The Virtual Goniometer: A new method for measuring angles on 3D models of fragmentary bone and lithics
Katrina Yezzi-Woodley, Jeff Calder, Peter J. Olver, Annie Melton, Paige Cody, Thomas Huffstutler, Alexander Terwilliger, Martha Tappen, Reed Coil, Gilbert Tostevin
TThe Virtual Goniometer: A new method for measuringangles on 3D models of fragmentary bone and lithics *Katrina Yezzi-Woodley † Jeff Calder ‡ Peter J. Olver ‡ Paige Cody § Thomas Huffstutler ‡ Alexander Terwilliger ‡ Annie Melton § Martha Tappen § Reed Coil ¶ Gilbert Tostevin § Abstract
The contact goniometer is a commonly used tool in lithic and zooarchaeological analysis,despite suffering from a number of shortcomings due to the physical interaction between themeasuring implement, the object being measured, and the individual taking the measurements.However, lacking a simple and efficient alternative, researchers in a variety of fields continueto use the contact goniometer to this day. In this paper, we present a new goniometric methodthat we call the virtual goniometer , which takes angle measurements virtually on a 3D modelof an object. The virtual goniometer allows for rapid data collection, and for the measurementof many angles that cannot be physically accessed by a manual goniometer. We comparethe intra-observer variability of the manual and virtual goniometers, and find that the virtualgoniometer is far more consistent and reliable. Furthermore, the virtual goniometer allows forprecise replication of angle measurements, even among multiple users, which is important forreproducibility of goniometric-based research. The virtual goniometer is available as a plug-inin the open source mesh processing packages Meshlab and Blender, making it easily accessibleto researchers exploring the potential for goniometry to improve archaeological methods andaddress anthropological questions.
Keywords— goniometer, taphonomy, zooarchaeology, fracture angle, lithics
Goniometry is an important aspect of archaeological and zooarchaeological analysis. The primary toolfor studying angles on objects such as bone fragments or lithics is the pocket, contact goniometer, inessence a metal protractor with a rotating arm. However, its reliability has come under question and al-ternative methods have been proposed (Archer, Pop, Gunz, & McPherron, 2016; Dibble & Bernard, 1980; * Source code for the virtual goniometer can be found here: https://amaaze.umn.edu/software † Anthropology, University of Minnesota, [email protected] (corresponding author) ‡ School of Mathematics, University of Minnesota § Anthropology, University of Minnesota ¶ Sociology and Anthropology, Nazarbayev University, Kazakhstan a r X i v : . [ m a t h . NA ] N ov orales, Lorenzo, & Vergès, 2015; Valletta, Smilansky, Goring-Morris, & Grosman, 2020). Nonetheless,it remains a constituent of current lithic analysis (Debert & Sherriff, 2007; Douglass, Lin, Braun, & Plum-mer, 2018; Muller & Clarkson, 2016; Scerri, Gravina, Blinkhorn, & Delagnes, 2016), and has expandedinto zooarchaeological and taphonomic research to study bone fragmentation. In particular it has beenapplied to differentiate anthropogenic bone fragmentation from fragmentation by carnivores (Alcántara-García et al., 2006; Capaldo & Blumenschine, 1994; Coil, Tappen, & Yezzi-Woodley, 2017; De Juana &Dominguez-Rodrigo, 2011; Moclán, Domínguez-Rodrigo, & Yravedra, 2019; Moclán et al., 2020; Picker-ing, Domínguez-Rodrigo, Egeland, & Brain, 2005; Pickering & Egeland, 2006).The goniometer has a long history. It was first described by Gemma Frisius (a doctor, mathematician,and cartographer) in 1538 and was derived from the astrolabe, which is the predecessor of the total station,an instrument used for surveying and cartography. The first pocket, contact goniometer was designed in the1780s to measure the angles on crystals, becoming an essential experimental tool for the burgeoning scienceof crystallography, enabling accurate measurement of the angles between crystal faces, thereby enablingearly researchers to establish the foundations of crystal symmetry classes and generate the classificationtables of modern crystallography. Although, at the time, the contact goniometer was revolutionary in itsprecision, effectively demonstrating the uniformity of crystals, it was largely abandoned in the early 1900swith the advent of yet more accurate methods based on x-ray diffraction (Burchard, 1998).The contact goniometer first appeared in archaeology when Barnes (1939) used the instrument to dif-ferentiate anthropogenically produced stone tools from naturally-occuring conchoidally fractured rocks.Taphonomic applications began when Capaldo and Blumenschine (1994) used the goniometer to distinguishbones broken by carnivores from those broken by hominins. They extrapolated directly from lithic methodsusing the goniometer to measure the internal platform angle on models of bone flakes created by taking im-pressions of notches and associated flake scars on experimentally broken bone. In 2006, Alcántara-Garcíaet al. (2006) introduced a method for identifying actors of breakage by using the goniometer to measure afew fracture angles — meaning the angle of transition from the periosteal surface to the fracture surface —on long bone shaft fragments. Prior to this, fracture angles were assessed by eye and categorized as oblique,right, or both, as a means of distinguishing green breaks from dry breaks (Villa & Mahieu, 1991). Usingthe goniometer in the analysis of fragmentary faunal assemblages is gaining traction because it permits re-searchers to collect seemingly more reliable, quantitative