AM 65-32 

PROBLEMS IN DEPTH PERCEPTION: 

A Method of Simulating Objects Moving in Depth 

Approved by 

J. RoBERT DILLE, M.D. 

CHIEF, CIVIL AEROMEDICAL 

INSTITUTE 

Walter C. Gogel 

Henry W. Mertens 

December 1965 

FEDERAL AVIATION AGENCY 

Office of Aviation Medicine 

Released by 

P. v. SIEGEL, M.D. 

FEDERAL AIR SuRGEON 
Qualified requestors may obtain Aviation Medical Reports from Defense Documentation 

Center. The general public may purchase from Clearinghouse for Federal Scientific and 

Technical Information, U.S. Department of Commerce, Springfield, Va., 22151. 
PROBLEMS IN DEPTH PERCEPTION: 

A Method of Simulating Objects Moving in Depth* 

Visual size is an important variable in a variety 

of experiments; for example, in studies involving 

size constancy, in studies of size as a cue to distance, and in studies concerned with the perceptual 

effects of the optical-expansion pattern associated 

with objects moving in depth. In many of these 

experiments, it would be useful to be able to vary 

continuously the visual angle of complex objects 

without changing other visual qualities of the objects, such as the stimulus to accommodation. A 

method of producing controlled and continuous 

*This study was conducted at the ·Civil Aeromedical Research 

Institute. The authors are now at the University of California, 

Santa Barbara, Calif(}rnia. 

eR 

1---- DRI ----;•~1 

/ 

/ 

/ 

/ 

variations in visual size can be achieved by applying principles involved in the "shadow transformer" described by Gibson,t or in the projection 

system described by the de Florez Company.2•

3 

The resulting apparatus, which might be called 

the moving-object simulator, is useful in simulating the motion of complex objects or surfaces in 

depth. 

A schematic side-view diagram that can be used 

in discussing the principles involved in the 

moving-object simulator is shown in Figure 1. In 

Figure 1, eR, fR, gR, hR represents a simulated, 

rigid, physical configuration (object) observed 

from the position 0. By the term "rigid configu-

/ 

/ 

/ 

w 

a 

/\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

..----------DR2 ----------:~ 

...----------Do----------~ ...,.....,_ ____ Dp -----1 

FIGURE 1. A side-view schematic diagram for considering factors involved with using a point source of light in 

the simulation of objects moving in depth. 

1 
ration" is meant a configuration whose angular 

and linear dimensions are constant. The front 

and rear object surfaces of size SR1 and SR2

, respectively, are parallel to the frontal plane of the 

observer (for simplicity of discussion), are separated by a distance LR measured along the line OQ 

perpendicular to a screen W, and are at distances 

DR1 and DR2 from 0. The top surface (fR, gR) 

of the object forms an angle f3R with the line OQ. 

The screen is a distance Do from 0 along OQ. 

Straight lines from the observer's position 0 

through fR and gR intersect the screen at a and b, 

respectively. At a distance DP on the other side 

of the screen is a point source (P) that transilluminates another configuration e-r, fT, gT, hT 

(called the transparency). The front and rear 

surfaces of the transparency are of sizes ST1 and 

ST2 , are parallel to the screen, are separated from 

each other by a distance Lr measured along the 

line PQ perpendicular to the screen W, and are 

at distances DT1 and DT2 from P, respectively. 

The top surface ( fT, gT) of the transparency configuration forms an angle f3T with the line PQ. 

The screen (W) is a rear-projection screen with, 

for example, the image of fT and gT appearing on 

the screen at a and b, respectively, and being 

visible from the position of 0. 

The problem of this paper is to determine the 

values of f3T, ST , ST , and LT so that, when the l 2 

transparency eT, fT, gT, hT is moved as a rigid 

object along PQ, it will produce a changing 

stimulus on the screen and, hence, an opticalexpansion pattern in the eye of the observer 

equivalent to the optical-expansion pattern that 

would have occurred from the rigid physical 

configuration eR, fR, gR, hR moving toward or away 

from 0 (along OQ or an extension of OQ). The 

simulated configuration eR, fR, gR, ha is assumed to 

be rigid because the purpose of the apparatus is to 

simulate physically rigid objects. The configuration eT, fT, gT, hT must be rigid if the apparatus is 

to be physically simple. The case in which the 

configurations have a depth component (La and 

LT) is selected for discussion because it represents 

a general formulation of the problem. 

