The reviewer is mistaken on this point. He says that what I propose is merely a change of coordinate system, and then observes (correctly) that coordinate system changes do not change the attributes of objects, only their coordinates. Consider a point p_xy in a two-dimensional cartesian space, as shown in Figure 3 (A).

Figure 3.

The position of that point could equally be expressed in polar coordinates of p_rq. This does not of course transform or move the point in any way, it only changes the way in which its location is expressed mathematically. What I propose is not a change of coordinate system, but an actual geometrical transformation from one space to another, i.e. every point p_rq is transformed to a corresponding point q_vq using the equation

which performs the nonlinear compression of radial distance from the range 0 to infinity in cartesian space, to the finite bounded range of 0 to p, as would be represented by a vergence angle. In other words all radial distances are compressed in this nonlinear way, while the original polar angles are preserved. This transformation converts the vertical and horizontal grid lines in Figure 3(A) into something like the curved grid lines shown in Figure 3(B). Again, the reviewer is mistaken when he says that "fronto-parallel horizontal and vertical lines, which are unaffected by depth compression, would remain straight and not curved". This would be true only of spherical shells centered on the observer. In fact, back in the 2-D example, the only lines that remain straight after this transformation are the ones that pass exactly through the origin. The curvature on all other lines can be explained intuitively by the fact that the middle of the line is near the observer, and therefore its range is changed very little by the compression, while the ends of the line are far from the observer, and their radial range is more compressed by the transformation, and therefore they approach closer to the observer than in Euclidean space.

A three-dimensional version of the described 2-D transformation works exactly the same way by simply adding an elevation angle. This transformation does not project points from three-dimensional space onto a spherical surface, as the reviewer suggests, but projects them from anywhere in infinite Cartesian space to somewhere within a finite bounded sphere, with nearer points mapping close to the center of the sphere, and farther points mapping closer to the periphery of the sphere. Points that were originally in a straignt line in cartesian space are no longer in a line in this transformed space, therefore collinearity is not preserved in this transformation. In the model I propose however, the neural connections that represent collinearity, i.e. the connections that duct the propagation of boundary and surface signals, themselves follow the curved lines in Figure 2(B), and thereby promote boundary completion in a manner that is consistent with cartesian collinearity, even though in the transformed space, they actually propagate along smooth curves, rather than collinear lines. This offers a system that can simulate collinear boundary completion in an infinite space, but using a finite transformed space.