data, and opens the possibility for other avenuesof analysis to address questions related to hominin and carnivore interactions at important paleoanthropo-logical sites (Coil et al., 2017; De Juana & Dominguez-Rodrigo, 2011; Domínguez-Rodrigo & Barba, 2006;Moclán et al., 2019, 2020; Pickering et al., 2005; Pickering & Egeland, 2006).Although goniometry is claimed to be useful in anthropology, the overall reliability of measurements,and hence subsequent inferences, depends on their accuracy, precision, and replicability. For the contactgoniometer to be accurate and precise, the target object must be a size that is compatible with the measuringimplement. The contact goniometer was originally designed to measure the angle between intersectingflat surfaces, such as are found on crystals, and is less well adapted to surfaces with curvature and otherfeatures, such as cylindrical long bones, or uneven surfaces that are found on bones and lithic flakes. Thepositioning of the goniometer on the object depends on the user, who must ensure that each arm of theinstrument lies flat against and perpendicular to the faces being measured. Curvature variations can makethis placement challenging. Furthermore, sometimes the goniometer cannot access the necessary locationfor taking the measurement. It might be blocked by adhering matrix or some other surface feature such asa protuberance on a bone fragment or the break on the opposite side of the fragment. Measurements takenby the goniometer can be expected to vary ± ◦ (Capaldo & Blumenschine, 1994; Draper et al., 2011) andextremely sharp edges cannot be measured by a goniometer (Dibble & Bernard, 1980), which, in some cases,has a lower limit of 20 ◦ . As a result of all these constraints, measurements taken at the edge angle of a singleflake or bone fragment will be inconsistent (Dibble & Bernard, 1980; Johnson, Dropps, & Yezzi-Woodley, virtual goniometer . This is avirtual method and can be used to measure angles on 3D models of any object. The virtual goniometer offersa precise and replicable way to measure angles that addresses all of the challenges highlighted above. Weprovide examples of using the virtual goniometer to measure angles on crystals and stone tools. We assessand compare intra-observer variability in fracture angle measurements taken on bone fragments using acontact goniometer versus the virtual goniometer, and find that the virtual goniometer is far more consistent,performing at least 3.6 degrees ( ± The Virtual Goniometer
In this section we describe the mathematical algorithms used to design the virtual goniometer. Its imple-mentation and applications to crystals, lithics, and bones will be described in the following section.The starting point is a mesh that represents (part of) the bounding surface of a solid object — a bonefragment, a lithic, a crystal, etc. Thus the mesh consists of a reasonably dense sample of points on thesurface of the scanned object. With each mesh point, we also require a unit (length one) vector that pointsin the outward normal direction to the surface at the point. The unit outward normals can be constructedfrom a triangulated mesh by averaging the normals over nearby triangles; for a point cloud one can use localPrincipal Component Analysis (PCA) or other convenient methods to compute the normal. Once we havethe mesh points and their associated unit outward normals, we do not require any further information suchas mesh connectivity to effect our algorithm.To take an angle measurement at a specified location on the surface, the user must set the location,either by clicking on the mesh representation in the chosen software platform, or by inputting its ( x , y , z ) coordinates directly. Both methods are supported in our implementation. The location is assumed to be on,or at least close to a break edge where an angle measurement has meaning. Choosing a location far awayfrom the edge can lead to spurious angle measurements that are of no use to the studies for which the virtualgoniometer is designed. The user then specifies a patch on the surface, that is, a subset of the vertices,centered at the specified location where the virtual goniometer angle measurement is to be computed. By apatch of radius r >
0, we mean the set of all mesh points that are within distance r of the specified location.In Meshlab, distance means geodesic distance measured (approximately) along the surface, and is computedusing built in code that is based on Dijkstra’s algorithm on a k -nearest neighbor graph constructed over thepoint cloud of vertices. In Blender, we simply use the Euclidean distance between vertices in the ambientthree-dimensional space. The user specifies the radius r by first clicking on the location and dragging thecursor using the graphical interface to see the resulting patch. Alternatively, the radius can be a priorispecified or entered manually.The virtual goniometer algorithm has two main steps. In the first step, we use unsupervised machinelearning algorithms, in particular, data clustering techniques, to separate the patch into two regions, eachcorresponding to one side of the break edge. This part of the algorithm relies on a parameter λ ≥ X will represent all the vertices in thepatch, while N will represent their corresponding outward unit normal vectors, while λ is the aforemen-tioned tuning parameter. The Virtual Goniometer is summarized in Algorithm 1, whose output is the anglemeasurement θ in degrees, and the goodness of fit ε > n be the number of vertices in the patch. We use = [ , . . . , ] T to denote the columnvector with n entries all equal to