2D and 3D visualizations of archosaur jaw muscle mechanics, ontogeny and phylogeny using ternary diagrams and 3D modeling

Recent advances in imaging and computational methods are enabling researchers to capture three-dimensional (3D) morphology at high resolutions. Researchers are ushering in a renaissance of imaging approaches in areas such as astronomy (Cohen et al., 2003; Preusker et al., 2015; Korsun et al., 2016), biochemistry (Lüthy et al., 1992; Zemla, 2003; Arnold et al., 2006), and archaeology and anthropology (Hughes et al., 2005; Lee et al., 2007; Carlson et al., 2011; Du Plessis et al., 2015). Similarly, biological sciences are greatly enhanced by morphological studies incorporating large amounts of high-resolution 3D data. Morphologists frequently model diverse morphological systems in entomology (Klaus et al., 2003; Friedrich and Beutel, 2008), physiology (Witmer et al., 1999; Schachner et al., 2013; Tsai and Holliday, 2015; Stephenson et al., 2017), neuroanatomy (Evans et al., 2009; Lautenschlager et al., 2012; Kawabe et al., 2013) and skeletal tissue biomechanics (Grosse et al., 2007; Cuff et al., 2015) using high-resolution data. Modeling musculoskeletal systems in 3D is now enabling researchers to investigate the underlying biomechanics of behaviors, such as feeding (Gans et al., 1985; Zusi, 1987; Witmer and Rose, 1991; Hoese and Westneat, 1996; Herrel et al., 1999; Dumont et al., 2005; Dawson et al., 2011; Snively et al., 2013; Sellers et al., 2017; Bates and Falkingham, 2018; Cost et al., 2020) and locomotion (Hutchinson, 2004; Charles et al., 2016; Manafzadeh and Padian, 2018), using computational and imaging methods such as finite element analysis (e.g. Keyak et al., 1993; Rayfield, 2007; Santana et al., 2010), XROMM (e.g. Brainerd et al., 2010; Baier et al., 2013) and multibody dynamics (e.g. Moazen et al., 2009; Curtis et al., 2010; Snively et al., 2015).

Shared among many biomechanical studies is the measurement and visualization of the magnitudes (e.g. Newtons) and orientations of forces (e.g. x, y, z), often across time (t). Biologists often report the most biomechanically and ecologically important portions of forces, such as the vertical (orthal) component of bite force or vertical component of ground-reaction force. Additionally, we also often disregard component forces in the transverse or axial planes, in part as a product of the historical focus on mammalian feeding and locomotor systems where muscles largely act in parasagittal planes (e.g. Maynard Smith and Savage, 1959; Dullemeijer, 1956; Cartmill, 1974, 1985) relative to the systems in other vertebrates. But also, pragmatically, it is simply challenging to visualize all force components simultaneously or in a fashion understandable by readers in traditional publishing. Thus, the complicated 3D nature of musculoskeletal anatomy remains difficult to convey in two-dimensional (2D) publications and other media (e.g. Greaves, 1982; Sinclair and Alexander, 1987; Bimber et al., 2002; Lockwood et al., 2002; Holliday and Witmer, 2007; Vincent et al., 2007; Huber et al., 2008; Sustaita, 2008; Holliday, 2009; Kolmann and Huber, 2009; Pfaller et al., 2011; Figueirido et al., 2013; Holliday et al., 2013; Cost et al., 2020). The solution has been to decompose multidimensional measurements into more simplified plots that are designed specifically for 2D publishing including iterative columns, box plots and line graphs. This, in turn, causes higher-order questions of behavior, ontogeny and phylogeny to be even more challenging to analyze and disseminate. One solution is to project 3D force components in a ternary diagram. Ternary diagrams have a long history of use in geology, chemistry and physics, wherein the relative contributions of three variables to a whole are studied. For example, phases of matter (Othmer and Tobias, 1942), soil composition (Norton, 1966), vertebrate limb proportions (Gatesy and Middleton, 1997, 2008; Middleton and Gatesy, 2000) and ventilation kinematics (Capano et al., 2019) have made use of ternary diagrams in the past.

