v&ionnes. vol. 30, No. 2, pp. 255-262. 1990 tinted in Great Britain. All rights rcIcNcd
0042-6989/9053.00+ 0.00 Copyright Q 1990Pqamon Praa pk
EVALUATION OF RETINAL ORIENTATION AND GAZE DIRECTION IN THE PERCEPTION OF THE VERTICAL WERNER HAU~TEXN and
HOIST MITTESTAEDT
Max-Planck-Institut fir Verhaltensphysiologie, D-8130’Seewiesen iiber Stamberg, FAG. (Received 3 March 1989; in revised form 8 July 1989) Al&me-The orientation of the median plane of the eye with respect to the head varies with gaze direction according to Listing’s Law. The subjective vertical (SV), however, is known to be only partially a&ted by these involuntary variations of eye orientation. In order to learn more about the compensatory process underlying this finding, six normal-sighted young subjects were tested monocularly and binccularly in eight directions of gaxe. The results show that: (1) the SVs, determined monocularly for both eyes, fall on corresponding retinal meridians, the binocular SV-settings generally lie between the monocular ones; (2) the tilt of the SV is not linearly related to the tilt angle of the median plane of the eye, as hitherto assumed. On theoretical considerations, the dependence of ocular tilt on gaze direction may be decomposed into three components, which are ah treated differently in the compensation process. We interpret these results in the following way: the SV is determined from the sensorially fused image of both eyes and the tilts of the eyes are accounted for by an extra-retinal signal which is common to both eyes. The charactmistics of the compensation mechanism may be explained by an extra-retinal signal which relies on information about gam direction and Listing’s Law. Such a signal might be derived from an efference copy of gaxe direction commands. Subjective vertical
Listing’s law
Ocular torsion
INTRODUCIION
When a vertical line is fixated with the head upright and the eyes straight ahead, its retinal image lies in the median plane of the eye. According to Listing’s law, this plane does not stay vertical in oblique directions of gaze, but is tilted with respect to the vertical by several degrees. Nakayama and Balliet (1977) investigated whether the subjective vertical (SV) were tilted accordingly. To this end, the tilt of the eye and the SV were measured monocularly at five positions of gaze along a horizontal line 27 deg above the horizon. In all three of their subjects, the tilt of the SV increased almost linearly with the tilt of the eye, but much less, namely about 50% of eye tilt on the average. In a qualitative experiment, the authors obtained similar results in other directions of gaze. They inferred that the partial compensation is due to an extraretinal signal which varies linearly with the torsion of the eye. This interpretation implies that the tilt angle of the median plane of the eye relative to the head is known to the brain. It is an open question how an extra-retinal signal may be gained which is proportional to tilt. We quantified the relation between the tilt of the “I
3012-F
Human
Lo&i&ion
Etferena copy
eye and the SV in a larger range of gaze directions, and found it to be more complex than previously known. METHOD5
Eye torsion and the subjective vertical were measured in the eight main directions of gaze at an eccentricity 6 of 25 deg relative to the horizontal “straight-ahead” gaze. In the following, we will denote these directions as shown in Fig. 1. For the quantitative description of tilt values we will use angles $ which we shall name “aberration”. The “aberration of the eye median*‘, tiE, is defined as the angle between the median plane of the eye and the vertical plane through the visual axis; positive when the former is tilted to the right, as seen from behind the subject. Essentially, JIEis the angle at which these two planes intersect at the retina. Correspondingly, the “aberration of the SV”, Jlv, is the angle between the plane through an uppurentiy vertical line, and the vertical plane, both passing through the visual axis. With Listing’s law, we can calculate the theoretical values of tiE which are to be expected
255
WERNERHALJSTEIN and HOIST MITTEUTAEDT
256 V
t
1
. 6
.2
. i
c
I I Y
7*
l
03
-h
‘4
6’ :
Fig. 1. Arrangasnmtand dcaigrdion ofused gaze dirccti~as and coordinates. Gare dimtiom arc ati by positions 1-S; shown as seen by the subject. Stmc~Smphic coordinatw u vertical. h horizontal. Spherical coordimtcs; c eccentricity from straight ahead, j4 polar angle, ckkwise positive.
if the eye is in its primary position when looking straight ahead (see Appendix). We obtain (for ddinition of coordinates see Fig. 1):
x sin(2p)/[l - fg2(6/2)+cos(2/r)]; or approximately, with an error of less than 0.1 deg at all positions with L < 27 deg, and no error at all at the chosen positions: ~g(~E.,lKO) 2 &/2)*M2Cc).
(1)
= 0 deg for direction 1, 3, 5 and 7, Thus )(IEJbro +2.81 deg for direction 2 and 6, and -2.81 deg for direction 4 and 8. In stereographic coordinates: h = fg(C/2)*sin(p)); u = tg(C/2)*cos@); equation (1) becomes: tg(JIE,,& = 2.h.v/(1 +h* - 0’) s 2.h ‘0. (la) iUeasurement of eye torsion Eye torsion was determined photographically by means of &era1 blood vessels in the vicinity of the limbus. The principle of the method is described in Howard and Evans (1963). However, a special apparatus was used which permitted to take accurately matchable photographs from different directions of gaze. Photographs were taken monocularly for the left and right eye at the eight directions of gaze in a scramhkd sequence (3, 8, 5, 2, 7, 4, 1, 6) which was run
through twice. The surrounding was totally dark; illumination during photography was provided by a ring flash. Four photographs were taken with the eyes straight ahead (6 = 0 deg) and the average taken as the reference value. The I//E value at each of the eight eccentric positions was calculated as the mean of the two measurements minus the reference value. An average standard error sE of I/I~ (v = 8) was calculated for each eye from the pair differences. The mean sE for all 12 eyes was 0.33 deg. Torsions were determined from the projected negatives with respect to the image boundaries provided by the camera. There was no headfixed reference in the photographs, but head movements relative to the camera were largely restricted by the use of a dental-impression biteboard and a forehead rest. From the small differences in consecutive measurements we conclude that the stability of head position was very good indeed. However, reproducibility of head position when the subject had left the biteboard, as was the case between the reference measurement and eccentric measurements, might not be equally good. This might have caused an offset in our JIE data. According to Listing’s law, the average I/I~ should be zero, but in five cases it differed significantly more from zero than expected from sE. During photography the subject looked at a fixation mark which was adjusted to an accommodative distance of 1.3 m, equal to the distance of the display for the determination of the subjective vertical. Measurement of the subjective vertical During the measurements of the SV, the subject sat in front of a tangent screen with the head fixed by the same biteboard and in the same position as in the torsion measurements. A display could be presented on the screen at 8 positions along a circle of 1.11 m diameter. The head of the subject was adjusted in such a way that the mid point between the two eyes formed a cone of 50deg aperture with this circle, and the median plane of the head was perpendicular to the screen. Thus, except for the small deviation caused by the interocular separation, the directions of gaze were the same as during the measurement of eye torsion. The display consisted of two red LEDs separated by 6 cm, shining from behind at a fine linen cloth at the surface of the screen. From the front, they appeared as two sharply limited red points of 5 mm diameter. The LEDs were rotatable via
Evaluation of retinal orientation
remote control around the midpoint between them by the subject (S) as well as by the experimenter (E). In otherwise total darkness, the E set the display at an arbitrary angle, and then asked the S to rotate the display until one of the two red points appeared exactly above the other to her or him. The set value was read out on an analog voltmeter. The eight gaze positions were again arranged in the scrambled order, but now the sequence was run through five times. As the angular values on the tangent screen are not identical to the aberration angles, they had to be converted beforehand (see Nakayama & Balliet, 1977). The average of the five SVsettings was taken as $“. An average standard error sy of Jly (v = 32) was calculated from the standard deviations at each position. Measurements were done monocularly right, monocularly left and binocularly on the same day. The sequence of those three conditions was different for each of the six Ss, hence every possible permutation occurred once. The mean sy for all Ss in the monocular condition was 0.65 deg, in the binocular condition 0.55 deg. The Ss were allowed to move their eyes freely between the two points. Thus, in additon to the orientation of the retinal image, the direction of eye movements could possibly have been essential for the perception of the vertical. Therefore eye movements were excluded in an additional binocular experiment with subject RW: she was asked to tixate a point in the center of the display which was continuously visible whereas the two test points were flashed for 20 msec at a rate of 1 Hz. The SV-settings in this situation (mean of 5 at every position) were indistinguishable from the SV-settings with permanently illuminated points and free eye movements (F-value = 1.265, v, = 8, v1= 32). Therefore we assume that our results did not depend critically on the type of display used.
