BULLETIN OF MATHEMATICAL BIOPHYSICS

VOLUME 2, 1940

A NOTE ON THE HOROPTER

ALSTON S. HOUSEHOLDER THE UNVE~SITY OF CHICAGO

By assuming the fixity (but not the symmetry) of correspond- ing points on the two retinae, it is possible to derive the equation of any horopter when one is known. In particular when, as experiment sh~)ws, one horopter is linear, then all horopters must be conics. These have the form given by Ogle, but whereas Ogle leaves one parameter undetermined at each fixation, on our assumption the only arbitrary parameter is determined by the position of the linear horopter.

I t is well known, and easily verified, tha t when any fixed object, not too f a r away, is fixated by the two eyes, then near objects as well as d is tant objects appear double, whereas objects in some region which includes the object fixated do not appear double. That is, when the eyes are held fixed, the two retinal images of any given object will or will not be fused, depending upon its position relative to the ob- server and to the object he is fixating. Therefore i t may be supposed tha t to each point on one ret ina there corresponds a unique point on the other ret ina, called its "corresponding point," which is such tha t if the same object is imaged on these two corresponding points the two images will fuse. Thus there is established a one-to-one corres- pondence between the points on the two retinae, and if the optical system of the eye were specified, and the correspondence were known, then, the eyes remaining fixed, one could determine the locus of points in space whose images would fuse, i.e., whose images would fall on corresponding retinal points. This locus would be a piece of an ordi- na ry surface in space, known as the horopter.

Actually it is known tha t to a given point on one ret ina there corresponds a region of points on the other re t ina in the sense tha t fusion can occur, although the fusion becomes more and more difficult as the region is departed from. Certainly the boundaries of this region are not clearly defined. We may therefore think of each point on one ret ina as having a unique, " t ru ly" corresponding, point on the other at which fusion is easiest, and tha t as this point is departed f rom in any direction the fusion becomes increasingly more difficult and the spli t t ing of the images increasingly easy. We shall therefore continue to speak of ret inal correspondence as a one-to-one correspondence. Images of the same object which fall upon points t ha t do not corres- pond in this sense will (as is customary) be said to be disparate, even

135

136 MATHEMATICAL BIOPHYSICS

though they may fuse for the observer. Also we may dist inguish among nasal or medial dispari ty, temporal or lateral d ispar i ty and vertical dispari ty, the adject ives being self-explanatory.

Depth perception may be "explained" in terms of the horopter by saying tha t objects whose images are medially d ispara te appear far ther , those whose images are laterally disparate appear closer, than points on the horopter , this being t rue whether or not fusion has occurred. This, however, is not a t rue explanation, and for several reasons. First , it says nothing about the subject ive fo rm of the horop- te r and how it is achieved. Second, while we may grant tha t medially d ispara te images should be quali tat ively different f rom lateral ly dis- para te images, whe ther or not these are fused, nothing is said about how these quali tat ive differences become organized into a unified, three-dimensional space with quant i ta t ive a t t r ibutes (Roelofs, 1935). Finally, the manner in which fusion occurs remains a mystery . Never- theless, the notion of the horopter is a useful descriptive concept in discussions of depth perception.

Let us res t r ic t ourselves in the present discussion to the con- siderat ion of the horopter curve, which is the intersection of the horopter surface with the plane of the visual axes. We need to con- sider then only those light rays which lie in this plane, and this restr ict ion will be taci t ly assumed hereafter . Now it is convenient to suppose, first, tha t the light rays f rom any point to its retinal image can be represented by a s t ra ight line passing through some point in the bulbus - - say the nodal point - - which is common to all rays, and second, tha t the position of this point does not va ry sensibly with respect to the ret ina as a result of accommodation. This is equivalent to saying that this common point can be regarded as the center of a pencil of lines through the points of the retina, tha t each of these lines gives the project ion of a point in space upon a retinal point which is the optical image of the space point, and tha t the Euclidean proper t ies of this pencil as determined by the retinal points are in- var ian t under changes in accommodation, as well as position. Where the i m a g e is not in focus on the re t ina we must regard the line as passing through the center of the circle of confusion on the retina.

We now make a second assumption to the effect tha t the relation of correspondence, as defined above, is invar iant under changes in position and accommodation. Thus if the retinal point P of the left eye corresponds to the retinal point Q of the r ight eye when the eyes are in one position, they will correspond when the eyes are in any position whatsoever.