Here, we use ternary diagrams to provide examples of how jaw muscle resultants (a summation vector composed of the contributing forces applied to a muscle across multiple planes and directions) of archosaurs (crocodylians, dinosaurs, and birds) change over time using three case studies of behavior, ontogeny and phylogeny. First, within a single individual, the 3D nature of the cranial musculature requires that muscle forces and resultants must vary with gape (Herring and Herring, 1974; Dumont and Herrel, 2003) during a feeding bout. We demonstrate how ternary plots enable researchers to visualize the changes in orientations through one open-to-close phase of a feeding cycle in Alligator mississippiensis. Second, archosaurs have a wide diversity of cranial morphologies and feeding ecologies that change during ontogeny (see Fig. 1; Erickson et al., 2003; Yanega and Rubega, 2004; Grigg and Kirshner, 2015). Here, we show that ternary plots are able to track the trajectory of muscle orientation change through ontogeny in alligators. Third, comparisons across geological time reveal broad patterns among Archosauria including repeated evolution of large body size (Turner and Nesbitt, 2013), convergent evolution of feeding ecologies (Burton, 1974) and relatively hard biting in diverse clades (e.g. Carril et al., 2015; Sellers et al., 2017; Bates and Falkingham, 2018, Cost et al., 2020). We plotted the orientations of jaw muscles across a sample of living and extinct dinosaurs to illustrate their morphological disparity and, as a proof of concept, to demonstrate how ternary diagrams can be used to compare related taxa across time. Finally, we also compare the jaw muscle resultants of hard-biting species to demonstrate how convergence in behavior can occur using very different jaw muscle configurations. We show that ternary diagrams are particularly useful for conveying components of jaw muscle resultants at different scales of organization, enabling comparisons of higher-order biomechanical data across behavior, ontogeny and phylogeny. More broadly, ternary diagrams can represent complex 3D structures or forces in a 2D space such that anatomical relationships and physiological parameters are retained as comparable data.

The skulls of four extant taxa (American alligator, Alligator mississippiensis: MUVCAL008, MUVCAL024, MUVCAL031, MUVCAL612, MUVCAL622, MUVCAL700; Common ostrich, Struthio camelus: OUVC10659; Domestic chicken, Gallus gallus: MUVCAV003; and Grey parrot, Psittacus erithacus: MUVCAV042, MUVCAV092) and one extinct taxon (Tyrannosaurus rex: BHI3033) were scanned using computed tomography (CT) or micro-computed tomography (µCT; Table 1). Specimens of alligator (MUVCAL031), chicken (MUVCAV003) and parrot (MUVCAV042) were µCT scanned at the Truman VA Biomolecular Imaging Center, in Columbia, MO (Siemens Inveon MicroCT, Siemens Medical Solutions USA Inc., Malvern, PA, USA). The largest alligator specimen (MUVCAL008) was CT scanned at the University of Missouri School of Medicine Department of Radiology (Siemens Somatom Definition Scanner, Siemens Medical Solutions USA Inc., Malvern, PA, USA). All other alligator specimens (MUVCAL024, MUVCAL612, MUVCAL622, MUVCAL700) were CT scanned at the University of Missouri School of Veterinary Medicine (GE LightSpeed VCT CT scanner, GE Medical, Milwaukee, WI, USA; Sellers et al., 2017). A second parrot specimen (MUVCAV092) was µCT scanned at the University of Missouri Department of Geological Sciences X-ray Microanalysis Core (Zeiss Xradia 510 Versa 3D x-ray microscope, Carl Zeiss Microscopy, LLC, Pleasanton, CA, USA). A 1/6-scale model of T. rex (BHI 3033) was scanned at OhioHealth O'Bleness Memorial Hospital, Athens, OH (General Electric LightSpeed Ultra Multislice CT scanner, Milwaukee, WI, USA; Cost et al., 2020).