257
Fig. 2. Upper graph: tilt of the eye in the eight directions of gaze (a). Ordinate: eye aberration ST; ab6cissa: gaze directioll.ThemeM!JtaIhdcrror0ftllc&ta~ints(left eye of ~OCsubject) Q given at the top. (o--ojnpr&ts a fit aazxdhg to Listing’s law. The cmdfkknts E,, E,, Eh ofthegtangivmontbeupperkR(~k~crint).Lower
graph: comaponding plot for the aberration Sv of the SV.
median. However, in this example, the relationship between tiy and tiE is not linear, as may be seen more clearly from Fig. 3, where the two
I
l-2
RESULTS
Monocular measurements
Figure 2 shows a typical example of the values of tiE and the monocular SV-settings tiy in the eight directions of gaze (subject MW, left eye). As in the experiments of Nakayama and Balliet (1977), the S does not simply adjust the apparent vertical to the median plane of the eye, but to a retinal meridian at an angle of #,r = ey - eB to this plane. Since JIR is apparently not constant, there must be an extra-retinal signal which accounts for the aberration of the eye
Fig. 3. Relation between eye tilt and tilt of the SV; mme subject as in Fig. 2. Ordinate: SV aberration &, rbacism eye aberration tir. with standard errors.
WERNERHAUSTEINand Horn
258
angles are plotted versus each other: There may be two significantly different values of I/J” at virtually the same value of JIE, e.g. at positions 2 and 7 (r-test, v = 8, P c 0.5%). This result is not only incompatible with a linear function, but with any direct functional relationship between $h and Jly (or $E and IfiR).The same is found with other Ss, though in some the correlation is better; in others, however, it is even worse. Obviously, $” is not a function of $E which in turn depends on the direction of gaze, but both $y and llih depend on the gaze direction, though each in a different way. Regarding $E, this dependence may be deduced from the well-known law of Listing. This law defines a so-called “primary position” of the eye. In normal-sighted subjects, the primary position is close to the position which the eye adopts when, with the head upright, the S looks straight ahead. But deviations of the primary gaze direction from “straight ahead” of several degrees are not uncommon. Ferman, Coliewijn & Van den Berg (1987) found that the primary directions of the two eyes diverge a bit on the average, and this is what we found too. The pitch of the primary direction in the vertical plane depends on the pitch of the head, and therefore on the position which was chosen as “upright head position”. For these reasons the primary direction is unlikely to exactly coincide with the straight-ahead direction in the chosen head position and this will lead to deviations of $E from equation (1). These deviations may be expressed mathematically. If, in its primary position, the eye does not look at h=O, Y=O, but at h=P,, Y = P,, we will theoretically find an additional aberration (see Appendix): t&W, r.+ro /2) = (P;h - P,*u)/(l + P,eh + P;v); or approximately, with an error of less than 0.1 deg at all positions with L < 34 deg, if the primary direction does not deviate more than 15 deg from straight ahead: t&WE. rkm/2) z P,,*h - P&W.
(21
In polar coordinates L and I( we obtain W E.rheO * (360 deg/x ). Wc. 0) x
(P;sin p - Phcos p).
(2a)
This means that a horizontally deviating primary direction will lead to an additional ~08pcomponent in (kE,a vertically deviating primary
MITIEISTAEDT
direction to an additional sin component, Thus, the dependence of d/E on gaze direction should be separable into three components: cI = sin(2p), ch= sin p, c, = cos p; so that *E=EI.c,~E~.ch+E,.c,+E,
(3)
where Et, E,, E, and Em are constants. The component c, follows from equation (I), with E, = 2.81 deg. Component ch is proportional to the horizontal angle of gaze; E,, is 2.22 deg if the primary direction points 1Odeg upwar&. c, is proportional to the amid angle of gaze; E, is 2.22 deg if the primary direction points 10deg to the left. The constant E,,, was allowed for because of possible offsets (see section “measurement of eye torsion”). As all four components are orthogonal, the best-fitting coefhcients E,, Eh, Es and E, may be debunk from the data by simple weighting procedures. In fact, these components allowed a good fit in most cases: only for two of the 12 eyes was the approximation significantly Werent from the data points (F-test, VI= 4, v, = 8, P c 5%). An example of the fit may be seen from Fig. 2 (open symbols). In order to see whether this decomposition might also fit the SV data, and if so, how the two results might compare, we decomposed ev into the same components, i.e. approximated it by the equation: J/“= v,.c,+ v,*c,+ F;c,+
v,*
(4)
As a matter of fact, in 9 of the 12 eyes, the values of J/” were approximated by equation (4) without a sign&ant rest (P-test, vi = 4, v2= 32, P > 5%). Moreover, there is an inviting relation between the components of ev and those of #E. This is illustrated in Fig. 4, where the coefficients V are plotted versus the coe&ients E. In the case of full compensation, the coefficients V shouid be zero, i.e. all points in the plot should lie near the abscissa. This applies to none of the components. If there were no compensation at all, then the coefficients V should be equal to the coefficients E, i.e. all points should lie near the main diagonal. This is indeed the case for the c,-component. Thus, this component is obviously not compensated by an extra-retinal signal. The ch-components, by contrast, do not show any correlation. This means that an extra-retinal signal does indeed exist which has an influence on this component. However, it dues not correctly account for it in general. Concerning cl, the E-coefficients correspond nicely to the value of 2.81 deg expected from Listing’s law (dotted line), whereas the
Evaluation of r&al
orkntation
259
“h
0
R.
EYE
0
L.
