COMPUTER GRAPHICS AND IMAGE PROCESSING (1975)4, (396-402)

A Three Dimensional Computer Display*

WILLIAM SIMON~"

Division of Biomathematics, The University of Rochester School of Medicine and Dentistt3~, Rochester, New York 14642

Communicated by A. Rosenfeld

Received March 20, 1975

A simple device for producing an autostereoscopic visual computer display is described.

For many applications it is useful to have a three-dimensional computer display device. A number of methods have been tried such as stereoscopic views and psychophysical tricks in which two-dimensional images simulate rotation to create the illusion of three dimensions but none of these has been completely satisfactory. The method to be described here produces a true three-dimensional image which can be viewed without optical aids of any kind and from a wide variety of observer positions. It is, in fact, truly three-dimensional. Paradoxically it cannot be satifactorily photographed (except with a stereoscopic camera) without losing its three-dimensional character, but neither can any three-dimen- sional object.

In 1969 the author published a technique [1] J of producing a three-dimen- sional display by reflecting an image of the face of a cathode ray tube in a ro- tating mirror (Fig. 1). As the mirror rotates, the image of the face of the CRT sweeps through a volume of space. By proper timing and positioning of the CRT spot, an image spot may be displayed at any point in the swept volume. This method, while producing a three-dimensional image, has the disadvantages of a narrow field of view obstructed by the oscilloscope and the fact that the observer sees the edge of the mirror passing through his field of view twice during each rotation.

In order to correct these difficulties a new version of the device has been developed. This improved design makes use of a mirror mounted on the end of a diagonally sliced cylinder (Fig. 2). Rotation is around the axis of the cylinder and the principal direction of observation is along the axis (a wide variety of others are possible) so that the edge of the mirror is never within the field of view. The axial view produces a three-dimensional image which is visible over a much wider range of observer positions than was the case with the earlier design.

The principle of operation is self-evident from the accompanying diagram (Fig. 2). The relationship between the position and timing of a display point PD on the face of the cathode ray tube and the apparent position of the image point PI as

* This work was supported by Grant 5 POI GM14928-03. t Dr. Simon is currently Visiting Associate Professor in the Research Laboratory of Electronics,

Massachusetts Institute of Technology, Cambridge, Massachusetts 02139. On source related techniques see [2-4].

396

Copyright @ 1975 by Acadernle Press, Ine. All rights of reproduetlon in any form reserved.

3-D COMPUTER DISPLAY 397

MOVING IMAGE OF C.RT. FACE

.~REVOLVIN6 MIRROR

C.RT.

Fro. I. A three-dimensional image is displayed by producing a moving reflection of the face of a CRT and timing the illumination of displayed points to coincide with the passage of the image plane through the desired image position.

j - ' - TOP VIEW

VIEWING DIRECTION

IMAGE OF CRT FACE M' O"o OS'T'ON

]Fie. 2. The sub 1 and sub 4 coordinate systems are stationary while the sub 2 and sub 3 systems rotate with the diagonal mirror, Sub ] and sub 2 refer to the coordinates of the point as it appears on the face of the CRT. Sub 3 and sub 4 refer to the reflected image. Also shown is the position of the image of the CRT when the position of the mirror is as shown and as if the face of the CRT were wholly illuminated.

398 WILLIAM SIMON

-I 2

,Z 3 e

"7 FI~. 3. The coordinate transformation of Eq. (3),

viewed by reflection in the mirror is less obvious. Let X~, Yj, Z~ be the coordi- nates of the spot on the face of the cathode ray tube. Let X,~, Y~, and Z., be a ro- tating coordinate system whose X axis is the projection of a perpendicular to the face of the mirror into tile plane defined by XI and YI (Fig. 2) and whose Z axis is the same as ZI. The conversion from sub 1 coordinates to sub 2 coordinates is simply a rotation given by

Y2 = s 0 cos0 Y~ , Z~ 0 \Z1/

(1)

(2)

sub 2 to (Fig. 3).

the sub 3 system is trivial

Y3 = 1 Y2. Za 0 Z2

if the slant angle of the mirror is 45°C

(3)

(Other angles are only slightly more complicated but since 45 ° is simple, we will continue to use it for the purpose of derivation.) The sub 3 coordinates are a ro- tating system and we wish to refer back to a stationary system X~, Y4, Z4 iden- tical to the sub 1 system but referring to the reflected image.