One a t t empt to give a rational derivation and mathematical rep- resentat ion of the horopter curve has resulted in the circles of Vieth

ALSTON S. HOUSEHOLDER 137

and Mfiller, and is made on the basis of the fu r ther assumption that the eyes are essentially symmetrical , wi th corresponding rays (lines leading to corresponding points on the two retinae) making equal angles with the visual axes. These circles are therefore coaxal circles - - the centers of the pencils being regarded as fixed in space - - the common intersections of the circles being the centers of the pencils. Unfor tunate ly , these circles fail to represent adequately the experi-

m e n t a l results, for i t is found that in all cases the empirical horopter has less curvature toward the observer, and in fac t for dis tant vision the horopter actually has its concavity directed away f rom the observer. Lying intermediate to the horopters with concavity directed toward, and those with concavity directed away f rom the observer, there is found to be one which is a s t ra ight line (Southall, 1937; Ogle, 1938).

The simplest family of curves possessing the propert ies here des- cribed is evidently a family of conics, and Ogle (1938) has derived the equations of the horopters on the assumption that they are conics passing through the centers of the visual pencils. Fo r horopters which are symmetr ic with respect to the frontal plane these conics depend upon a single parameter , aside f rom those which are directly accessible, viz., the distance between the eyes and the location of the fixation point. This parameter , H, would be zero fo r the Vieth-Mtiller circles, bu t empirically it is in general positive. Ogle's derivation does not relate the values of H for different fixations of the same subject.

This note is intended to point out tha t on the assumptions as to the invariance of the form of the pencils to the ret inae and of the relation of correspondence, it is possible, given the horopter curve for one fixation, to predict the form of the curve for any fixation. In part icular, f rom the fact tha t one horopter curve is linear, it follows easily tha t the other curves must be conics of the type described. This is t rue whether or not the linear horopter is perpendicular to the frontal plane.

Let the centers of the pencils at the two eyes be located a t (_+a~, 0) ; let a and n - - a be the inclinations (in the sense of elemen- t a ry analytic geometry) of the visual axes for the par t icular sym- metr ic fixation which yields the linear horopter ; let fl be the inclina- tion of the linear horopter ; let 01 and 02 be the angles made with the visual axes by a pair of corresponding rays (lines of the pencil passing through corresponding points on the two re t inae) . The assumption or invariance means simply that the functional relation between 01 and 6~. is independent of the position of the two eyes. Hence, since in this position the two pencils are in perspective (in the sense of project ive geometry) , the pencils remain project ively related, and therefore the the locus of their intersections is a lways a conic.

138 MATHEMATICAL BIOPHYSICS

A n y r a y of t he one penci l can be w r i t t e n in the f o r m

- - x t a n a + (1 -~ ~ l )y - - a t a n a = 0 ,

a n d a n y r a y of the o t h e r in t he f o r m

x t a n a ~ - ( l ~ - ~ 5 ) y - - a t a n a - - - - 0 .

T h e condi t ion t h a t the r a y s be c o r r e s p o n d i n g is t h a t t h e y i n t e r s e c t on the l ine

x t a n fi - - y - ~ a t a n a = 0 ,

a n d hence t h a t t he d e t e r m i n a n t of the t h r e e l i n e a r equa t ions shal l van i sh . Th i s condi t ion is equ iva l en t to t h e condi t ion

~ l / s i n ( a - fi) = ~ / s i n ( a + fi) = ~,

w h e r e , is a n a u x i l i a r y p a r a m e t e r . H e n c e we h a v e the equa t ions f o r c o r r e s p o n d i n g l ines in the f o r m

- - x t a n a - t - [1 - [ - ~ s in ( a - - f i ) ] y - - - a t a n a = O ,

x t a n a + [ 1 - - ~ s i n ( a + fl ) ] y - - a t a n a = O .

T h e r e f o r e

- - ~ s in (a - - fl) t a n a t a n ~1 . . . .

secfa -[- ~ s in (a - - fl) '

(1)

(2)

t a n 0~ = . - - ~ s in (a -[- fi) t a n a sec 5 a - - ~ s in (a q- fi) "

F r o m th i s i t fo l lows t h a t

s in (a-{- fl) cot 0.~ - - s in (a - - fi) cot ~1 = 2 cos a cos fl , (3)

wh ich is a n a l o g o u s to Ogle ' s r e l a t i on (10) . In t h e spec ia l case t h a t fl = 0 , th i s r e l a t i o n b e c o m e s

cot 05 - - cot 01 = 2 cot a . (4)

On the bas i s o f o u r a s s u m p t i o n s , Ogle ' s p a r a m e t e r H f o r the s y m m e t r i c h o r o p t e r is equal to 2 cot a .