EYE
Fig. 4. Relation between the ala&al coc&itats Y (ordinate) and E (&scima) for IIUsix suw. Epch gr8phshowsthecodtkknUfmoneofthe compomnta c,, c,, c, into which $a and &were decomposed on theorctiall grounds (see text). coMaa data pointa reprant kR and right eye of each subject. Bars indicate fstudud errors. Dotted lines io tbc upper right graph indicate the 99% confidcncz interval for the nptim th8t V, = 4, calculated tiom ths rmagc standud error of )(IR(r-test, v = %). Dotted line in the lower graph indicatea expected value of E, according to Listing’s kw.
V-coeflicients, which for full compensation should be zero, are about 20% of that value on average. Thus, this component is compensated to about 80%.
Another interesting result may be drawn from Fig. 4: the ditkences (V - E) are essentially the same for the right and left eye. These difkences appear in the Ggure as the distances of the data points from the 45 deg-line which, taking the standard deviations into account, do not differ signifkantly for both eyes of the same subject (r-test, v = 16, P < l%), except in one case concerning c,.. This means that, but for a constant value, the retinal angle tir = Sy- tiE,
which signifies on which meridian of the eye the SV is imaged, is practically the same for both eyes. In fact, the differences of *a between the right and the left eye do not vary more between eye positions than would be expected from the measurement variances of tiy and SE (F-test, v, = 7, v2= 16, P < 5%) and therefore may be assumed to be constant for a subject. These individual constants, estimated as the mean right-left differences of tis, range from - 0.2 to + 3.5 deg, with a mean of 1.9 deg. It is known (Holmholtz, 19lQ Cogan, 1979) that the corresponding “vertical” meridians of the two eyes while looking straight ahead diverge by about
WERNER HAUSTEIN and HORSTMITTELSTMDT
260
2 deg, opening upwards. Thus the display, set to vertical in both eyes separately, obviously stimulates corresponding retinal meridians. The relatively large range of the mean AJIRis partly due to the offsets in some of our torsion data (see, for instance, the example in Fig. 2), which we believe to contain artefacts. When these offsets are subtracted, the range becomes more narrow, going from 0.1 to 2.4 deg, with an average of 1.68 deg. Summarizing our foregoing results, we should expect that the retinal angle eR consists of the components:
+(~*-~*)*c*+(~,-El).
(5)
The c,-component vanishes because we found that V, = E,. The coefficients V,- E, and V, - E& should be the same for both eyes. In Fig. 5 we fitted the average JI,-values for the right and left eyes of all subjects with two curves according to these restrictions. With one exception in the gaze-down position of the left eye, the fit is very good. Binocular measurements The binocular SV-settings show essentially the same dependence on eye position as the monocular ones. Moreover, they were generally found to lie in between the SV-settings for the left and the right eye. This is also reelected in Fig. 6, where the mean values of tiy of all six Ss are shown for the right eye, left eye and the binocular condition. The binocular values are essentially equal to the average values of the left and the right eye. They are not closer to veridicality than would be expected by this
p- \ 0 \
I\
2
,’ /“-
h,
4
Fig. 6. W-tilts averaged for ail six subjects: right eye monocular, left eye mwmcular (dots like in Fig. 4). and binocular condition (0). Ordinate: mean SV-aberration; abscissa: gaze directions. Dashed curves: fit by theoretical function.
averaging. The same is found if the components c,, c,, and c, are regarded separately. Obviously, no extra information upon the orientation of the display to the vertical (e.g. due to the fact that. in the upright head position, the eyes lie in a horizontal plane) is gained by the usage of both eyes. The simplest assumption which is in agreement with all data is that the orientation of the display is determined from the fused image of both eyes and that the eye position is accounted for by an extraretinal signal common to both eyes. In Fig. 6, the fit by the theoretical curves is also shown. Though the expected deviations (sy/JE) are very small (0.19 deg monocularly, 0.16 deg binocularly), the actual deviations for the right eye and the binocular condition are not significantly larger than those expected. The same holds for the mean torsion values (not shown graphically). However, the fit for the mean tiy of the left eye is much worse than expected, again especially at position 5. Moreover, all three cases where the individual fit by equation (4) was imperfect were left eyes. At present, we have no explanation for this finding. DiSCUSSKlN
\
9
\p’&
,c
8 ,
‘, PO& I
P -
-r
‘0’
Fig. 5. Retinal angles tir averaged for all aix &jccts; right and left eye mpurtoly (dots like in Fig. 4). CMhxatc mean rctinnl angles, abscissa: gaze dirsctions. Dotted curves:iit by theoretical functions with four pnramctcrs estimated from the 16 data points.
I. Comparison with the results of Nakayama and Balkt
At first glance, the present results seem to be at variance with the findings of Nakayama and BalIlet (1977). However, in cases where (see equations 3 and 4):
v,= E,V,I4); and
Evaluation of retinal
(%O because we also found that V,,2 E,, and V,/E,xO.2; we get
orientation
261
(Ph, P,,). On average, 12:is found to be 0.8 from our data, which is not too far away from this condition. tiy = tjE.(V,/EI) + const; We suggest, therefore, that the compensatory i.e. in fact a linear relation between tiy and JIE, mechanism does not rely on information about as was found by Nakayama and Balliet in their the actual tilt of the eye, but on (correct) qualitative experiment, and also in some of our information about gaze direction and on the subjects* eyes. In view of our present findings assumption (1) that the eye follows Listing’s law this linear relation turns out to be a special case and (2) that the primary direction lies in a of a more complex general relation. The above sagittal plane of the head. calculation holds for those positions where our Since the fifties (Mittelstaedt, 1960; for a component decomposition is valid; so far, review see Howard, 1982; Mittelstaedt, 1989)an strictly speaking, at the eight measurement increasing body of evidence has been accumudirections. However, we aiready know that lated to show that the extra-retinal information equation (3), which was derived from Listing’s about the eye position in the head which parlaw, may be extended to the whole field of gaze takes in absolute localization is provided by an in the form (see equations la and 2): efference copy. The assumption that the compensatory signal(s) are derived from an efirence JIEx (360 deg/x) copy of gaze direction commands also fits our x @,./I at, + P;h - Ph.u) + E,; (3a) present data: an efference copy yields the required information only if the motor system with 1, x 1 and E,,,z 0. If we assume that this provides a constant and unique relation between relation may be transferred to the aberration of commands and eye positions. Given that the the vertical we obtain: motor system yields the correct gaze direction, ey 2 (360 deg/n) then the ocular torsion may also be deduced from the commands, if the relation between x (A;*h ‘0 + Pie/l - Pi-u) + v,. (4a) gaze direction and torsion is rather fixed. This We found that A; z 0.2 and P; z Ph. is indeed the case, and the best a priori estiThis extension also fits the quantitative results mation of this relation for both eyes is given by of Nakayama and Balliet which were obtained Listing’s law with a centered primary direction. at gaze positions along a line with u R const. In Additionally, visual localization with respect this case, tiE and tiy in equations (3a) and (4a) to gravity requires that head-centric inforbecome linear functions of the remaining mation is combined with information upon the variable h and thus also linear functions of each orientation of the head to gravity. It is known other indeed. that considerable deviations of the SV from the physical gravity vector occur with larger body 2. Interpretation of the present results tilts (e.g. Mittelstaedt, 1986). However, in the We found that the retinal angle Jlr consists of present study with upright head position, the components according to equation (5). If we main source of aberrations is assumed to be a again extend this result to the whole field of discrepancy between the factual and the gaze, we would expect to find a dependence of assumed primary position, which is compatible with the typical SV-aberrations actually found t,QRon the gaze direction h, u in the form: (except for the small q-components). $Rx(360deg/n)*(k:*h-u +Pt-h)+R,. (5a) At larger body tilts, an additional compliThe constant R,,, means that the SV is on cation arises from the effect that an altered head average not set to the median plane of the eye, position in turn has on eye position, like the but at an angle R,,, to this plane. The actual ocular counterroll at roll tilts of the head. It is dependence of er on eye position, which must still unclear whether this effect is explicitly acoriginate from an extra-retinal signal, is thus counted for in the perception of the SV (Udo de given by (36Odeg/n)-(IF-h-u + P:-h). If A: Haes, 1970). If it is, the compensation may be was 1, such a function would be well-suited (see located at the visual side, transforming retinal equation 3a) to compensate the aberration of an into headcentric coordinates, but also at the eye with a primary direction (0, P,‘)-a hypo- vestibular side or at an intermediate stage, as thetical direction which in general does not has been favoured previously by Mittelstaedt coincide with the actual primary direction (1986, see Fig. 9, 1988).Notably, such a solution
WERNERHAIJSTEINand HORSTM~LSTAEDT
262
would allow extraction of visual information about the vertical in retinal coordinates.
