Supporting Information

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Fig. 6. Extrapolation data for the circular-arc inducers. The mean settings of angular position and orientation are shown in the Cartesian plane, at each of the six radial distances, with curved SD bars for angular position and SD cones for orientation. The solid curves show the actual extensions of the circular arcs used to define the inducers. The dashed lines correspond to the linear extensions of the inducer tangents at the point of occlusion.

Fig. 7. SDs in probe position and orientation plotted as a function of curvature of the parabolic inducers. Tests of heteroscedasticity revealed a significant dependence of setting variability on inducer curvature in all six cases. This dependence demonstrates a cost of curvature, i.e., a systematic decline in the overall precision with which angular position (Left) and orientation (Right) are represented, with increasing inducer curvature.

Fig. 8. Contour plots of the standardized likelihood surfaces for the fits of the Euler-spiral model to the extrapolation data. The four surfaces in each plot correspond to the four different inducer curvatures. The maximum-likelihood estimates of the parameters (k, g ) are marked with ‘+’ symbols. Twenty-three of the 24 cases yielded negative estimates for g, thereby indicating decreasing curvature along extrapolated contours with increasing distance from the point of occlusion. (See also the statistical tests reported in Table 2.)

Fig. 9. Demonstrating the influence of the spread of the likelihood on the Bayesian posterior. (Upper) Prior distribution on curvature centered on 0, and likelihoods centered on the estimated inducer curvature. The likelihoods have increasing SDs (with increasing distance from the point of occlusion). (Lower) The posterior distributions of extrapolation curvature corresponding to the different likelihoods shown in Upper. The maximum a posteriori estimate of extrapolation curvature asymptotes to 0.

Values of Bayes factors (ratio of marginal likelihoods) for the parabolic model M_p against the circular model M_c. All values are >1, indicating that the parabolic model provides a consistently better fit to the observers’ extrapolation data than the circular model, irrespective of whether the inducers themselves are parabolic or circular.

Values of the log-likelihood ratio statistic for the Euler-spiral model M_es against the circular model M_c (i.e., degenerate case of the Euler-spiral model with g = 0). This statistic is given by twice the log of the ratio of the maximized likelihood values under the two models, i.e., D = 2 log(L_es/L_c), where L_c = maxk l_c(k |D) and L_es = max_{(k , g )} l_es((k , g )|D). Because M_c is nested within M_es, the ratio of the likelihoods cannot be smaller than 1 (and therefore the statistic D cannot be negative). To set a cutoff value, we use the fact that, under the null hypothesis that model M_c is correct, D is asymptotically distributed as a c ² with 1 degree of freedom (the difference in the number of parameters between the two models). Cases where the null hypothesis of the circular model is rejected at the .05 level, and thus the g estimate is significantly different from 0, are marked with asterisks.

Values of Bayes factors (ratio of marginal likelihoods) for the logarithmic-spiral model M_ls against the Euler-spiral model M_es, for the extrapolation data with parabolic and circular inducers. All values are >1, indicating that the log-spiral model consistently provides a better fit to the observers’ extrapolation data than the Euler-spiral model.