reconstructed with three-dimensional analysis of the centrum morphology and zygapophyseal rticular facets for the cervical vertebrae, centrum morphology only for the dorsal vertebrae, and published reconstructions for the caudal vertebrae. As a first step towards creating a whole body reconstruction, the articulated fore- and hindlimbs establish the height of the glenoid and acetabulum with some certainty particularly for graviportal animals. The acetabular axis constitutes a pivot about which the axial skeleton rotates according to the placement and orientation of the pectoral irdles relative to the dorsal ribs. While reconstruction of the limbs and axial skeleton is well constrained by the osteology of their component elements, the pectoral girdles are problematic for they are suspended by soft tissue. Analysis of modern nalogs for the pectoral girdle, plus specific geometric constraints provided by the digital models of the ribcage, permits xploration of a range of reconstructions. The fundamental implications for feeding and lifestyle resulting from the whole ody posture are discussed.

2001 SVP (Bozeman)
Gracile versus Robust Cervical Vertebral Designs in Sauropods
Stevens, K.A. & E.D. Wills

Sauropod necks with broad, robust vertebrae (e.g. Camarasaurus and Apatosaurus) differ in biomechanical design compared to the more gracile forms (e.g. Diplodocus, Barosaurus, and Euhelopus). Caudally, where the design differences are generally most pronounced, the robust form has wide-set, flat, nearly horizontal zygapophyses, while the gracile form has narrow-set zygapophyses that curve through nearly 90 degrees (horizontal laterally yet near vertical medially). Quantitatively, diapophysis width can be compared to posterior centrum width wc and to centrum length lc. In Apatosaurus, maximum wd/wc is about 3.0 at the tenth cervical vertebra, C10, while in the more gracile Diplodocus and Barosaurus vertebrae that ratio is substantially less, with a maximum of 1.9. Likewise, wd/lc in Apatosaurus ranges from just under 1.0 at C5 to 2.2 at C15 compared to a range of 0.4 (at C5) to 0.7 caudally. If the pronounced width in the robust form provides stabilization against axial torsion load during mediolateral deflection (the Chinese toy snake problem), how was stability achieved in the longer yet narrower 6 m neck of Diplodocus or the extraordinary 9 m neck of Barosaurus? Biomechanical modeling and articulation of the cervical vertebrae of gracile design using the DinoMorph™ software reveals how torsional stability might have been achieved at the extremes of mediolateral deflection without the need for massive buttressing at the base of the neck.

2001 IVCM
Neck Mobility in Long-Necked Vertebrates: From Modern Mammals to Sauropods
Parrish, J.M. & K.A. Stevens

Modern long necked mammals, and especially giraffes, have often been considered appropriate contemporary analogs for sauropod dinosaurs. As part of our studies on sauropod neck mobility, we have done dissections and osteological manipulations of both llamas and giraffids. Several striking differences are apparent between sauropods and the long necked mammals. First, both mammalian groups have the plesiomorphic mammalian cervical count of seven rather than the expanded sauropod count of 11-19. Nonetheless, the mammals are capable of far greater flexibility than found in avian and reptilian necks, and presumably greater flexibility than achieved by auropods. The vertebrae may deflect further when attempting to combine ventriflexion with mediolateral flexion. The large angular deflection in the mammals is achieved as a result of the zygapophyses being large and placed close to the midline. Although sauropods have more vertebral segments than mammals, the fact that their zygapophyses are located further from the midline and are generally obliquely inclined results in less mobility between segments and throughout the neck. Vertical lexion and mediolateral deflection are often functionally decoupled, particularly at the proximal end of the neck.

The performance available in 'modest' personal computers is now sufficient to run software which until recently required expensive and highly specialized hardware. The graphics-intensive DinoMorph system, which has been used to reconstruct the pose and flexibility of sauropod dinosaur necks, will soon be able to be run on desktop and laptop computers. DinoMorph is being proposed as a means to create, modify, articulate, animate, and exchange digital models of quadrupedal skeletons. It will permit digital publishing (e.g. on the internet) of morphology and joint angle data, allowing exchange of, and independent verification of, proposed reconstructions of morphology and pose. Interactive animation from arbitrary viewpoint offers further advantages over conventional 2D silhouette renditions. In contrast to 3D animation software that requires sophisticated computer skills, DinoMorph's graphical user interface is intuitive and little computer expertise is needed to edit bone dimesions, joint articular surface shape and to adjust joint angles. Overall pose can then be reconstructed in 3D and extremes of movement can be explored and subsequently animated. The name DinoMorph derives from its ability to 'morph' or interpolate between two models, allowing one to visualize the differences in morphology between two presumably related taxa, and to view intermediate forms. Future releases will permit importing of 3D scanned data and exporting of articulation data to other digital animation software.