Jaw muscles of interest (m. adductor mandibulae externus medialis, mAMEM; m. adductor mandibulae externus profundus, mAMEP; m. adductor mandibulae externus superficialis, mAMES; m. adductor mandibulae posterior, mAMP; m. pterygoideus dorsalis, mPTd; m. pseudotemporalis superficialis, mPSTs; m. pseudotemporalis profundus, mPSTp; m. ethmomandibularis, mEM; m. depressor mandibulae, mDM) were identified through physical dissection and processed diffusible iodine contrast-enhanced CT (DiceCT; Fig. 2; Holliday et al., 2013; Gignac and Erickson, 2016). Scans were rendered and segmented using Avizo v9.5 (Thermo Fisher Scientific, Waltham, MA, USA; Fig. 3A). Defects were repaired, segmenting artifacts were removed, and a constant universal axis was applied to all models in Geomagic Studio 2013 (3D Systems, Rock Hills, SC, USA). Axes were oriented such that the x-axis corresponded to the mediolateral axis, the y-axis to the dorsoventral axis and the z-axis to the rostrocaudal axis. Models were then imported into Strand7 (Strand7 Pty Ltd, Sydney, NSW, AUS) where muscle attachment areas were identified and mapped on 3D finite element model (Fig. 3B and Fig. 4). Finite element analysis (FEA) software was used for muscle mapping to facilitate incorporation of muscle reconstruction data into subsequent analyses following the methods of Grosse et al. (2007) and Davis et al. (2010). Muscle maps (Fig. 4) were estimated from direct observations of material (dissections, DiceCT; see Fig. 2), interpretations of osteological correlates and the literature, when necessary (Baumel et al., 1993; Holliday and Witmer, 2007; Holliday, 2009).

Muscle resultants were computed from physiological cross-sectional area (PCSA) estimations, areas of muscle attachments and the centroids of the attachments (Fig. 3C). Physiological cross-sectional areas of muscles were determined by physical dissection or extant phylogenetic bracketing of taxa, as in T. rex (Witmer, 1995; Sellers et al., 2017; Cost et al., 2020). Muscle parameters such as pennation angles were estimated as falling within known ranges for alligators and birds, and osteological correlates of representative fossils informed PCSA estimates in T. rex (Cost et al., 2020). Three-dimensional muscle orientations were calculated from the centroid of one attachment to the centroid of the opposite attachment. The centroids and muscle areas were calculated from stereolithographic (STL) models using the script Area_Centroid_From_STL (Davis et al., 2010; Santana et al., 2010).

Fig. 3E shows a ternary diagram with a point representing this vector. The lengths of the colored lines are proportional to the proportions of the vector. This vector represents a muscle with a low mediolateral component; as such, the point is far from the corner labeled ‘ML’ (Fig. 3E, bottom left) and the corresponding yellow line is short. The dorsoventral and rostrocaudal components are several times that of the mediolateral component, and the dorsoventral component is a bit higher, so the point is slightly closer to the corner labeled ‘DV’ (top) than to the corner labeled ‘RC’ (bottom-right). Ternary diagrams were used to represent the component vectors of all of the jaw muscles in two dimensions, with size of the point scaled to each muscle's force (Fig. 3F). All calculations and plots were solved and created, respectively, in R (v. 3.5.1; https://www.r-project.org/) using the custom-written and freely available R package MuscleTernary (https://github.com/Middleton-Lab/MuscleTernary), which extends the R packages ggtern (v. 3.0.0; Hamilton and Ferry, 2018) and ggplot2 (v. 3.1.1; https://CRAN.R-project.org/package=ggplot2).