vectors w, and wvImay be replaced by one rotation with vector w, where: w = (w, + w*-w,
Acknowledgemenr-We wish to thank Thomas Eggert for his helpful support in statistical questions.
REFERENCES Cogan, A. I. (1979). The relationship between the apparent vertical and the vertical horopter. Vision Research, 19, 655-665.
Fennan, L., Collewijn, H. & Van den Berg, A. V. (1987). A direct test of Listing’s law-I. Human ocular torsion measured in static tertiary positions. Vision Reseurch, 27, 929-938.
Haustein, W. (1989). Considerations on Listing’s law and the primary position by means of a matrix description of eye position control. Biological Cybernetics, 60,41 I-420. Hehnholtz, H. von (1910). Hundbuch der physioiogischen Optik. Vol. III, p. 3. Hamburg: Voss. Howard, I. P. (1982). Human visuuf orientation. New York: Wiley. Howard, I. P. d Evans, J. A. (1963). The measurement of eye torsion. Vision Research, 3, 447-455. Mittelstaedt, H. (1960). The analysis of behavior in terms of control systems. In ScbatTner, B. (Ed.), Transactions o/the jijih cwrference on group processes (pp. 45-83) (see also revised version in Mittelstaedt 1989 below). Mittelsmedt, H. (1986). The subjective vertical as a function of visual and extraretinal cues. Aclu Psychologieu, 63, 63-85.
MitteIstaadt, H. (1988). The information processing structure of the subjective vertical. A cybernetic bridge between its psychophysics and its neurobiology. In Marko, H., Hauske, G. 8; Struppler, A. (Eds.), Processkg strutlures for perception and action (pp. 217-263). Weinheim: VCH. Mittistaedt, H. (1989). Basic solutions to the problem of head-centric visual localization. In Warren, R. & Wertheim, A. H. (Eds.), The perception und conrrol of egomofion. Hillsdale, New Jersey: Erlbaum. Nakayama, K. & Balliet, R. (1977). Listing’s law, eye position sense, and the perception of the vertical. Vision Research,
17, 453-457.
Udo de Haes, H. A. (1970). Stability of apparent vertical and ocular countertorsion as a function of lateral tilt. Perception and Psychophysics, 8, 137-142.
An advantageous way to treat rotations mathematically is to use rotation vectors. Their application for the description of eye positions and Listing’s law is described in detail in Haustein (1989). Let the “reference position” of the eye .be the straight ahead position. Any other eye position may then be deBned by the axis about which and the angle a by which the eye may be rotated directly from the reference position into thii position. A vector which points into the direction of this axis and has a length of fg(a/Z) is ddined as the rotation vector. The rotation vector is especiaIly useful for the caIculation of combined rotations: two consecutive rotations with
(Al)
w=(r-pPq.p+tq,q-fP)l(l+pqf). One may check that
2r/(l - r2) = rg(S,) = 2(w, + w,w,)/(l - w; - w$ + wf,.
(A2)
This formula gives the general relation between an eye position in the rotation vector representation and its abcrration #r. During distant fixation with the head 6xed, eye positions follow Listing’s law. This law states that from a certain eye position, the so-called “primary position”, any other actually occurring eye position may be reached by a rotation about an axis orthogonal IO the visual axis. If the primary position coincides with the reference position, then Listing’s law impIied that the rot&ion vectors a for all occurring eye positions tie in the yz-plane, i.e. have no x-components. In the chosen coordinate systems, the rotation vector a therefore reads: a = (0, -v, -h)=(O,
-cosp,
-sinr).rg(t/2).
Substituting a for w in equation (A2) we obtain: tg(+,) = 2.h .v/(l + hz - vz). If the primary position does not coincide with the reference position, the rotation vectors are no longer orthogonal to the x-axis in general. Rather they may be expfesscd as a combination of a rotation (0, p, q) and a following rotation (0, -P,* - P*). where Pb and P, are the coordinates of gaze direction in the primary position (see Haustein, 19%9).This combination yields the rotation vector: b=(p.P,-q.P,,,p
APPENDIX
x w&(1 -w,,w,).
We will use the rotation vector to calculate the aberrations $E which are to be expected according to Listing’s law. The median plane of the eye remains perpendicular (i.e. aberration tid = 0) if, starting from the straight ahead position, the eye is first rotated about a horizontal axis which is orthogonal to the median plane (y-axis) and then about the vertical axis (z-axis). An aberration qE occurs ii the eye is beforehand rotated about the visual axis (x-axis) by an angle tiE. These three rotations may readily be described by rotation vectors: in the cartesian head-fixed xyzcoordinate system, w, = (I, 0.0) with r = rg(tiE/2)+ w2 = (O,p, 0) and w, = (O,O, q). Applying equation (Al) two times we obtain for the total rotation:
-P,,q
-P,)l(l
-p.P,-q.Ph).
We find that: b,-b;P,+b;P,=O.
(A3)
This may be compared to an “iw’ Listing rotation (0, -v, -h) whkh is pm&cd by an additional tomional rotation (d, 0.0). with d = rg(A# /2), yielding a rotation
vector: c=(d,
-v - hd, -h
+ vd).
Substituting c for b in equation (A3) we obtain: d+(v+hd).P,+(-h
tvd).P,,=O,
or d = tg(A+/Z) ==(P;h
- P,.v)/(l
+ Ph.h + P;u).
