Three-dimensional studies of range of motion currently plot joint poses in an "Euler space" whose axes are angles measured in the joint's three rotational degrees of freedom. Researchers then compute the volume of a pose cloud to measure rotational mobility. However, pairs of poses that are equally different from one another in orientation are not always plotted equally far apart in Euler space. This distortion causes a single joint's mobility to change when measured based on different joint coordinate systems and precludes fair comparisons among joints. Here we present two alternative spaces inspired by a 16th century map projection -- cosine-corrected and sine-corrected Euler spaces -- that allow coordinate-system-independent comparisons of joint rotational mobilities. When tested with data from a bird hip joint, cosine-corrected Euler space demonstrated a ten-fold reduction in variation among mobilities measured from three joint coordinate systems. This new quantitative framework enables previously intractable, comparative studies of articular function.

Data Table 1. Density of uniformly distributed random poses throughout ZYX Tait-Bryan Euler space, ZYZ proper Euler space, cosine-corrected Euler space, and sine-corrected Euler space. 1 million uniformly random quaternions were generated using MATLAB r_2019a and converted to each space. All complete 10x10x10 degree cubes in each space were surveyed and the number of poses in each was counted. Averages and standard deviations were calculated for each 10-degree strip of cubes along the second axis.

Data Table 2. Helmeted guineafowl hip poses measured from JCS1-3.