We demonstrate this approach using three case studies that explore jaw muscle resultants over three scales of time: during biting, through ontogeny, and across phylogeny. To visualize changes in muscle orientation between high and low gapes during an orthally biting feeding behavior, we plotted muscle parameters in ternary space for two specimens (1 juvenile, 1 adult) of A. mississippiensis by manipulating the mandibles to 5 deg and 30 deg of gape. A gape of 5 deg allows the animal to exert near its peak bite force, and at 30 deg, the jaws are at more extreme separation. Ternary coordinates for these models can be found in Table S1. To visualize changes in muscle orientation through ontogeny, we plotted the muscle parameters in ternary space for individuals of A. mississippiensis ranging from juvenile (head length=4.9 cm) to adult (head length=45.4 cm). Ternary coordinates for these models can be found in Table S2. To visualize changes in muscle orientation across a phylogeny, three avian taxa and one non-avian dinosaur were plotted in ternary space and the patterns of muscle orientation were analyzed and their differences described. Non-avian theropod and avian taxa were constructed with a gape of 20 deg. This value was chosen because it was the lowest gape determined to be of optimal performance in theropods by Lautenschlager (2015). Muscle resultants were plotted in ternary space (Table S3) to show patterns within the dinosaur lineage.

As gape decreases in A. mississippiensis, muscle insertion points on the lower jaw rotate caudoventrally and cause most muscles to exchange dorsoventral for rostrocaudal orientations (Fig. 5). If the jaws rotate symmetrically about the two quadratoarticular joints in an orthal jaw closing bite, the mediolateral component of most jaw closing muscles necessarily increases. The dorsoventral components of certain muscles (mAMES, mAMEM, mAMEP and mAMP) are reoriented mediolaterally as gape decreases during a bite. In ternary space, this is represented by points moving away from the top of the triangle and toward the bottom left corner. The pseudotemporalis muscles mPSTs and mPSTp are reoriented more rostrocaudally and dorsoventrally, respectively, at a low gape. The resultant of mPTd decreased rostrocaudally and increased mediolaterally and dorsoventrally as the gape decreased. In this instance, the point in ternary space moves away from the bottom right corner and toward the top and left, settling in the middle area of the ternary plot (Fig. 5). The jaw opening muscle mDM becomes more dorsoventrally oriented as the jaw is closed and does not reorient mediolaterally.

Overall, temporal muscles in individual alligators become less dorsoventral throughout ontogeny (Fig. 6). A rostrocaudal increase in muscle orientation is most appreciable in the external temporal muscles (mAMES, mAMEP and mAMEM). Rostrocaudal increases in temporal muscle orientations move the corresponding point in ternary space toward the lower right region of the ternary plot. The muscle mPSTs increases the mediolateral and rostrocaudal components but still retains a highly dorsoventral orientation whereas mPSTp increases the dorsoventral component and loses some rostrocaudal orientation. The other large adductor muscle, mAMP, experiences a slight ontogenetic decrease in its dorsoventral component and corresponding increase in its rostrocaudal component. The point representing mAMP therefore moves away from the top of the ternary plot and closer to the bottom right corner as animals increase in size. However, this change is not large, and mAMP largely retains its juvenile orientation overall as an adult. Musculus pterygoideus dorsalis (mPTd) retains and increases its rostrocaudal orientation; however, the changes associated with ontogenetic development are not as great as in the temporal muscles. The point for mPTd moves farther into the bottom right corner of the ternary diagram in this instance. The least change over ontogeny is seen in mAMP and mPTd. These two muscles retain large dorsoventral (mAMP) and rostrocaudal (mPTd) orientations. Musculus depressor mandibulae (mDM) does not change its orientation throughout ontogeny, remaining nearly equally dorsoventrally (≈47.5%) and rostrocaudally (≈47.5%) oriented with very small mediolateral components (<5%).

The effects of evolutionary changes on jaw muscle resultants are illustrated here in ternary space using a non-avian dinosaur (T. rex; Fig. 7B) and three avians (S. camelus, G. gallus and P. erithacus; Fig. 7C–E); adult alligator is also presented here as an outgroup (Fig. 7A). The temporal muscles (mAMEP, mAMES, mAMEM, mPSTs, mPSTp) as well as mAMP of T. rex are largely dorsoventrally oriented (Fig. 7B). The mPTd of T. rex is mixed, with contributions from all directions causing the resultant to reside nearer the center of the plot. In the paleognath S. camelus (Fig. 7C), the temporal muscles are almost entirely rostrocaudally oriented, but all possess dorsoventral aspects as well. Unique to S. camelus among other avian taxa sampled here, the resultant for mPSTp is centrally located in ternary space. The mPTd of S. camelus is largely rostrocaudal with a small dorsoventral component.