0042-6989/9053.00+ 0.00 Copyright Q 1990Pqamon Praa pk
EVALUATION OF RETINAL ORIENTATION AND GAZE DIRECTION IN THE PERCEPTION OF THE VERTICAL WERNER HAU~TEXN and
HOIST MITTESTAEDT
Max-Planck-Institut fir Verhaltensphysiologie, D-8130’Seewiesen iiber Stamberg, FAG. (Received 3 March 1989; in revised form 8 July 1989) Al&me-The orientation of the median plane of the eye with respect to the head varies with gaze direction according to Listing’s Law. The subjective vertical (SV), however, is known to be only partially a&ted by these involuntary variations of eye orientation. In order to learn more about the compensatory process underlying this finding, six normal-sighted young subjects were tested monocularly and binccularly in eight directions of gaxe. The results show that: (1) the SVs, determined monocularly for both eyes, fall on corresponding retinal meridians, the binocular SV-settings generally lie between the monocular ones; (2) the tilt of the SV is not linearly related to the tilt angle of the median plane of the eye, as hitherto assumed. On theoretical considerations, the dependence of ocular tilt on gaze direction may be decomposed into three components, which are ah treated differently in the compensation process. We interpret these results in the following way: the SV is determined from the sensorially fused image of both eyes and the tilts of the eyes are accounted for by an extra-retinal signal which is common to both eyes. The charactmistics of the compensation mechanism may be explained by an extra-retinal signal which relies on information about gam direction and Listing’s Law. Such a signal might be derived from an efference copy of gaxe direction commands. Subjective vertical
Listing’s law
Ocular torsion
INTRODUCIION
When a vertical line is fixated with the head upright and the eyes straight ahead, its retinal image lies in the median plane of the eye. According to Listing’s law, this plane does not stay vertical in oblique directions of gaze, but is tilted with respect to the vertical by several degrees. Nakayama and Balliet (1977) investigated whether the subjective vertical (SV) were tilted accordingly. To this end, the tilt of the eye and the SV were measured monocularly at five positions of gaze along a horizontal line 27 deg above the horizon. In all three of their subjects, the tilt of the SV increased almost linearly with the tilt of the eye, but much less, namely about 50% of eye tilt on the average. In a qualitative experiment, the authors obtained similar results in other directions of gaze. They inferred that the partial compensation is due to an extraretinal signal which varies linearly with the torsion of the eye. This interpretation implies that the tilt angle of the median plane of the eye relative to the head is known to the brain. It is an open question how an extra-retinal signal may be gained which is proportional to tilt. We quantified the relation between the tilt of the “I
3012-F
Human
Lo&i&ion
Etferena copy
eye and the SV in a larger range of gaze directions, and found it to be more complex than previously known. METHOD5
Eye torsion and the subjective vertical were measured in the eight main directions of gaze at an eccentricity 6 of 25 deg relative to the horizontal “straight-ahead” gaze. In the following, we will denote these directions as shown in Fig. 1. For the quantitative description of tilt values we will use angles $ which we shall name “aberration”. The “aberration of the eye median*‘, tiE, is defined as the angle between the median plane of the eye and the vertical plane through the visual axis; positive when the former is tilted to the right, as seen from behind the subject. Essentially, JIEis the angle at which these two planes intersect at the retina. Correspondingly, the “aberration of the SV”, Jlv, is the angle between the plane through an uppurentiy vertical line, and the vertical plane, both passing through the visual axis. With Listing’s law, we can calculate the theoretical values of tiE which are to be expected
255
WERNERHALJSTEIN and HOIST MITTEUTAEDT
256 V
t
1
. 6
.2
. i
c
I I Y
7*
l
03
-h
‘4
6’ :
Fig. 1. Arrangasnmtand dcaigrdion ofused gaze dirccti~as and coordinates. Gare dimtiom arc ati by positions 1-S; shown as seen by the subject. Stmc~Smphic coordinatw u vertical. h horizontal. Spherical coordimtcs; c eccentricity from straight ahead, j4 polar angle, ckkwise positive.
if the eye is in its primary position when looking straight ahead (see Appendix). We obtain (for ddinition of coordinates see Fig. 1):
x sin(2p)/[l - fg2(6/2)+cos(2/r)]; or approximately, with an error of less than 0.1 deg at all positions with L < 27 deg, and no error at all at the chosen positions: ~g(~E.,lKO) 2 &/2)*M2Cc).
(1)
= 0 deg for direction 1, 3, 5 and 7, Thus )(IEJbro +2.81 deg for direction 2 and 6, and -2.81 deg for direction 4 and 8. In stereographic coordinates: h = fg(C/2)*sin(p)); u = tg(C/2)*cos@); equation (1) becomes: tg(JIE,,& = 2.h.v/(1 +h* - 0’) s 2.h ‘0. (la) iUeasurement of eye torsion Eye torsion was determined photographically by means of &era1 blood vessels in the vicinity of the limbus. The principle of the method is described in Howard and Evans (1963). However, a special apparatus was used which permitted to take accurately matchable photographs from different directions of gaze. Photographs were taken monocularly for the left and right eye at the eight directions of gaze in a scramhkd sequence (3, 8, 5, 2, 7, 4, 1, 6) which was run
through twice. The surrounding was totally dark; illumination during photography was provided by a ring flash. Four photographs were taken with the eyes straight ahead (6 = 0 deg) and the average taken as the reference value. The I//E value at each of the eight eccentric positions was calculated as the mean of the two measurements minus the reference value. An average standard error sE of I/I~ (v = 8) was calculated for each eye from the pair differences. The mean sE for all 12 eyes was 0.33 deg. Torsions were determined from the projected negatives with respect to the image boundaries provided by the camera. There was no headfixed reference in the photographs, but head movements relative to the camera were largely restricted by the use of a dental-impression biteboard and a forehead rest. From the small differences in consecutive measurements we conclude that the stability of head position was very good indeed. However, reproducibility of head position when the subject had left the biteboard, as was the case between the reference measurement and eccentric measurements, might not be equally good. This might have caused an offset in our JIE data. According to Listing’s law, the average I/I~ should be zero, but in five cases it differed significantly more from zero than expected from sE. During photography the subject looked at a fixation mark which was adjusted to an accommodative distance of 1.3 m, equal to the distance of the display for the determination of the subjective vertical. Measurement of the subjective vertical During the measurements of the SV, the subject sat in front of a tangent screen with the head fixed by the same biteboard and in the same position as in the torsion measurements. A display could be presented on the screen at 8 positions along a circle of 1.11 m diameter. The head of the subject was adjusted in such a way that the mid point between the two eyes formed a cone of 50deg aperture with this circle, and the median plane of the head was perpendicular to the screen. Thus, except for the small deviation caused by the interocular separation, the directions of gaze were the same as during the measurement of eye torsion. The display consisted of two red LEDs separated by 6 cm, shining from behind at a fine linen cloth at the surface of the screen. From the front, they appeared as two sharply limited red points of 5 mm diameter. The LEDs were rotatable via
Evaluation of retinal orientation
remote control around the midpoint between them by the subject (S) as well as by the experimenter (E). In otherwise total darkness, the E set the display at an arbitrary angle, and then asked the S to rotate the display until one of the two red points appeared exactly above the other to her or him. The set value was read out on an analog voltmeter. The eight gaze positions were again arranged in the scrambled order, but now the sequence was run through five times. As the angular values on the tangent screen are not identical to the aberration angles, they had to be converted beforehand (see Nakayama & Balliet, 1977). The average of the five SVsettings was taken as $“. An average standard error sy of Jly (v = 32) was calculated from the standard deviations at each position. Measurements were done monocularly right, monocularly left and binocularly on the same day. The sequence of those three conditions was different for each of the six Ss, hence every possible permutation occurred once. The mean sy for all Ss in the monocular condition was 0.65 deg, in the binocular condition 0.55 deg. The Ss were allowed to move their eyes freely between the two points. Thus, in additon to the orientation of the retinal image, the direction of eye movements could possibly have been essential for the perception of the vertical. Therefore eye movements were excluded in an additional binocular experiment with subject RW: she was asked to tixate a point in the center of the display which was continuously visible whereas the two test points were flashed for 20 msec at a rate of 1 Hz. The SV-settings in this situation (mean of 5 at every position) were indistinguishable from the SV-settings with permanently illuminated points and free eye movements (F-value = 1.265, v, = 8, v1= 32). Therefore we assume that our results did not depend critically on the type of display used.