It will be noted that, in Figure 1, parts of each 

of the configurations can obscure other parts of 

the same configuration. For example, if the 

surface ea, fa in the simulated configuration were 

opaque with the surface ga, ha directly behind 

ea, fa, the front surface ea, fa would obscure the 

remaining surfaces of this configuration. In this 

2 

case, only the surface ~. fT in the trnsparency 

would be required for the simulation. Thus, it 

is not intended that all parts of Figure 1 necessarily 

represent an actual simulation. Figure 1 is 

useful, however, in considering the relations involved in any simulation of a rigid configuration 

by a rigid transparency using a point-source 

system of projection. For example, conclusions 

involving the relation between f3T and f3a or the 

relation between LT and La can be expected to 

apply respectively to any angles and any depth 

extents in the configurations. Also, although 

only the vertical components of frontal extents 

are represented in Figure 1, conclusions concerning 

the relation between simulated and transparency 

configurations with respect to such vertical 

components can be applied to horizontal components as well. 

Using Figure 1, it can be shown that 

and 

Or, more generally, if 

Sat= Sa2= Sa=C 

STt ST2 ST ' 

it follows that, for any value of Da and DT 

(1) 

(2) 

(3) 

(4) 

Equation 3 imposes the restriction that the 

size (ST) of any frontoparallel extent in the 

transparency configuratio:n is a constant proportion of the size (Sa) of the corresponding frontoparallel extent in the physical object simulated. 

The only case that will be considered in this 

paper is the situation in which the viewing position 

and the point source are at fixed distances (although not necessarily the same distances) from 

the screen. Therefore, D0 /DP is a constant, and 

Equation 4 states that the distance from 0 of 

any point on the simulated configuration is 

directly proportional to the distance from P of a 

corresponding point on the transparency configuration. It follows from Equations 3 and 4 

that 

(5) 
where LR=DR2

-DR1 and LT=DT2

-DT1" 

Furthermore, 

KT 

tan iST=LT (6) 

and, 

KR 

tan iSR=LR (7) 

where, 

KT=STt-ST2 (8) 

and, 

KR=SRt-SR2 (9) 

From Equations 3, 5, 6, 7, 8, and 9, it follows that 

tan f3R =(~:) tan iST· (10) 

Equations 3, 5, and 10 express the size and 

shape relations between the rigid transparency 

configuration eT, fT, gT, hT and the rigid simulated 

configuration eR, fR, gR, hR. Values of SR and ST 

are related by the constant C, values of LR and LT 

by the constant CD0 /DP and values of tan PR 

and tan f3T by the constant D0 /Dp. Equation 4 

expresses the relation between the simulated 

distance DT from the observer of a point on the 

simulated rigid configuration and the distance DR 

from the point source of a corresponding point on 

the rigid transparency configuration. Values of 

DR along OQ and DT along PQ are related by the 

constant CD0 /Dp. 

If the apparatus is to simulate rigid objects 

moving in depth by means of a-rigid transparency 

configuration, it must be demonstrated also that 

the movement of the rigid transparency configuration toward or away from P will have a 

unique relationship to the movement of the rigid 

simulated object away from or toward 0. If the 

successive derivations of DR and DT with respect 

to time are considered in Equation 4, it follows 

that 

In Equation 11, d indicates derivatives not 

distances, t indicates time not transparencies, 

dnDR/dtn and dnDT/dtn indicate any derivative 

of DR and DT respectively with respect to time 

(t), and OD0 /DP is a constant. According to 

Equation 11, any velocity (dDT/dt) or acceleration (d2DT/dt2

), etc., in the movement of the 

rigid transparency configuration will produce a 

3 

proportional although not necessarily an identical 

velocity (dDR/dt) or acceleration (d2DR/dt2), etc., 

in the movement of the simulated rigid object. 

An advantage of the moving-object simulator 

is that it can simulate complex objects moving in 

distance at any specified rates. Suppose, for example, that the object to be simulated is a playing 

card presented in a frontoparallel position and 

moving directly toward or away from 0.4 The 

positive transparency would be produced from 

photographing a real playing card oriented frontoparallel to the camera. As required by the suit 

of the playing card, either a black-and-white positive transparency or a transparency containing 

colors can be used. The positive transparency is 

mounted ( e.r, fT, Figure 1) perpendicular to its 

direction of travel. Some device must be used to 

prevent light originating from the point source 

from reaching the rear-projection screen except by 

passing through the positive transparency. For 

example, the transparency might be mounted on an 

aperture cut in an otherwise opaque baffle. The 

baffle and the attached transparency can be 

mounted on a cart that is moved back and forth between the point source and the rear-projection 

screen by means of a mechanical linkage to a motor. The result is that, if the image on the rearprojection screen is viewed in an otherwise dark 

room from the position 0, it will appear as a real 

playing card having appropriate colors and moving in distance in an otherwise black visual field. 

The movement of the transparency can be 

arranged to be under the control of either the 

experimenter or the observer as required by the 

experiment. The chamcteristics of the simulated 

movement will be related to the characteristics of 

the movement of the transparency by the constant 

C Do/DP as shown in Equations 3 and 11. 

If the moving transparency is tilted toward or 

away from the screen to form a constant angle f3T 

with PQ its direction of movement, it will simulate 

a moving playing card at a constant tilt iSR· From 

Equation 10, f3T will equal PR only when Dp=D0 . 

