Quantification of range of motion and wing shape

We quantified range of motion in the avian wing using three metrics: (i) extension ROM, (ii) bending capability, and (iii) twisting capability. Extension ROM was defined as the space occupied by the outer boundary of possible elbow angles against possible manus angles. To determine extension ROM for each species, the set of all possible elbow angles (X axis; ranging from 0° to 180°) was plotted against all possible manus angles (Y axis; ranging from 0° to 180°). Because we found intraspecific variation to be minimal, we pooled data from all specimens of a species. We then generated a convex hull that encapsulated all possible elbow and manus angles. To facilitate comparison of these shapes across species, we converted all convex hulls into Coe objects (12) and then used elliptical Fourier analysis (49) to quantify outline shapes without the use of homologous landmarks. Because we sought to test hypotheses of extension ROM shape, we first standardized all outlines to the same size and location using generalized Procrustes analysis (50). Elliptical Fourier coefficients that described >95% of shape variance across species were retained for further analyses involving extension ROM. The area of each extension ROM shape was also computed by taking the total area of occupation in the elbow angle × manus angle plots.

Bending capability was defined as the combined capability to elevate or depress the hand-wing at the manus joint, evaluated at each point of extension ROM. For each species, elevation (positive) and depression (negative) (both on Z axis) were simultaneously plotted in 3D against elbow angle (X axis) and manus angle (Y axis). To encapsulate the ROM, an α-hull [3D generalization of a convex hull (51)] was fit to the data (fig. S3). To describe, visualize, and compare bending capabilities across species, the vertices of each α-hull were extracted and then separated according to whether they corresponded to elevation or depression. We fit regularized neural networks (52, 53) to each set of vertices to resolve the relationship between elevation (or depression) capability and wing extension (elbow angle and manus angle, jointly). This method not only guards against potential outliers resulting from digitization error (amounting to a low-pass filter) but also allows for the interpolation of data missing at any point within the extension ROM. Each regularized neural network was trained and then cross-validated within its dataset following a machine-learning framework using the Caret package (54). In cases where datasets were too sparse to use regularized neural networks, locally estimated scatterplot smoothing (LOESS), or linear models were used. Last, bending capability was determined by summing the predictions of elevation and depression from trained models for every point of extension ROM. Two key metrics were collected: (i) bending capacity at full extension (the combined ability to elevate or depress the wing when it is fully extended) and (ii) maximum value of bending capacity (the highest value of bending capacity achieved regardless of wing extension).

Twisting capability follows a similar definition to that of bending capability. The combined capability to pronate or supinate the hand-wing was evaluated at each point of extension ROM. All methods of α-hull and model fitting follow the above. Twisting capability across extension ROM, including its value at full extension and maximum value, was computed in a similar fashion.

Wing shape was quantified via a method akin to that used to quantify extension ROM. We first standardized all wing shape Coe objects to the same size and location using generalized Procrustes analysis (50). To standardize wing orientation, the distalmost tip of the wing was set as the “principal point” (49). Elliptical Fourier coefficients that described >95% of shape variance across species were retained for further analyses involving wing shape.

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