257
Fig. 2. Upper graph: tilt of the eye in the eight directions of gaze (a). Ordinate: eye aberration ST; ab6cissa: gaze directioll.ThemeM!JtaIhdcrror0ftllc&ta~ints(left eye of ~OCsubject) Q given at the top. (o--ojnpr&ts a fit aazxdhg to Listing’s law. The cmdfkknts E,, E,, Eh ofthegtangivmontbeupperkR(~k~crint).Lower
graph: comaponding plot for the aberration Sv of the SV.
median. However, in this example, the relationship between tiy and tiE is not linear, as may be seen more clearly from Fig. 3, where the two
I
l-2
RESULTS
Monocular measurements
Figure 2 shows a typical example of the values of tiE and the monocular SV-settings tiy in the eight directions of gaze (subject MW, left eye). As in the experiments of Nakayama and Balliet (1977), the S does not simply adjust the apparent vertical to the median plane of the eye, but to a retinal meridian at an angle of #,r = ey - eB to this plane. Since JIR is apparently not constant, there must be an extra-retinal signal which accounts for the aberration of the eye
Fig. 3. Relation between eye tilt and tilt of the SV; mme subject as in Fig. 2. Ordinate: SV aberration &, rbacism eye aberration tir. with standard errors.
WERNERHAUSTEINand Horn
258
angles are plotted versus each other: There may be two significantly different values of I/J” at virtually the same value of JIE, e.g. at positions 2 and 7 (r-test, v = 8, P c 0.5%). This result is not only incompatible with a linear function, but with any direct functional relationship between $h and Jly (or $E and IfiR).The same is found with other Ss, though in some the correlation is better; in others, however, it is even worse. Obviously, $” is not a function of $E which in turn depends on the direction of gaze, but both $y and llih depend on the gaze direction, though each in a different way. Regarding $E, this dependence may be deduced from the well-known law of Listing. This law defines a so-called “primary position” of the eye. In normal-sighted subjects, the primary position is close to the position which the eye adopts when, with the head upright, the S looks straight ahead. But deviations of the primary gaze direction from “straight ahead” of several degrees are not uncommon. Ferman, Coliewijn & Van den Berg (1987) found that the primary directions of the two eyes diverge a bit on the average, and this is what we found too. The pitch of the primary direction in the vertical plane depends on the pitch of the head, and therefore on the position which was chosen as “upright head position”. For these reasons the primary direction is unlikely to exactly coincide with the straight-ahead direction in the chosen head position and this will lead to deviations of $E from equation (1). These deviations may be expressed mathematically. If, in its primary position, the eye does not look at h=O, Y=O, but at h=P,, Y = P,, we will theoretically find an additional aberration (see Appendix): t&W, r.+ro /2) = (P;h - P,*u)/(l + P,eh + P;v); or approximately, with an error of less than 0.1 deg at all positions with L < 34 deg, if the primary direction does not deviate more than 15 deg from straight ahead: t&WE. rkm/2) z P,,*h - P&W.
(21
In polar coordinates L and I( we obtain W E.rheO * (360 deg/x ). Wc. 0) x
(P;sin p - Phcos p).
(2a)
This means that a horizontally deviating primary direction will lead to an additional ~08pcomponent in (kE,a vertically deviating primary
MITIEISTAEDT
direction to an additional sin component, Thus, the dependence of d/E on gaze direction should be separable into three components: cI = sin(2p), ch= sin p, c, = cos p; so that *E=EI.c,~E~.ch+E,.c,+E,
(3)
where Et, E,, E, and Em are constants. The component c, follows from equation (I), with E, = 2.81 deg. Component ch is proportional to the horizontal angle of gaze; E,, is 2.22 deg if the primary direction points 1Odeg upwar&. c, is proportional to the amid angle of gaze; E, is 2.22 deg if the primary direction points 10deg to the left. The constant E,,, was allowed for because of possible offsets (see section “measurement of eye torsion”). As all four components are orthogonal, the best-fitting coefhcients E,, Eh, Es and E, may be debunk from the data by simple weighting procedures. In fact, these components allowed a good fit in most cases: only for two of the 12 eyes was the approximation significantly Werent from the data points (F-test, VI= 4, v, = 8, P c 5%). An example of the fit may be seen from Fig. 2 (open symbols). In order to see whether this decomposition might also fit the SV data, and if so, how the two results might compare, we decomposed ev into the same components, i.e. approximated it by the equation: J/“= v,.c,+ v,*c,+ F;c,+
v,*
(4)
As a matter of fact, in 9 of the 12 eyes, the values of J/” were approximated by equation (4) without a sign&ant rest (P-test, vi = 4, v2= 32, P > 5%). Moreover, there is an inviting relation between the components of ev and those of #E. This is illustrated in Fig. 4, where the coefficients V are plotted versus the coe&ients E. In the case of full compensation, the coefficients V shouid be zero, i.e. all points in the plot should lie near the abscissa. This applies to none of the components. If there were no compensation at all, then the coefficients V should be equal to the coefficients E, i.e. all points should lie near the main diagonal. This is indeed the case for the c,-component. Thus, this component is obviously not compensated by an extra-retinal signal. The ch-components, by contrast, do not show any correlation. This means that an extra-retinal signal does indeed exist which has an influence on this component. However, it dues not correctly account for it in general. Concerning cl, the E-coefficients correspond nicely to the value of 2.81 deg expected from Listing’s law (dotted line), whereas the
Evaluation of r&al
orkntation
259
“h
0
R.
EYE
0
L.