If the photographic transparency is of a flat surface 

and is produced by pointing the camera along a 

perpendicular to the surface, the optical expansion 

pattern involved in approaching the surface at a 

constant angle can be simulated 2

•

5 by orienting the 

transparency at the appropriate angle as determined by Equation 10. 

The development of the optical expansion 

pattern of a rigid moving surface at a particular 
constant slant is readily achieved by using the 

appropriate value of DP/D0 ; however, the simulation with this apparatus of a rigid surface rotating 

at a prescribed volocity is not easily achieved 

unless D0 =Dp. This is demonstrated by differentiating tan f3R and tan f3T with respect to time 

(t) in Equation 10. It follows that 

(12) 

where dn tan f3R/d tn and dn tan f3T/dtn signify 

any derivative of tan f3R and tan f3T, respectively, 

with respect to time (t), and where DP/Do is a 

constant. It follows from Equation 12 that. a 

proportional relation between the transparency 

and the simulated object occurs with respect to 

tangents of angles, not with respect to angles. 

Only when DP=Do will angular displacements, 

velocities, and accelerations be proportional between the transparency and simulated object. 

For the simulation of linear depth movement of 

rigid three-dimensional objects or rigid surfaces 

at a constant slant, the permissible inequality of 

DP and Do and of STand SR has the advantage that 

the proper selection of these terms can result in the 

simulation of a large linear movement in depth 

from a much smaller linear movement of a 

transparency. 

The range of experimental problems to which 

the moving-object simulator can be applied will be 

determined by the quality of the image rendition 

available from the use of point-source projection 

systems. Several of the factors involved in image 

sharpness, image brightness, and the permissible 

complexity of the object simulated will be considered in relation to the type of system illustrated 

in Figure 1. The sharpness of the image on the 

4 

screen is increased (a) by covering reflecting surfaces inside the apparatus with black velveteen or 

some other nonreflecting material, (b) hy using, 

whenever possible, glass photographic plates of 

high contrast for the transparencies, (c) by decreasing the diameter of the point source, and (d) 

by using ·an appropriate rear-projection screen. 

With the proper precautions, a good quality of 

image can be achieved. 

It is clear from Figure 1 that light from the 

point source will strike different parts of the 

transparency and the screen at different angles. 

Since some reflection as well as transmission of 

light will occur at each surface, and since the 

amount reflected is a function of the incident 

angle, the brightness of the image across the screen 

will not be completely uniform. Also, any point 

on the transparency not on the line PQ of Figure 1 

will occupy different positions on the screen as 

a function of its distance from the point source. 

Therefore, the intensity of the image on the screen 

related to a particular point on the transparency 

will change as the transparency moves toward or 

away from the point source. Clearly, this nonuniformity of brightness can be limited by physical conditions that restrict the range of incidence 

angles on the transparency and screen. 

Some linear movements of some three-dimensional objects cannot be simulated by the apparatus schematically illustrated in Figure 1. 

These are objects with parts or sides that, during 

the movement of the object, appear or disappear 

from the view of the observer. In addition, when 

complex transparencies are used, special care must 

be taken to insure that the opaque bafBe on which 

the transparency is mounted does not interfere 

with the production of the image on the screen.8 
SUMMARY 

A set of equations was developed for the simulation on a screen of the movement of an object or 

surface toward or away from an observer by the 

movement of a positive photographic transparency of the object or surface away or toward a 

point source. The general case was developed for 

simulating objects in which the distance of the 

observer from the screen was constant but not 

necessarily equal to the distance of the point source 

from the screen. Equations were developed relating the dimensions of the rigid transparency to 

tho~ of the rigid simulated object. These equations, under a wide variety of conditions, permit 

the simulation of surfaces or objects moving in 

depth at any designated linear speed or acceleration with respect to the observer. 

REFERENCES 

1. GmsoN, J. J.: Optical motions and transformations as 

stimuli for visual perception, P-sychoZ. Rev., 64: 291-

292, 1957. 

2. The de Florez Company, Study af point light source 

projection system components, US Naval Training Device Center, Port Washington, N.Y., Tech. Report No. 

1628-1, 1951. 

3. The de Florez Company, Methods of presenting moving 

objects in point light source visual displays, US Naval 

5 

Training Device Center, Port Washington, N.Y. Tech. 

Report No. 1628-5, 1959. 

4. GOGEL, W. C., and MERTENS, H. W.: Perceived size and 

distance of familiar objects, 1966 (In preparation). 

5. HocHBERG, J., and SMITH, 0. : Landing strip markings 

and ;the "expansion pa·ttern" : I. Prog·ram, preliminary 

analysiiS, and apparatus, Percept. mot. SkiZZs, 5: 81-92. 

1955. 

U.S. GOVERNMENT PRINTING OFFICE: 1966 Q-211-399