one, and (cid:107) x (cid:107) to denote the Euclidean norm of the vector x . Let X be the3 × n matrix whose i th column is the vector of x , y , z coordinates of the i th vertex in the selected patch. Let N be the 3 × n matrix whose i th column is the outward unit normal vector to the surface at the i th vertex.The Virtual Goniometer Algorithm uses both the normal vectors N and the vertices X to segment thepatch into two regions. The parameter λ controls the tradeoff between how much to rely on the normalsversus the vertices. Setting λ = N . If the surface isnoisy, this can give a poor segmentation, since some normal vectors could point in a similar direction even ifthey are on opposite sides of a break. Increasing λ encourages the segmentation to put points that are closetogether into the same region, and can help to improve the segmentation on noisy meshes. Figure 1 shows lgorithm 1 Virtual Goniometer Input:
Points X , normals N , and tuning parameter λ ≥ Output:
Angle θ and goodness of fit ε x ← Centroid of vertices X . {See notes.} r ← Radius of patch X . {See notes.} t ← Eigenvector of NN T with smallest eigenvalue. {Tangent to break curve.} n ← n N {Average outward normal vector.} b ← t × n (cid:107) t × n (cid:107) {Cross product, normalized — binormal vector.} p ← b T [ N + ( λ / r )( X − x ) ] {Dot product of all rows with b .} s ← WithinSS ( p ) {Find best clustering of 1D data p ; see notes.} X − = X [ p < s ] , X + = X [ p ≥ s ] {Split into subpatches divided by break curve.} ( v ± , ε ± ) ← PCA ( X ± ) {Principal component with smallest variance, mean squared error.} θ ← − arccos (cid:0) sign ( n T v + ) sign ( n T v − ) v T + v − (cid:1) {VG angle — arccos returns degrees.} ε ← ε + + ε − {Goodness of fit.} the effect of λ on the segmentation, and there are generally three instances, when it needs to be adjusted:(1a) sharp curves in the ridge; (1b) subtle ridges, usually associated with obtuse angles; and (1c) rugosesurfaces. In our implementation, we take λ = Figure 1: Examples of how the tuning parameter λ affects how the measurement is taken. Let us describe the individual steps in our algorithm in some detail. In Step 3, one can use for x anyreasonable notion of centroid of the patch X , and either the mean value of the coordinates or a geodesiccentroid work well. In our implementation in Meshlab, we set x to be the vertex selected by the user in theclick-and-drag selection method. In Step 4, we use the geodesic radius of the patch for r ; that is the largestgeodesic distance from x to any point in X as measured along the surface. It is also possible to use theEuclidean radius of the patch, and the only difference is a minor change in the effect of the tuning parameter λ . However, the algorithm is not overly sensitive to this effect.The vector t produced in Step 5 is to be interpreted as the tangent vector to the “break curve” thatseparates the two approximately planar regions on the surface. The vector n in Step 6 is the averagedoutward normal vector to the patch. Thus, in Step 7, the vector b given by normalizing the cross product n bN i N j Figure 1.
Patch. patch
Figure 1 blankf 10/15/20 1
Figure 2:
Depiction of a surface patch and the tangent, normal and binormal vectors t , n , b in Algorithm 1. Thevector t is tangent to the break curve, which is a line in this example, n is the average of the outward normals on bothsides of the break edge, denoted N i and N j in the figure, and the binormal b is orthogonal to n and t , and hence points across the break edge. The basis for our segmentation algorithm is that the sign of the dot product b · N i will indicateon which side of the break edge a particular vertex falls. of the break curve tangent and the surface normal is orthogonal to both and hence can be interpreted as theunit binormal vector of the patch X , pointing across the break curve, i.e., a unit vector that is both tangentto the surface and normal to the break curve. The binormal vector b is used to quickly obtain the correctsegmentation, as described below. See Figure 2 for a depiction of the vectors t , b , and n when the surface is(approximately) planar on either side of the (approximately) straight break edge.Our clustering method used in Step 9 to divide the surface into two classes was inspired by the randomprojection clustering methods of Han and Boutin (2014); Yellamraju and Boutin (2018), which involverepeatedly randomly projecting the data to one dimension, and then using the function WithinSS describedbelow to perform the clustering of the resulting one-dimensional projected data. Given one-dimensionaldata represented by p = ( p , p , . . . , p n ) where p ≤ p ≤ · · · ≤ p n , this function computes a real number s for which the quantity f ( s ) = ∑ p i ≥ s ( p i − c ) + ∑ p i < s ( p i − c ) where c = ∑ p i ≥ s p i ∑ p i ≥ s , c = ∑ p i < s p i ∑ p i < s . is minimized. The value of s that minimizes f ( s ) gives the optimal clustering of the one-dimensionaldata p = ( p , . . . , p n ) into two groups { p i ≥ s } and { p i < s } . Note that f ( s ) is constant on each interval p i < s ≤ p i + , and hence we can the global minimizer of f simply by computing f ( p i ) for i = , . . . , n , andchoosing s = p i that gives the smallest value. (And hence there is also no need to actually sort the datapoints p .) This highlights the advantage of working with one-dimensional data; it is very simple and fast tocompute optimal clusterings.Step 8 in the Virtual Goniometer Algorithm implements a one-dimensional projection of a λ weightedcombination of the unit normals to the vertices in the patch and the