EYE
Fig. 4. Relation between the ala&al coc&itats Y (ordinate) and E (&scima) for IIUsix suw. Epch gr8phshowsthecodtkknUfmoneofthe compomnta c,, c,, c, into which $a and &were decomposed on theorctiall grounds (see text). coMaa data pointa reprant kR and right eye of each subject. Bars indicate fstudud errors. Dotted lines io tbc upper right graph indicate the 99% confidcncz interval for the nptim th8t V, = 4, calculated tiom ths rmagc standud error of )(IR(r-test, v = %). Dotted line in the lower graph indicatea expected value of E, according to Listing’s kw.
V-coeflicients, which for full compensation should be zero, are about 20% of that value on average. Thus, this component is compensated to about 80%.
Another interesting result may be drawn from Fig. 4: the ditkences (V - E) are essentially the same for the right and left eye. These difkences appear in the Ggure as the distances of the data points from the 45 deg-line which, taking the standard deviations into account, do not differ signifkantly for both eyes of the same subject (r-test, v = 16, P < l%), except in one case concerning c,.. This means that, but for a constant value, the retinal angle tir = Sy- tiE,
which signifies on which meridian of the eye the SV is imaged, is practically the same for both eyes. In fact, the differences of *a between the right and the left eye do not vary more between eye positions than would be expected from the measurement variances of tiy and SE (F-test, v, = 7, v2= 16, P < 5%) and therefore may be assumed to be constant for a subject. These individual constants, estimated as the mean right-left differences of tis, range from - 0.2 to + 3.5 deg, with a mean of 1.9 deg. It is known (Holmholtz, 19lQ Cogan, 1979) that the corresponding “vertical” meridians of the two eyes while looking straight ahead diverge by about
WERNER HAUSTEIN and HORSTMITTELSTMDT
260
2 deg, opening upwards. Thus the display, set to vertical in both eyes separately, obviously stimulates corresponding retinal meridians. The relatively large range of the mean AJIRis partly due to the offsets in some of our torsion data (see, for instance, the example in Fig. 2), which we believe to contain artefacts. When these offsets are subtracted, the range becomes more narrow, going from 0.1 to 2.4 deg, with an average of 1.68 deg. Summarizing our foregoing results, we should expect that the retinal angle eR consists of the components:
+(~*-~*)*c*+(~,-El).
(5)
The c,-component vanishes because we found that V, = E,. The coefficients V,- E, and V, - E& should be the same for both eyes. In Fig. 5 we fitted the average JI,-values for the right and left eyes of all subjects with two curves according to these restrictions. With one exception in the gaze-down position of the left eye, the fit is very good. Binocular measurements The binocular SV-settings show essentially the same dependence on eye position as the monocular ones. Moreover, they were generally found to lie in between the SV-settings for the left and the right eye. This is also reelected in Fig. 6, where the mean values of tiy of all six Ss are shown for the right eye, left eye and the binocular condition. The binocular values are essentially equal to the average values of the left and the right eye. They are not closer to veridicality than would be expected by this
p- \ 0 \
I\
2
,’ /“-
h,
4
Fig. 6. W-tilts averaged for ail six subjects: right eye monocular, left eye mwmcular (dots like in Fig. 4). and binocular condition (0). Ordinate: mean SV-aberration; abscissa: gaze directions. Dashed curves: fit by theoretical function.
averaging. The same is found if the components c,, c,, and c, are regarded separately. Obviously, no extra information upon the orientation of the display to the vertical (e.g. due to the fact that. in the upright head position, the eyes lie in a horizontal plane) is gained by the usage of both eyes. The simplest assumption which is in agreement with all data is that the orientation of the display is determined from the fused image of both eyes and that the eye position is accounted for by an extraretinal signal common to both eyes. In Fig. 6, the fit by the theoretical curves is also shown. Though the expected deviations (sy/JE) are very small (0.19 deg monocularly, 0.16 deg binocularly), the actual deviations for the right eye and the binocular condition are not significantly larger than those expected. The same holds for the mean torsion values (not shown graphically). However, the fit for the mean tiy of the left eye is much worse than expected, again especially at position 5. Moreover, all three cases where the individual fit by equation (4) was imperfect were left eyes. At present, we have no explanation for this finding. DiSCUSSKlN
\
9
\p’&
,c
8 ,
‘, PO& I
P -
-r
‘0’
Fig. 5. Retinal angles tir averaged for all aix &jccts; right and left eye mpurtoly (dots like in Fig. 4). CMhxatc mean rctinnl angles, abscissa: gaze dirsctions. Dotted curves:iit by theoretical functions with four pnramctcrs estimated from the 16 data points.
I. Comparison with the results of Nakayama and Balkt
At first glance, the present results seem to be at variance with the findings of Nakayama and BalIlet (1977). However, in cases where (see equations 3 and 4):
v,= E,V,I4); and
Evaluation of retinal
(%O because we also found that V,,2 E,, and V,/E,xO.2; we get
orientation
261
(Ph, P,,). On average, 12:is found to be 0.8 from our data, which is not too far away from this condition. tiy = tjE.(V,/EI) + const; We suggest, therefore, that the compensatory i.e. in fact a linear relation between tiy and JIE, mechanism does not rely on information about as was found by Nakayama and Balliet in their the actual tilt of the eye, but on (correct) qualitative experiment, and also in some of our information about gaze direction and on the subjects* eyes. In view of our present findings assumption (1) that the eye follows Listing’s law this linear relation turns out to be a special case and (2) that the primary direction lies in a of a more complex general relation. The above sagittal plane of the head. calculation holds for those positions where our Since the fifties (Mittelstaedt, 1960; for a component decomposition is valid; so far, review see Howard, 1982; Mittelstaedt, 1989)an strictly speaking, at the eight measurement increasing body of evidence has been accumudirections. However, we aiready know that lated to show that the extra-retinal information equation (3), which was derived from Listing’s about the eye position in the head which parlaw, may be extended to the whole field of gaze takes in absolute localization is provided by an in the form (see equations la and 2): efference copy. The assumption that the compensatory signal(s) are derived from an efirence JIEx (360 deg/x) copy of gaze direction commands also fits our x @,./I at, + P;h - Ph.u) + E,; (3a) present data: an efference copy yields the required information only if the motor system with 1, x 1 and E,,,z 0. If we assume that this provides a constant and unique relation between relation may be transferred to the aberration of commands and eye positions. Given that the the vertical we obtain: motor system yields the correct gaze direction, ey 2 (360 deg/n) then the ocular torsion may also be deduced from the commands, if the relation between x (A;*h ‘0 + Pie/l - Pi-u) + v,. (4a) gaze direction and torsion is rather fixed. This We found that A; z 0.2 and P; z Ph. is indeed the case, and the best a priori estiThis extension also fits the quantitative results mation of this relation for both eyes is given by of Nakayama and Balliet which were obtained Listing’s law with a centered primary direction. at gaze positions along a line with u R const. In Additionally, visual localization with respect this case, tiE and tiy in equations (3a) and (4a) to gravity requires that head-centric inforbecome linear functions of the remaining mation is combined with information upon the variable h and thus also linear functions of each orientation of the head to gravity. It is known other indeed. that considerable deviations of the SV from the physical gravity vector occur with larger body 2. Interpretation of the present results tilts (e.g. Mittelstaedt, 1986). However, in the We found that the retinal angle Jlr consists of present study with upright head position, the components according to equation (5). If we main source of aberrations is assumed to be a again extend this result to the whole field of discrepancy between the factual and the gaze, we would expect to find a dependence of assumed primary position, which is compatible with the typical SV-aberrations actually found t,QRon the gaze direction h, u in the form: (except for the small q-components). $Rx(360deg/n)*(k:*h-u +Pt-h)+R,. (5a) At larger body tilts, an additional compliThe constant R,,, means that the SV is on cation arises from the effect that an altered head average not set to the median plane of the eye, position in turn has on eye position, like the but at an angle R,,, to this plane. The actual ocular counterroll at roll tilts of the head. It is dependence of er on eye position, which must still unclear whether this effect is explicitly acoriginate from an extra-retinal signal, is thus counted for in the perception of the SV (Udo de given by (36Odeg/n)-(IF-h-u + P:-h). If A: Haes, 1970). If it is, the compensation may be was 1, such a function would be well-suited (see located at the visual side, transforming retinal equation 3a) to compensate the aberration of an into headcentric coordinates, but also at the eye with a primary direction (0, P,‘)-a hypo- vestibular side or at an intermediate stage, as thetical direction which in general does not has been favoured previously by Mittelstaedt coincide with the actual primary direction (1986, see Fig. 9, 1988).Notably, such a solution
WERNERHAIJSTEINand HORSTM~LSTAEDT
262
would allow extraction of visual information about the vertical in retinal coordinates.