Specimens like the juvenile skeleton of Camarasaurus from Dinosaur National Monument (Carnegie Museum 11388) and the numerous specimens of Coelophysis from Ghost Ranch exhibit a post-mortem postural deformation familiar in skeletons of once-living tetrapods wherein the cervical series has been flexed dorsally such that the vertebrae are no longer in articulation. In his classic work on terrestrial taphonomy, Weigelt attributed this postural deflection to dewatering and contraction of the nuchal ligament, and this also seems the most likely explanation for these dinosaur skeletons. We first manipulated images of the articulated skeleton of CM 11388 manually and in Photoshop to determine the extent to which the cervical series had been disarticulated. Next, the metric parameters of the neck were fed into DinoMorph to restore the juvenile skeleton to the neutral pose. The result was a sauropod with greater flexibility than in the diplodocids, but one clearly incapable of the drastic dorsiflexion of the head observed in the death pose.

Two diplodocids, Apatosaurus louisae and Diplodocus carnegii, have been modelled with three-dimensional graphics software, with particular attention directed to rendering the osteology of their necks as accurately as possible. Using techniques described at the last annual meeting of the SVP, these articulated models permit exploration of the maximum ranges of movement llowed by osteological constraints. Of critical importance is the total flexibility of a tetrapod neck is the geometry of the paired zygapophyseal joints and their placement to the center of rotation of the intervertebral joint. Relatively subtle differences in the size and location of the articular facets of the zygapophyses had substantial effect on neck mobility. By working from several sources of material including measurements taken directly off mounted specimens, we can model the neutral pose (one which centeres the prezygapophyses above the postzygapophyses throughout the neck) and, extremes of mediolateral and dorsoventral flexion, and the range and extent of compound movements such as torsion. This work revises the traditional views on neck posture in these diplodocids, refines estimates of the geometry of their feeding envelopes, suggests peculiarities of bending in order to achieve extremes of flexure, and demonstrates the potential (on the basis of osteology) for extraordinary ventriflexion, which is consistent with the hypothesis of tripodal feeding.

Stereoscopic vision requires that the two eyes see a common region of space, i.e. that the two monocular fields of view overlap. The shape and extent of the binocular field of view (BFoV) is governed primarily by the placement of the eyes within the cranium and on the facial structure surrounding the eyes, especially rostrally. While many predators have forward facing eyes and prey have laterally placed eyes, the relationship between BFoV and behavior is more complex. A review of binocularity across diverse extant species forms the basis for examining binocularity in theropod dinosaurs. Life restorations (with taxidermic eyes) of Allosaurus, Carcharodontosaurus, Daspletosaurus, Nanotyrannus, Stenonychosaurus, Tyrannosaurus, and Velociraptor weres culpted by Garfield Minott for this study. A novel 'inverse perimetry' technique was developed to map their BFoV in polar coordinates, producing plots similar to those used in ophthalmology. Structures such as lacrymal crests were found to strongly restrict binocularity; the interpupillary separation and optic axis divergence much less so. The shape of the BFoV across these taxa varied considerably, with some predators afforded only a narrow region, others provided a fair panorama for stereoscopic vision comparable to that of some modern raptors and greater than that of any extant reptile. Two modes of visually guided predation are suggested by the BFoV. Furthermore, the distribution of BFoV width vertically has implications for head pose. Finally, two models of spatial resolution, an avian and a crocodilian, were used to provide quantitative estimates of the depth perceptionin the great theropods.

Read more here.

The structurally diverse cervical vertebral patterns in the great sauropods appear to represent distinct engineering solutions to a series of different biological challenges. A fundamental, and largely unstudied, aspect of sauropod paleobiology is the variation in ranges of motion that these different patterns afford. Directly studying neck mobility joint-by-joint using original material is extremely difficult, given its weight, bulk, and delicacy. Even manipulations using lightweight casts are problematical because of post-mortem distortions of the bones, and exploring complex poses over several joints is virtually impossible using full-sized specimens. Controlled, comparative studies of the ranges of motion afforded by the cervical seriews of different dinosaurs can be performed on the computer, provided the shapes, dimensions, and relative positions of the articular surfaces are modeled accurately. DinoMorph, a computer graphics system for quantitative study of joint articulation, has been extended from manipulating two-dimensional silhouettes (useful for earlier study of dorsoventral flexion) to animating full three-dimensional models, which can be articulated and rotated through all angular degrees of freedom. The software allows precise sculpting of the compound curvatures of the articular surfaces. The idealized vertebrae can then be articulated to measure the resultant extremes of movement, subject to particular assumptions about the size and flexiblilty of the joint capsules. Experimental articulations of models based on Apatosaurus, Diplodocus, Dicraeosaurus, and Euhelopus demonstrate distinctive and characteristic patterns of allowable curvature thar are direct consequences of their osteology.