vertices themselves, shifted by thecentroid so as to center the patch at the origin. However, the direction of the projection b is not random butis very carefully chosen as the binormal of the patch. The random projection algorithm advocated in Han andBoutin (2014) also works very well in this application; however, it requires around 100 random projectionsto obtain reliable results and is thus significantly slower. We have also experimented with other clusteringalgorithms, such as the hyperspace clustering method of Zhang, Szlam, Wang, and Lerman (2010), which lso gives very good results at the expense of longer computation times. Our method presented in Algorithm1 is very efficient and is suitable for real-time computations in mesh processing software such as Meshlab.The resulting pair of clusters X − = X [ p < s ] = { X i | p i < s } and X + X [ p ≥ s ] : = { X i | p i ≥ s } contain the vertices belonging to the two components of the surface patch on either side of the break curve.(However, we do not construct the actual break curve; nor do we require that the centroid or selected location x be thereon.) In Steps 10 and 11, the PCA function returns each principal component, denoted v ± , withthe smallest variance and the corresponding eigenvalues ε ± , which represents the mean squared error in thefitting. We also experimented with robust versions of PCA (see Lerman and Maunu (2018) for an overview),but did not find the results were any more consistent. Finally, in Step 13, arccos is the inverse cosine function,measured in degrees, not radians, and we use the branch with values between 0 and 180 degrees. In our applications, the process begins by importing a meshed surface into Meshlab. There are two methodsused to take virtual goniometer angle measurements at selected points on the surface mesh, which we callthe click-and-drag method and the xyz method . For the click-and-drag method, one simply clicks the desiredlocation on the mesh for the measurement (e.g. the break ridge) and then drags to select the area of thesurface to be used for the measurement. The chosen area, or surface patch, represents a disk of radius r > xyz -coordinates of the center of the patch. There is also an option to input aspecified radius and change the tuning parameter λ . The plug-in documentation which provides detailed,step-by-step instructions is available at https://amaaze.umn.edu/ After the algorithm is run (almost instantaneously), the selected patch is color-coded into two contrast-ing colors showing the two vertex clusters representing the two sides of the break curve that are used todetermine the PCA planes, and hence the angle based on their normals. The colors rotate through a palletlist of pairs of contrasting colors; see Figure 3 for examples of the resulting visual output in Meshlab. Fur-ther measurements can be taken at or near the original location; see Figure 3b. The user can also choose toadvance to a new location at which point the colors change; see Figures 3b and 3c. Numerical data for eachmeasurement are automatically recorded in a .csv file.(a) Lithic Flake (b) Crystal (c) Bone Fragment
Figure 3: Examples of the Virtual Goniometer on Different Materials
We compared the manual and virtual goniometers for computing fracture angles on a sample of bonefragments consisting of 537 breaks taken from 86 appendicular long bone shaft fragments ( ≥ imension) randomly chosen from a collection of experimentally broken Cervus canadensis nelsoni (RockyMountain elk) and
Odocoileus virginianus (white-tailed deer) limb bones. Fragments were scanned usinga medical CT scanner at the University of Minnesota’s Center for Magenetic Resonance Research (CMRR)(slice thickness: 0.6, reconstruction interval: 0.6 mm, KV: 80, MA:28, rotation time: .05 sec, pitch: 0.8,alogrithm: bone window, convolution kernel: B60f sharp) and then surfaced using MATLAB. The breaksurfaces on each fragment were manually subdivided into separate break planes (Gifford-Gonzalez, 1989;Haynes, 1983; Pickering et al., 2005). All breaks were measured, and we did not impose a minimumbreak length requirement. We measured the fracture angle — which is defined as the angle of transitionbetween the periosteal surface and break surface — on each break face of each fragment (Alcántara-Garcíaet al., 2006; Capaldo & Blumenschine, 1994; Villa & Mahieu, 1991). Following the method established byAlcántara-García et al. (2006) and further described by Pickering et al. (2005), we chose to measure at thecenter along the break length.Each break was measured using both a contact goniometer ("man") and the virtual goniometer usingour Meshlab plug-in. Two methods were employed using the virtual goniometer: the click-and-drag method(“drag") and the xyz-coordinate input method (“xyz"). The fragments were first measured using the dragmethod. After the first round of drag angle measurements, denoted θ drag , screen shots were taken of allthe colorized models to create a 2D map of the measurements that served as a visual guide for subse-quent drag angle measurements, which we denote by ϕ drag and ψ drag . Three manual angle measurements, θ man , ϕ man , ψ man were then taken using the contact goniometer, again using the map as a guide.The xyz method was used to replicate the first set of drag measurements, so θ xyz = θ drag . The valuesfor the radius and location (i.e. xyz-coordinates) of θ drag were input into the plug-in for the xyz method toproduce two further angle measurements ϕ xyz , ψ xyz ). Thus, the xyz method was only executed twice, and, intotal, the same person measured the angle of each break eight