vectors w, and wvImay be replaced by one rotation with vector w, where: w = (w, + w*-w,
Acknowledgemenr-We wish to thank Thomas Eggert for his helpful support in statistical questions.
REFERENCES Cogan, A. I. (1979). The relationship between the apparent vertical and the vertical horopter. Vision Research, 19, 655-665.
Fennan, L., Collewijn, H. & Van den Berg, A. V. (1987). A direct test of Listing’s law-I. Human ocular torsion measured in static tertiary positions. Vision Reseurch, 27, 929-938.
Haustein, W. (1989). Considerations on Listing’s law and the primary position by means of a matrix description of eye position control. Biological Cybernetics, 60,41 I-420. Hehnholtz, H. von (1910). Hundbuch der physioiogischen Optik. Vol. III, p. 3. Hamburg: Voss. Howard, I. P. (1982). Human visuuf orientation. New York: Wiley. Howard, I. P. d Evans, J. A. (1963). The measurement of eye torsion. Vision Research, 3, 447-455. Mittelstaedt, H. (1960). The analysis of behavior in terms of control systems. In ScbatTner, B. (Ed.), Transactions o/the jijih cwrference on group processes (pp. 45-83) (see also revised version in Mittelstaedt 1989 below). Mittelsmedt, H. (1986). The subjective vertical as a function of visual and extraretinal cues. Aclu Psychologieu, 63, 63-85.
MitteIstaadt, H. (1988). The information processing structure of the subjective vertical. A cybernetic bridge between its psychophysics and its neurobiology. In Marko, H., Hauske, G. 8; Struppler, A. (Eds.), Processkg strutlures for perception and action (pp. 217-263). Weinheim: VCH. Mittistaedt, H. (1989). Basic solutions to the problem of head-centric visual localization. In Warren, R. & Wertheim, A. H. (Eds.), The perception und conrrol of egomofion. Hillsdale, New Jersey: Erlbaum. Nakayama, K. & Balliet, R. (1977). Listing’s law, eye position sense, and the perception of the vertical. Vision Research,
17, 453-457.
Udo de Haes, H. A. (1970). Stability of apparent vertical and ocular countertorsion as a function of lateral tilt. Perception and Psychophysics, 8, 137-142.
An advantageous way to treat rotations mathematically is to use rotation vectors. Their application for the description of eye positions and Listing’s law is described in detail in Haustein (1989). Let the “reference position” of the eye .be the straight ahead position. Any other eye position may then be deBned by the axis about which and the angle a by which the eye may be rotated directly from the reference position into thii position. A vector which points into the direction of this axis and has a length of fg(a/Z) is ddined as the rotation vector. The rotation vector is especiaIly useful for the caIculation of combined rotations: two consecutive rotations with
(Al)
w=(r-pPq.p+tq,q-fP)l(l+pqf). One may check that
2r/(l - r2) = rg(S,) = 2(w, + w,w,)/(l - w; - w$ + wf,.
(A2)
This formula gives the general relation between an eye position in the rotation vector representation and its abcrration #r. During distant fixation with the head 6xed, eye positions follow Listing’s law. This law states that from a certain eye position, the so-called “primary position”, any other actually occurring eye position may be reached by a rotation about an axis orthogonal IO the visual axis. If the primary position coincides with the reference position, then Listing’s law impIied that the rot&ion vectors a for all occurring eye positions tie in the yz-plane, i.e. have no x-components. In the chosen coordinate systems, the rotation vector a therefore reads: a = (0, -v, -h)=(O,
-cosp,
-sinr).rg(t/2).
Substituting a for w in equation (A2) we obtain: tg(+,) = 2.h .v/(l + hz - vz). If the primary position does not coincide with the reference position, the rotation vectors are no longer orthogonal to the x-axis in general. Rather they may be expfesscd as a combination of a rotation (0, p, q) and a following rotation (0, -P,* - P*). where Pb and P, are the coordinates of gaze direction in the primary position (see Haustein, 19%9).This combination yields the rotation vector: b=(p.P,-q.P,,,p
APPENDIX
x w&(1 -w,,w,).
We will use the rotation vector to calculate the aberrations $E which are to be expected according to Listing’s law. The median plane of the eye remains perpendicular (i.e. aberration tid = 0) if, starting from the straight ahead position, the eye is first rotated about a horizontal axis which is orthogonal to the median plane (y-axis) and then about the vertical axis (z-axis). An aberration qE occurs ii the eye is beforehand rotated about the visual axis (x-axis) by an angle tiE. These three rotations may readily be described by rotation vectors: in the cartesian head-fixed xyzcoordinate system, w, = (I, 0.0) with r = rg(tiE/2)+ w2 = (O,p, 0) and w, = (O,O, q). Applying equation (Al) two times we obtain for the total rotation:
-P,,q
-P,)l(l
-p.P,-q.Ph).
We find that: b,-b;P,+b;P,=O.
(A3)
This may be compared to an “iw’ Listing rotation (0, -v, -h) whkh is pm&cd by an additional tomional rotation (d, 0.0). with d = rg(A# /2), yielding a rotation
vector: c=(d,
-v - hd, -h
+ vd).
Substituting c for b in equation (A3) we obtain: d+(v+hd).P,+(-h
tvd).P,,=O,
or d = tg(A+/Z) ==(P;h
- P,.v)/(l
+ Ph.h + P;u).
