382 John R. Platt 



the analog position-sensing circuits. With mosaic detectors, the sensitivity to 

 imperfections in a hne pattern is not affected by analog errors. 



In these theorems, a discriminated boundary will be perfectly straight if 

 the longitudinal Z-axis of the eyeball passes through it and if there is no Z 

 rotation during the scanning motion. Eye movements about the Z axis during 

 fixation seem not to have been measured as yet, but it seems unlikely that 

 physiological tremor about X and Y would be unaccompanied by tremor about 

 Z. It will be convenient here to consider the analogue of these theorems for 

 pure rotation about Z, with no X and Y components, and to come back later to 

 the question of how the present theorems will be changed, and what new theorems 

 will be valid if all three rotations are present. 



3. Circularity — During pure rotational scanning about the Z-axis over visual 

 fields containing sharp boundaries, if all the elements in a certain large set 

 fgh • • • k are stimulated in the same way and if the stimulation continues 

 unchanged throughout this kind of scan, then: 



(3a) there is a boundary in the field which is a circle or circular arc as seen 



from the eye ; 

 (3b) elements y^/? • • • k lie on the image of that boundary; and 

 (3c) the-Z axis of the rotation passes through the center of the circle. 



3'. Concentricity — If the elements/g/; • • • A: of Operation 3 are grouped into 

 /• subsets whose elements can be re-excited in the same local patterns by discrete 

 sets of X, Y rotations (in analogy with the Operation of 2') then: 



(3a') there are r concentric circular boundaries in the field; and so on. 



Concentricity is also one of our very delicate discriminations, as is shown 

 by many gunsight designs. 



4. Translational Comparison — During a random scanning drift movement 

 (in X and Y alone) over visual fields containing sharp boundaries, if a certain 

 time pattern of excitation of elements bed • • • is repeated after a certain fixed 

 time-delay (that is, a certain displacement) by elements y^/j • • • in a one-to-one 

 correspondence with the bed • • • excitation pattern, then : 



(4a) there is a stable pattern, fixed or undergoing translation, in the external 



field; and 

 (4b) there is a constant translational separation in the field between points 



whose images fall on elements b and/, e and g, d and //; and so on. 



4'. Translational Periodicity — If the elements bed • • -fgh • • • of Operation 4 

 can be divided into r subsets, where r is greater than 2, each of whose excitations 

 can be duplicated for r different displacements, with the excitations of several 

 subsets simultaneously duplicated for certain displacements, then 



(4a') there is a stable translationally periodic pattern in the field, with r 

 repetitions; and so on. 



Translational comparison is of course the theoretical procedure for estab- 

 lishing a metric in a space of unknown geometry. The precision of translational 

 inter-comparisons between the patterns in the two eyes is the basis of depth 

 perception. Under the most favorable conditions it approaches the same high 

 angular precision that was found in vernier acuity. This fact alone seems to 

 require a physical operation that can create a high-precision translational metric 