times (3 manual, 3 drag, and 2 xyz) therebyallowing us to test for intra-observer error. Using additional data automatically provided by the two virtualmethods, we analyzed the degree to which the location of the measurement, the distance from the edge, andthe number of vertices used impact how the angle measurement varies. Of the 537 breaks in our randomly selected sample of bone fragments, 500 (93.1%) could be measuredmanually using the contact goniometer, while the other 37 breaks could not be physically measured. For 34of those breaks, one or both arms of the contact goniometer were blocked from contacting the break face orperiosteal face. The 3 remaining breaks that could not be measured manually broke off of a fragment thatsuffered lab damage after being scanned. The 3D mesh constructed prior to the lab break made it possible touse these measurements for analysis of the virtual goniometer. When comparing the manual measurementsto the click-and-drag and xyz methods, we only used the 500 breaks that could be measured by all threemethods.To test variability, we computed three angle measurements for each of the three methods, which wedenote by θ , ϕ , ψ and use subscripts man, drag, xyz to indicate which method is employed. When using themanual and drag method, the location of the measurement is selected by eye. For the xyz method, the userenters the x , y , z coordinates directly.To assess how much each method varied, we calculated the Intra-Observer Variability (IOV) for the threeangle measurements ( θ , ϕ , ψ ) for each break under the three different methods. The IOV is the average ofthe absolute value of the differences between each of the three angle measurements, all taken at the same ocation: IOV man = | θ man − ϕ man | + | θ man − ψ man | + | ϕ man − ψ man | , IOV drag = | θ drag − ϕ drag | + | θ drag − ψ drag | + | ϕ drag − ψ drag | , IOV xyz = | θ xyz − ϕ xyz | + | θ xyz − ψ xyz | + | ϕ xyz − ψ xyz | . (4.1)Ideally, there should be no variation, so that IOV = ◦ ) is marginally better than the expected variation (5 ◦ ) describedin Capaldo and Blumenschine (1994) and Draper et al. (2011) but the mean for the manual IOV (7.08 ◦ ) isover 2 ◦ higher than expected and the standard deviation (8.48 ◦ ) is high. All but seven of the manual IOVsare < ◦ . The remaining seven are > ◦ and could be considered outliers to which the mean and standarddeviation are sensitive. However, removing those seven IOVs would not sufficiently reduce these valuesbecause the median IOV for the drag method (2.28 ◦ ) and the standard deviation (3.34 ◦ ) are considerablysmaller. The xyz method has a consistently smaller IOV compared to the other methods, with a median of0.001 ◦ , mean 0.006 ◦ , and standard deviation 0.011 ◦ . Table 1: Summary statistics for angle IOV ◦ S TATISTICS M ANUAL D RAG XYZ
N 500 500 500
MIN
MEAN
MEDIAN
MAX SD We calculated 95% and 99% confidence intervals, the results of which can be found in Table 2 and Figure 5(Weisberg, 2005). Even with an increase in the confidence level to 99%, the difference is striking — indeed,we have to apply a magnification to observe its confidence interval. The range for the manual method is2.5 times larger than the drag method and between 760.5 and 772.9 times larger than the xyz method. Thedrag method is a little over 300 times larger than the xyz method. The small range IOV of the xyz methodindicates the method is exceptionally precise, especially compared to the drag and manual methods.We ran Tukey’s HSD (honestly significant difference) test with confidence level α = .
05 to assess thesignificance of the difference in means of the IOV scores for each method (Barnette & McLean, 1998).Table 3 shows that the virtual goniometer’s drag method is 3.6 degrees ( ± a) manual (b) click-and-drag (c) xyz-coordinate Figure 4: Histograms of angle IOV ◦ using all three methods (n=500)Table 2: Confidence Intervals
95% CI 99% CIM
ETHOD M EAN I NTERVAL R ANGE M EAN E RROR I NTERVAL R ANGE M EAN E RROR M ANUAL ± ± RAG ± ± XYZ ± ± than the contact goniometer and the virtual goniometer’s xyz method is 7.1 degrees ( ± ± p < . Table 3: Results of the Tukey HSD Test
GROUP GROUP MEANDIFF P - ADJ LOWER UPPER REJECTDRAG
IOV
MAN
IOV 3.637 0.001 2.873 4.402 T
RUEXYZ
IOV
DRAG
IOV 3.436 0.001 2.685 4.187 T
RUEXYZ
IOV
MAN
IOV 7.073 0.001 6.309 7.838 T
RUE
Since the center, radius, and vertex data cannot be collected using the manual method, we compare the dragand xyz methods for the entire sample using calculations of the IOV for each variable. The IOV of theangle measurement for the drag method features numerous large values and has a standard deviation of 3.7 ◦ ,varying as much as 28.2 ◦ (see Figure 6). The xyz method has a standard deviation of 0.01 ◦ with a maximumvariation of 0.06 ◦ . Most of the xyz IOVs fall below 0.02 ◦ . It is clear that the variation in the angle IOV is theresult of variation in the location (represented by the center of the patch), the patch’s radius, and the numberof vertices in the patch (see Table 4).Table 4 gives summary statistics for the IOV for the drag and xyz methods based on IOVs for the anglemeasurement, the number of vertices in the selected patch, the radius of the patch, and the center locationof the patch. ◦ C LICK - AND -D RAG XYZ -C OORDINATES S TATISTICS A NGLE V ERTICES R ADIUS C ENTER A NGLE V ERTICES R ADIUS C ENTER
N 537 537 537 537 537 537 537 537
MIN
MEAN
MEDIAN
MAX SD To better understand how changes in the location, the radius, and the number of vertices in the patch affectthe angle measurement we ran a multiple regression using a log transformation (Weisberg, 2005).