The temporal and pterygoideus muscles of the neognath G. gallus are largely rostrocaudally oriented but also possess dorsoventral components relatively greater than those observed in S. camelus (Fig. 7D). The orientations of mAMEP, mAMES, and mAMP possess very limited mediolateral components whereas mAMEM, mPSTs, and mPSTp exhibit more dorsoventral orientation at the expense of rostrocaudal orientation. This is shown through the resultants residing in the upper central area of the ternary. The mPTd possesses the most evenly distributed orientation in G. gallus. The muscle orientations of P. erithacus, another neognath, are different from G. gallus (Fig. 7E). Psittacus possesses rostrocaudally oriented temporal muscles but also exhibits more mediolateral muscle resultant components. Additionally, P. erithacus possesses a parrot-specific muscle mEM which is extremely dorsoventrally oriented. Musculus depressor mandibulae is highly dorsoventral across all avian taxa sampled. A small mediolateral component is also present. The muscle orientations of the temporal muscles contain appreciable mediolateral components, especially in mAMES and mAMEP. In P. erithacus, mPTd is highly dorsoventrally and mediolaterally oriented. The rostrocaudal component of mPTd in P. erithacus is relatively comparable to G. gallus and T. rex as well.

This case study considers the hardest biting fossil and extant taxa (T. rex and A. mississippiensis). A third taxon, P. erithacus, also produces relatively high bite forces compared with other birds (e.g. Sustaita and Hertel, 2010; Carril et al., 2015). The three taxa produce high bite forces with different cranial configurations: A. mississippiensis possesses a dorsoventrally short, mediolaterally wide skull whereas T. rex and P. erithacus both possess dorsoventrally tall and mediolaterally wide skulls.

Despite possessing a dorsoventrally short skull, A. mississippiensis temporal muscles possess appreciable dorsoventral and mediolateral components (Fig. 7A). The temporal muscles are relatively weaker than mPTd and mAMP (see Figs 5 and 6), and these two muscles possess extensive rostrocaudal and dorsoventral components, respectively. The force of mPTd and mAMP together constitute approximately 35% of the total bite force produced by A. mississippiensis (Sellers et al., 2017). T. rex and P. erithacus also both exhibit very few mediolateral muscle components except in mPTd (Fig. 7B,E). The temporal muscles are largely dorsoventral and rostrocaudal in T. rex and P. erithacus, respectively. The exceptions to large rostrocaudal components in P. erithacus are in mPTd, which exhibits a highly dorsoventral and appreciable mediolateral orientation and the extremely dorsoventral mEM (Fig. 7E). In T. rex, the mPTd has a greater rostrocaudal orientation than any other muscle. The dorsoventral components of mPTd are relatively greater than the dorsoventral components of mPTd in P. erithacus compared with those of T. rex.

Movement of the mandibles with respect to the cranium during feeding translates muscle insertions relative to their origins, leading to differing muscle orientations at different gape angles. As some muscle orientations are less effective at producing a given mandibular movement (e.g. jaw adduction), the gape angle can influence a system's ability to generate forces in a given direction (e.g. Eng et al., 2009; Lautenschlager, 2015). The results of the gape case study in A. mississippiensis confirm the general pattern that jaw closing muscles studied here increased their mediolateral components as gape decreased. Increases in mediolateral components decreases the relative vertical force that mAMEM contributes to bite force whereas mPTd relatively increases its contributions to bite force because it is not reoriented mediolaterally.