Suppose , now, t h a t t he l e f t eye is r o t a t e d t h r o u g h a n ang le coi, a n d the r i g h t eye t h r o u g h a n ang le 0)5. F o r s y m m e t r i c f ixa t ion these ang l e s a r e equal a n d oppos i te . The equa t ion of t h e c o r r e s p o n d i n g l ines a r e t h e n

t a n a + t a n c o l + ~ s in (a - - fl) t a n ~ol y t a n (a + 01 + COl) -~ - - _ _

1 - - t a n a t a n c o l + v s i n ( a - - f l ) x + ~ '

and

ALSTON S. HOUSEHOLDER 139

- - t a n a + t a n ~o~--~ sin (a+f l ) t an ~o~ y t an (n--a+0~+~o2) -

1 + t a n ~ t an ~o~--v sin ( a+f l ) x- -a ' o r

(x -F a) ( t an a ~- t an ~o~) - - (1 - - t an a t an ~o~)y + v sin (a - - fi)

}( [ (x + a) t an o) 1 - - y ] : O, (5)

(x ~ - - y: - - a :) t an fi -~ 2xy = 0 , (8)

which is a h y p e r b o l a whose cen t e r is a t the o r ig in and whose t r a n s - ve r se axis has the inc l ina t ion n / 4 - - fl/2.

F o r s y m m e t r i c convergence a t the po in t [0, a t an ( a - - ( o ~ ) ] , w h e r e col---- - - ~a~ = co, the t a n g e n t to the h o r o p t e r a t th i s po in t is

y ~-- (x t an fi cot a - - a) t an ( a + ( o ) ,

so t h a t the slope h e r e is t an fl cot a t an (a - - co). All these t a n g e n t s pass t h r o u g h the po in t ( - - a t an a cot f l , 0 ) , which is the i n t e r c e p t of t he l i nea r h o r o p t e r w i th the x-axis .

Essen t i a l l y t h r e e a s sumpt ions a re involved in the d e r i v a t i o n of

(x - - a) ( t an a - - t an ~o2) + (1 + t an a t an o~_~) y + v sin (a + fl)

X [ ( x - - a ) t an c o ~ - - y ] = 0 .

B y e l imina t ing ~ be tween these two equa t ions we ob ta in the equa t ion of the locus of in te r sec t ions of c o r r e s p o n d i n g l ines which is the horop- te r . Th i s can be w r i t t e n in the f o r m

I (x+a)sin(a+a)l)--ycos(a+col) [ (x +a)sin(o1--ycose91]sin(a--fl)

( x - - a ) sin (a- -co , ) + ycos (a--~o2) [ (x--a) sin(o2--ycosco2] sin (a + fi)

= 0 . (6)

Thus whe n the l i nea r h o r o p t e r is known, the h o r o p t e r c o r r e s p o n d i n g to a n y o the r f ixa t ion po in t can be de t e rmined .

The h o r o p t e r is s y m m e t r i c w h e n and only w h en the coefficient of xy van i shes in the expans ion of (6 ) . Th is coefficient is found, a f t e r some t r i g o n o m e t r i c t r a n s f o r m a t i o n s , to be

sin 2 ~ sin (fi - - o ) 1 - - ~ o ~ ) .

Hence the h o r o p t e r is s y m m e t r i c w h e n

fi - - ~)1 - - 0)2 = 0 (7)

Th is is t rue , in pa r t i cu l a r , when the f ixat ion is s y m m e t r i c and fl = 0. In genera l t he locus of f ixa t ion poin ts f o r s y m m e t r i c h o r o p t e r s is the cu rve

140 MATHEMATICAL BIOFHYSICS

equation (6). One is the empirical fact that some horopter curve is linear. Another is tha t accommodative movements can be neglected. Presumably the validity of this assumption can be estimated by a consideration of the optical system of the eye.

The thi rd assumption is to the effect tha t the relation of corres- pondence is a fixed relation. This cannot be verified directly, but can only be inferred f rom theoretical considerations - - unless, indeed, histological examinations can reveal the individual fiber pa thways to the fusion center, wherever this may be. The theoretical neural mech- anism proposed by Verhoeff (1925) seems to imply such a fixity. And if equation (6) is found to agree with experiment, this fact would lend indirect empirical support to the notion.

Actually equation (6) is based upon a fu r the r approximation, for it has been made on the assumption tha t the centers o2 the pencils of rays were also the centers of rotat ion of the two eyes. This, of course, is not the case, though the discrepancy would be negligible un- less the fixation point is near. The exact equation to replace (6) can be worked out in the same way, but is considerably more complicated.

This work was aided by a g ran t f rom the Dr. Wallace C. and Clara A. Abbott Memorial Fund of the Universi ty of Chicago.

LITERATURE

Ogle, Kenneth N. 1938. "Die mathematische Analyse des L~ingshoropters," ArcK f.d. ges. Physiol., 239, 748-66.

l~oelofs, C. Otto. 1935. "Die optische Lokalisation." Arch. Augenheilk, 109, 395- 415.

Southall, James P. C. 1937. Introduction to physiological optics. New York: Oxford University Press.

Verhoeff, F. H. 1925. "A theory of binocular perspective." Am. J. Physiol. Op- tics, 6, 416-48.