Table 5: Multiple Regression Results (n=537)Estimate Std. Error t value Pr( > | t | )(Intercept) 0.3995 0.0717 5.58 3.94e-08vertices IOV -0.0009 0.0003 -3.6 0.000346radius IOV 0.9997 0.1993 5.015 7.23e-07location IOV 0.0573 0.0197 2.909 0.003772 Adjusted R-squared: 0.0872, p-value: 2.242e-10
We chose to log-transform the response variable, using the standard natural log, in order to make thedata satisfy the assumption of normal error terms (see Figure 7). On the original data the residuals did notfollow a normal distribution and were quite skewed (Figure 7a). The residuals for the log-transformed datafollow a normal distribution (Figure 7b). Though the residuals do not have an obvious pattern, the Cook’sdistance plot shows that none of the datapoints are overly influencing the model (Figure 8) (Cook, 1977).None of the observations have high Cook’s values which is indicated by the absence of datapoints in theupper or lower right-hand region of the plot. a) click-and-drag (b) xyz-coordinate Figure 6: Histograms of angle IOV using all virtual methods (n=537) −3 −2 −1 0 1 2 3
Theoretical Quantiles S t anda r d i z ed r e s i dua l s lm(drag_IOV ~ 1 + drgvt_IOV + drgrd_IOV + drgctr_IOV) Normal Q−Q (a) Q-Q plot of the original data −3 −2 −1 0 1 2 3 − − − Theoretical Quantiles S t anda r d i z ed r e s i dua l s lm(log(drag_IOV) ~ 1 + drgvt_IOV + drgrd_IOV + drgctr_IOV) Normal Q−Q (b) Q-Q plot of the log transformed data
Figure 7: Q-Q plots of the residuals(n=537)
The results of the multiple regression are presented in Table 5. The p-value of the model’s fit is sig-nificant (p=2.242e-10) as are the p-values for each of the explanatory variables indicating that all threevariables are influencing the angle IOV. In this model, the radius (estimate = 0.9997) has the biggest impact.The estimate for the vertices (-0.0009) shows that this has the least impact. The data are highly scattered(R = The salient questions are whether or not goniometry can be useful for addressing anthropological questionsand, if so, in what ways. To answer these questions, measurement methods must result in values thataccurately capture the angle of interest with reasonable precision. Methods that do not accurately reflect thedesired information undermine the ability to make credible anthropological inferences.Being able to take a measurement is the first priority. The primary issue with the contact goniometer .00 0.05 0.10 0.15 0.20 0.25 − − − − Leverage S t anda r d i z ed r e s i dua l s lm(log(drag_IOV) ~ 1 + drgvt_IOV + drgrd_IOV + drgctr_IOV)Cook's distance Residuals vs Leverage
174 401
Figure 8: Cook’s Plot (log-transformed) is the physical interaction between the instrument and its target. When measuring bone fragments it canbe difficult to reach the desired location for the measurement. This is often a matter of scale in that thegoniometer is too large relative to the size and shape of the bone fragment. While a tiny goniometer would,at least in principle, be able to measure smaller or hard-to-reach locations, in practice it would be difficultfor the user to handle comfortably and accurately. In some instances, the arms are too long and are blockedby features on the bone fragment or the break on the opposite side. The latter happens when more thanhalf of the circumference of the fragment is present or the angle of interest is acute and descends into themedullary cavity. When measuring notches, Capaldo and Blumenschine (1994) chose to take molds becausethe goniometer was unable to reach the notch surfaces. However, making molds is often not an option andcan even damage the specimen.Even if an acute angle is accessible, some goniometers cannot measure anything less than 20 ◦ becausethe arms overlap, blocking the location where the specimen is supposed to fit. To test the accuracy of thevirtual goniometer, we created synthetic meshes of intersecting flat planes with known angles. Given asufficient number of vertices, which depends on scan resolution and radius size, the virtual goniometer canaccurately measure an angle as small as 1 ◦ , which is far superior to any version of the contact goniometer ineveryday use.The only instance in which the virtual goniometer cannot measure a fracture angle is when a sharpnatural curve on the bone fragment is close to the fracture edge. This is only an issue when the radius islarge. Reducing the radius such that it captures more of the ridge and less of the natural curve rectifies theproblem. A smaller radius might be better when measuring angles on bone fragments. Limiting the radiussuch that it is local to the transition but still sufficiently large to not be hampered by imaging artifacts couldprovide a more informative measure. As the radius increases in size, it captures changes in the topography.Though the topographical changes on the break surface may also be of interest as it pertains to examiningbone fragments, this should remain separate from the angle of transition between faces.The ability to take a measurement does not ensure its appropriateness. Because the surfaces are not flat,the angle value depends on the distance from the edge where the measurement is taken on both faces. Stop-ping 3 mm from the edge of an object or stopping 5 mm from the edge can result in a different angle value.When using the contact goniometer, the distance