Additionally, in both the juvenile and adult, mAMP retained much of its dorsoventral components (more in the juvenile than the adult). As gape decreased, the dorsoventral component of mPTd increased. Sellers et al. (2017) determined that over one-third of total bite force is derived from the individual muscle forces of mPTd and mAMP. Retention of the dorsoventral components of the muscles enables these muscles to maintain high vertical bite forces even at low gapes. These results show that the decrease in dorsoventral components in many of the temporal muscles is potentially compensated for at low gapes by the increase in mPTd dorsoventral orientation.

Morphological changes of the skull are known to impact the feeding performance and ecology of the alligator (Erickson et al., 2003; Sellers et al., 2017). These morphological changes manifest as the skulls of alligators flatten considerably during their thousand-fold ontogenetic increase in body size (Dodson, 1975; Busbey, 1997; Monteiro et al., 1997; Brochu, 2001; Piras et al., 2014). Compared with the rest of the skull, the braincase and skull roof of adult alligators are more mediolaterally positioned relative to the jaw joints and mandibles than the same structures in juveniles (Mook, 1921a,b; Iordansky, 1973; Monteiro et al., 1997), which displaces the cranial attachments of temporal muscles mediolaterally. The ramus of the quadrate also shifts caudolaterally (Mook, 1921a,b; Iordansky, 1973; Monteiro et al., 1997), which gives some muscles a more rostrocaudal orientation in addition to the mediolateral dimension. As expected, shape changes in the skull cause a range of shifts in the resultants of some jaw muscles like the temporal muscles. However, changes over ontogeny are less evident in the pterygoideus muscles. Similarities in mAMP and mPTd throughout ontogeny likely reflect the integral roles that these two muscles play in increasing the bite force of alligators throughout the gape cycle regardless of specimen age.

The phylogenetic results are presented as a small-scale proof-of-concept that could be elaborated upon in future comparative studies across Archosauria. This study shows that jaw muscle resultants vary considerably across taxa but that ternary plots offer new ways to visually compare these differences and make predictions about evolutionary transformations, such as those along the lines to extant birds from non-avian theropod dinosaurs. Not only does the increased encephalization and expanded braincase (Balanoff et al., 2016; Fabbri et al., 2017) likely alter resultants of temporal muscles across the clade, but changes in the linkages and shapes of the palatal elements (Huxley, 1867; Bock, 1964, 1999; Gussekloo and Zweers, 1999) and the origin of avian-style powered kinesis likely altered the organization of palatal muscles.

Muscle resultants in T. rex are highly dorsoventrally oriented and possess few mediolateral or rostrocaudal components. Although not tested, this muscle morphology seems similar to what we envision in many other non-avian dinosaurs like Allosaurus spp. Birds, however, exhibit muscle resultants that are more rostrocaudal and, in neognathe taxa, possess some mediolateral components in the non-temporal mPTd. In G. gallus there are also mediolateral components evident in mAMEM, mPSTs, and mPSTp. Holliday (2006) and Holliday and Witmer (2008) described the lateral expansion of the braincase and reduction of the temporal region in birds as major features driving the evolution of the feeding apparatus of birds. This combination of braincase inflation and kinesis requires that the temporal muscles not only have a more dorsoventral than mediolateral orientation, but also more rostrocaudally oriented forces. This is driven by the derived propulsive nature of kinesis present in neognathe birds, particularly parrots as shown here, relative to other avian taxa. The translation of muscle resultants to a more rostrocaudal orientation is likely related to the evolution of powered, propulsive cranial kinesis in neognathe birds.

Differences between paleognathe and neognathe birds are subtler than those between sampled non-avian and avian theropods. The temporal muscles, though, are fairly similar in their rostrocaudal orientations. The trends described in this proof-of-concept study are shown with a small sample size; however, we are confident that an increase in sample size will only serve to bolster these trends in the dinosaur to bird evolutionary line.

