In symmetric occlusion, where
α =
β, the geometry of the model is simplified. As a consequence of the mirror symmetry of left and right vector fields, the generated trajectories are symmetric with respect to the GC-MP triangle median, which lies on the y-axis. The V
_{x} coordinate becomes 0. Equations
z_{L} (
Equation 7) and
z_{R}(
Equation 9) as well as
k_{L} (
Equation 8) and
k_{R} (
Equation 10) become equal. Therefore, the symmetric left and right trajectories meet on the y-axis at the cross-point with the GC-MP triangle median. Such a property holds also in asymmetric occlusion, where
α ≠
β.
The location of the IT maximum on the GC-MP triangle median is an important general property of field-model trajectories, consistent with empirical data obtained by Fantoni and Gerbino (
2001), who used a multiple probe procedure to study amodally perceived trajectories in the symmetric occlusion case (
α =
β= 45°) and in two asymmetric occlusion cases (
α = 35°,
β = 55°;
α = 25°,
β= 65°). In each case, observers adjusted three differently oriented probes tangent to the amodal trajectory. The orientations of the three probes (relative to the MP line) were as follows: 0° (i.e., parallel to the MP line), −
α/2 (i.e., the bisector of the
α angle);
β/2 (i.e., the bisector of the
β angle). Two conditions of retinal size were studied. We used long thin lines as probes to prevent a risk that is present with short probes (or dots). If a short probe were treated as a fragment belonging to the same contour defined by T-junction stems, the interpolated trajectory would be perturbed. Predictions of various models were evaluated by comparing the average intersections of tangent probes with the GC-MP triangle median. Independent of symmetry, probes intersected the median in the following order of closeness to MP: first, the MP-parallel probe; second, the −
α/2 probe; third, the
β/2 probe.
Figure 26 illustrates that such an ordering is predicted by our field model but not by the circular-arc model and by the Hermite spline, which, for small asymmetry, is a good approximation of elastica (as suggested by
Sharon et al., 1997).
Fantoni and Gerbino (
2001, slide 18) compared the distribution of 18 average localizations of tangent probes (3 probe orientations × 3 occlusion cases × 2 retinal gaps) to predictions of the field model, the circular-arc model and the Hermite spline. Other models that generate approximations to elastica curves do not make obvious predictions and were not considered. A small percentage of variance was explained by the Hermite spline (3%) and the circular-arc model (9%). Instead, more than 75% of the variance was explained by our field model, depending on the specific value of GC-MP contrast (see slide 17 to evaluate the model’s robustness). To be fair with competing models, we set GC-MP contrast to different values: 0.95 for the comparison with the Hermite spline and 0.45 for the comparison with the circular arc. The corresponding percentages of variance explained by our field model were 85% and 76%, respectively.
Like circular-arc models, the Bézier spline, and possibly some elastica-based models (but differently from the Hermite spline), the field model predicts that in all occlusion cases the whole family of interpolated trajectories corresponding to different GC-MP contrast values lies inside the GC-MP triangle. It is easy to demonstrate that when (α − β) is large, the Hermite spline goes outside the GC-MP triangle.
Another important property of our model is that the penetration of the trajectory decreases as the sum of the two GC-MP angles increases (
Figure 27). This is consistent with a quantitative formalization of relatability and with empirical data by Gerbino and Fantoni (
2000). We found that the amount of penetration into the GC-MP triangle, estimated by the probe localization procedure, increases as the interpolation angle
ϑ increases. Such a feature is shared by circular-arc models but not by spline models that maintain a constant penetration across different sizes of the interpolation angle.
Our algorithm generalizes from symmetric to asymmetric occlusion cases by approximating a shearing transformation, very common in nature (
Thompson D’Arcy, 1917/1942). The simple shear of a given symmetric trajectory determines a whole set of asymmetric trajectories referred to
x_{new},
y_{new} coordinates obtained by tilting the y-axis of original rectangular coordinates by a variable angle
w (
Figure 28). The algorithm does not implement a perfect simple shear, although the approximation is high at small asymmetries. In the present version of the algorithm, an increasing asymmetry produces an increase of the GC-MP contrast value, which makes the interpolated angle sharper. This feature is desirable given that extreme asymmetric cases correspond to the reduction of the IR and to short distances between a fragment endpoint and the GC vertex.