from the edge cannot be controlled. Due to variations intopography among break faces on bone fragments, the arm of the goniometer does not consistently makecontact on the surface at the same distance from the edge. When the surface is concave, the goniometer willnot rest against the concavity, and will reach across that expanse to the other side.Conversely, a convex surface prevents the goniometer from reaching the other side of the break. Ca- aldo and Blumenschine (1994) bypassed this by cutting into the molds featuring convex surfaces so thegoniometer could reach the full extent of the surface, which is not a viable solution when measuring theobject directly. In this case, measuring an angle that extends across the whole surface is not possible. Whenthe surface is rugose, the goniometer will make contact with the highest point within its reach at that lo-cation. Such topographical variations on the surface of the break prevent the goniometer from connectingwith most of the surface and it fails to capture a significant amount of intermediate information at a singlelocation.If we were uncertain about a break, we flagged it and categorized it as concave, hinged, surface, edge,or other. If we were unable to measure the break, it was categorized as blocked or other (see Table 6). Ofthe flagged fragments, 108 were measured. Arguably, these measurements do not accurately capture thefracture angle and may not be useful for fracture edge analysis. Table 6: Categorized and uncategorized breaks C ATEGORY C OUNT P ERCENT A LL BREAKS
537 100%N
OT CATEGORIZED
392 73%C
ATEGORIZED
145 27%B
LOCKED
34 6.33%C
ONCAVE
70 13.04%H
INGE
21 3.91%S
URFACE DGE
11 2.05%O
THER
ATEGORIZED ( MEASURED ) 108 20.11%C
ATEGORIZED ( NOT MEASURED ) 37 6.89% T HE GONIOMETER ARM CANNOT MAKE CONTACT WITH THE DESIRED SURFACES T HE SURFACE IS RUGOSE AND HAS A LOT TOPOGRAPHICAL RELIEF T HE EDGE IS ROUNDED OR HINDERS EFFORTS TO MEASURE
Not only does the surface of a break change across the width of the break, angle values can vary alongthe length of the break edge. Some researchers measure at the midpoint along the ridge (Coil et al., 2017;Dibble & Bernard, 1980; Pickering et al., 2005) whereas others use the most extreme angle (Moclán et al.,2019). Capaldo and Blumenschine (1994) define a midpoint, but, it only applies to notches and cannot beextrapolated to all fracture edges because it depends on features that are specific to notches, specificallyinflection points.Finding the midpoint on a break edge is less clear. The midpoint on a bone fracture edge as describedby Pickering et al. (2005) and Coil et al. (2017) is likely the approximate midpoint as opposed to an exactmidpoint. The break edge on each break face on a bone fragment can be viewed as a contour. Establishingthe endpoints for the contour is the first challenge. Typically, the full length of the break face does notterminate at the same location that the ridge between the periosteal surface and break surface terminates.Once a decision is made as to where the endpoints of the contour will be located, then one can decide whereto take the angle measurement. In regard to the midpoint, one could choose the midpoint of the Euclideandistance between the two end points of that contour or one could choose the midpoint of the full contourlength. In either case, finding the exact midpoint on the physical object is challenging, if not impossible,and time consuming. For example, calipers could be used in the case of the Euclidean midpoint. However,this would require a consistent orientation of the specimen in relationship to the calipers. Finding the most xtreme angle would require approximation as well unless one measures many angles along the edge inorder to identify which one is the most extreme. This requires that the angle can be measured at all locationsand the goniometer is not blocked by features on the specimen at any location.When using 3D models, specific points can be chosen as endpoints for the contour. The Euclideandistance or the contour length can be calculated and a midpoint can be consistently defined and extracted.Though this is beyond the purview of the virtual goniometer, the xyz-coordinates for the exact midpoint canbe input into the virtual goniometer.Whether choosing the most extreme measure or the midpoint, taking a measurement at a single locationis anthropologically arbitrary. Bones are not limited to one instance of fragmentation. If the objective is toidentify the first actor of breakage, then the extrema or midpoints will not be comparable among specimensif additional fragmentation takes place. An alternative approach, that could provide a richer dataset, wouldbe to take multiple measurements along the edge of the break which can be done quickly using the virtualgoniometer.Assuming the angle can be taken and is appropriate to take, it also needs to be precise enough thatit is useful. When measuring platform angles of notch molds using the contact goniometer, Capaldo andBlumenschine (1994) expected measurements to vary up to 5 ◦ . This may not be precise enough. Forexample, Alcántara-García et al. (2006) state that, in general, carnivores produce fracture angles between80 ◦ and 110 ◦ , whereas fracture angles on bones broken by percussion will be <80 ◦ and >110 ◦ , offering a30 ◦ range for assigning carnivores