Our final case study indicates that species that can generate high bite forces can do so using variable muscle resultant orientations. The ternary diagrams in this section also illustrate how ecomorphs might converge in function despite disparate cranial shapes. Numerous vertebrate species have evolved increased bite forces in order to dispatch prey (e.g. A. mississippiensis, T. rex) or to husk tough food items (e.g. P. erithacus), and in these cases, we expect the skulls to have increased vertical components of jaw muscle resultants, as vertical forces are more optimally in line with orthal forces driven into food items.

The highly vertical muscle orientations found here in diverse adductor muscles of A. mississippiensis (mAMP), T. rex (mAMEP, mAMEM, mAMES, mPSTs, mPSTp and mAMP) and P. erithacus (mPTd, mEM) show that hard biting archosaurs employ different biomechanical strategies to generate relatively high bite forces. As in gape (Fig. 5), mPTd in A. mississippiensis increases its dorsoventral component as gape decreases, contributing a larger dorsoventral component at low gape. A. mississippiensis produces high bite forces using one highly dorsoventrally oriented muscle (mAMP) and one rostrocaudally oriented muscle (mPTd) that increases its dorsoventral orientation at low gapes. This reorientation of mPTd appears to compensate for the decreases in dorsoventral orientations in other muscles.

Dorsoventral bite force production of T. rex is largely the result of the temporal muscles (mAMEP, mAMEM, mAMES, mPSTs, mPSTp and mAMP). This dorsoventral orientation of high force producing muscles is in line with expectations for hard-biting taxa mentioned previously. P. erithacus, however, employs a pterygoideus muscle (mPTd) and a parrot-specific muscle (mEM) to produce high bite forces. The temporal muscles of P. erithacus contribute less overall force to the bite force of the animal as these muscles (mAMEP, mAMES, and mAMP) are oriented more rostrocaudally.

Ternary diagrams are a powerful means of conveying complex muscle orientation data in comparative contexts across behavior, ontogeny, and phylogeny. All of our case studies provide solid proof-of-concept work that lays a solid foundation for future studies using ternary diagrams to visualize and interpret complex biomechanical data and concepts. We identify trends in muscle orientation changes across gape in juvenile and adult specimens as well as ontogenetic changes within the same lineage (A. mississippiensis). Decreases in gape during biting in A. mississippiensis change the resultant orientation of muscles in both juvenile and adult specimens to a more mediolateral orientation. Over ontogeny in A. mississippiensis, however, the orientations of the muscles shift rostrocaudally and mediolaterally. In the dinosaur-avian lineage, muscle resultant orientations were shifted rostrocaudally and mediolaterally along the dinosaur-to-bird axis. More dinosaur taxa are needed to better illustrate this pattern. This case study shows that future comparison studies across lineages of extinct and extant taxa can be conducted in a meaningful way that incorporates complex and informative 2D visualizations of 3D muscle systems. Finally, we showed that among archosaurs, hard-biting taxa generate high bite forces using disparate arrangements of jaw muscles. Data like these will enable us to follow the evolutionary changes that resulted in different anatomical solutions to functional demands. Future studies can make use of these types of ecomorphological variables to study the convergence of bones and muscles across guilds of animals. Studies that estimate ancestral states and evaluate the tempo of adaptive radiations of animals that modify the feeding apparatus (e.g. cichlids) could also benefit from using ternary diagrams to present data and results.

We thank M. Scott Echols, Lawrence Witmer, and Ruth Elsey and the staff at the Rockefeller Wildlife Refuge for specimens. We thank Julian Davis, Stephen Gatesy, Emily Rayfield, Ryan Ridgely and Lawrence Witmer for discussions throughout the development of this paper. We thank the Truman VA Biomolecular Imaging Center (Columbia, MO), University Hospital Department of Radiology (University of Missouri, Columbia, MO), School of Veterinary Medicine (University of Missouri, Columbia, MO), Geological Sciences X-ray Microanalysis Core (University of Missouri, Columbia, MO), and OhioHealth O'Bleness Hospital (Athens, OH) for scanning specimens used in this study. We are grateful for the input of two anonymous reviewers whose thoughtful suggestions improved the manuscript.