as the actors responsible for breakage. If one chooses a more stringentapproach where fracture angles between 85 ◦ and 95 ◦ (Pickering et al., 2005), generally labeled as dry breaks,are excluded, this diminishes the ranges for carnivores to 5 ◦ and 15 ◦ . These are small windows for such alarge error range. In fact, one of those ranges is equal to the expected error range. In a test that looked at thereliability of the contact goniometer when measuring knee angle in a medical context, Draper et al. (2011)noted that in order to remain within an error range of 5 ◦ , the location of the measurement had to be within2mm of the actual center of the patella. Not only is this equal to the error range presented by Capaldo andBlumenschine, these results also highlight the importance of the location where the measurement is taken.Being able to control how a measurement is taken is key to extracting anthropologically useful informa-tion. The virtual goniometer does not suffer from the limitations of the contact goniometer. It is flexible andcan better meet the specific needs of the user who can select the location and which portion of the objectto measure. It captures the entire surface in the specified region. The radius (i.e. the distance from theedge) of the patch can be specified. The patch can be selected by eye using the click-and-drag method or byspecifying the parameters through numerical inputs. The option to select the location by clicking and thenspecifying the radius is also available when a consistent radius is required. The segmentation parameter canalso be adjusted in order to tweak how the virtual goniometer segments the specified patch into two planes.A final consideration is the individual taking the measurement. The way in which individual analystsmitigate topographical changes on the surface introduces variation to the measurements taken. This canresult in major inconsistencies across analysts and even among a single analyst’s repeated measurements.Though we did not test inter-observer error, we have demonstrated that the precision and accuracy of the vir-tual goniometer far surpasses the capabilities of the contact goniometer. Additionally, the virtual goniometerautomatically exports the segmentation parameter, a measure of the goodness of fit, the measurement loca-tion, radius, and vertices, none of which can be garnered by using the contact goniometer. By providingnumerical metadata for the measurement and 3D visualizations, inter- and intra-observer discrepancies canbe easily identified and measurements can be replicated with precision. The virtual goniometer resolves theinherent limitations of the contact goniometer and gives researchers a tangible way to discuss how best toemploy goniometry to address questions in anthropology. Conclusion
The virtual goniometer offers a more precise and consistent way to measure angles and extract data that can-not be extracted manually. Details of the data are recorded such that they can be clearly communicated andreplicated. The virtual goniometer provides flexibility that allows the user to adjust parameters and choosean approach that is dependable and useful. Scanning objects is becoming mainstream in anthropology andthe virtual goniometer is a logical next step that is easy to integrate into research that uses 3D models. Thepurpose of this project was to introduce the virtual goniometer and demonstrate its capabilities. The nextstep for future research is to explore where and how to take measurements on different material types, in-cluding analysis of appropriate scan resolution, to address specific anthroplogical questions.
Acknowledgments
Thank you to Scott Salonek with the Elk Marketing Council and Christine Kvapil with CrescentQuality Meats for the bones used in this research. We thank the hyenas and their caretakers at the Milwaukee CountyZoo and Irvine Park Zoo in Chippewa Falls, Wisconsin and the various math and anthropology student volunteers whobroke bones using stone tools. Thank you to Sevin Antley, Chloe Siewart, Mckenzie Sweno, Alexa Krahn, MonicaMsechu, Fiona Statz, Emily Sponsel, Kameron Kopps, and Kyra Johnson for helping to break, clean, curate and pre-pare fragments for scanning. Thank you to Cassandra Koldenhoven and Todd Kes in the Department of Radiologyat the Center for Magnetic Resonance Research (CMRR) for CT scanning the fragments. Thank you to SamanthaPorter with the University of Minnesota’s Advanced Imaging Service for Objects and Spaces (AISOS) who scannedthe crystal. Bo Hessburg and Pedro Angulo-Umaña worked on the virtual goniometer. Pedro and Carter Chain workedon surfacing the CT scans. Thank you Matt Edling and the University of Minnesota’s Evolutionary AnthropologyLabs for support in coordinating sessions for bone breakage and guidance for curation. Thank you Abby Brown andthe Anatomy Laboratory in the University of Minnesota’s College of Veterinary Medicine for providing protocolsand a facility to clean bones. Thank you Henry Wyneken and the Liberal Arts Technologies and Innovation Services(LATIS) for statistical consulting.
Funding Information
We would like to thank the National Science Foundation NSF Grant DMS-1816917 and theUniversity of Minnesota’s Department of Anthropology for funding this research.
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