114 = s 0 c o s 0 Y3 . (4) Z,I 0 Z3

Multiplying the matrix transformations, we go directly from the sub 1 coordi- nates of the spot on the face of the cathode ray tube and the mirror angle 0 to the coordinates of the reflected image of the spot in the sub 4 system.

0 = rot,

where 0 is the angle between X1 and X2 and is time dependent since the mirror is rotating. Let X3, Y3, Z.~ be identical to the X2, Y2, Z2 system but let it refer to the reflection P~ of the point PD as seen in the mirror. The transformation from the

¥{

3-D COMPUTER DISPLAY

Xt ~FAC ×° E OF" CRT

399

Fro. 4. Position of the CRT relative to the sub I coordinate system.

Y4 = s 0 cos 0 Z4 0

i)(! 0 s,.0 !)(:> 1 s 0 cos 0 Y, 0 0

= I - - s i n 0 cos 0 cos 2 0 sin Y1 • \ cos 0 sin 0 0 Z,

(5)

T h e coord ina tes X,, II1, and Z1 are not independent but are constrained to lie on the face o f the C R T . Al though a variety of positions o f the CRT can be used it is conven ien t to have one d imens ion of the C R T parallel to the axis of the cyl- inder and the o ther at an angle d~ wi th the X, axis (Fig. 4). Then X1 and Y~ are re la ted as follows.

Y, = ( X 0 - X1)tan q5. (6)

Al ternat ively , in one ve ry useful case q) is set to 7r/2, Xt is a constant, and II1 is independen t o f X1.

In order to display a point at posit ion X.~, Y4, Z4 one must solve Eq. (5) (sub-

FIG. 5. Photograph of the mirror used in the present device.

400 WILLIAM SIMON

Fro. 6. Photograph showing mirror and CRT display.

j ec t to Eq. (6) or the condition • = ~r/2) for a set o f coordinates X1, Y1, Z1, and 0. Unfor tunately , this is complicated but since 0 is small, over the angles in which reflections can be seen, it can be done by successive approximations usually requiring at most one iteration. Consider the case in which X, is a con- stant equal to X0 and Y, is independent of X~.

First approximation:

cos 0 = Z4/Xo, Z, cos 0 - Y~ sin 0 cos 0 = X 4 - X0 sin260, (7)

Zt sin 0 + Y1 cos20 = Y.t + Xo sin 0 cos 0.

The latter pair are now solved for Y~ and Z,. T h e resulting value for Y1 is then inserted in the equat ion for Z4,

Z4 = X,, cos 0 + Y~ sin 0, (8)

3-D COMPUTER DISPLAY 401

Fro. 7. Stereopair of a three-dimensional pattern obtained by photographing the display.

and a better value obtained for 0 which may then be used to find second-approx- imation values for Y1 and Z~.

In use the coordinates of the point on the face of the CRT are entered into a timing table. The display of the table is synchronized with the angle of rotation of the mirror. This is most easily done by reflecting a deep red or infrared light beam from the face of the mirror and detecting it with one of the many red-sensi- tive solid state detectors. If conflicts for position in the table arise, they can usually be resolved by shifting one or two locations. Assuming a reasonable rate of rotation of the mirror of l0 rps and a display rate of 20/~sec per point, a dis- placement of one position in the table corresponds to an era-or in the angle of the mirror of less than one-tenth o f a degree so that a particular entry can be shifted a couple of places in either direction without causing an objectionable error in the position of the displayed image. Where conflicting demands for table space cannot be resolved by small shifts in location, it may be necessary to display tables with different entries alternately.

In its present form, this device consists of a diagonally sliced 6 in. diameter aluminum cylinder (Fig. 5). It rotates at 600 rpm, which is fast enough for flicker fusion and allows about 20 msec of useful display time per revolution. With the existing computer facility, a P D P 9 with Hewlett Packard 1360 A Display (Fig. 6), this allows about 1000 points to be displayed during each revolution. This is sufficient for many uses but a faster display is probably desirable.

It is difficult to describe the subjective effect of viewing a true three-dimen- sional computer display. Most people feel initially that it must be a psychophys- ical trick. Except for persistence of vision, it is not. The display really is three- dimensional. It has, however, one serious drawback, which is that only transpar- ent structures can be displayed. In some cases, such as the display of molecules (Fig. 7), this is desirable. In others it limits the usefulness of this device. Fortu- nately there are many such transparent structures of interest as computer outputs.

402 WILLIAM SIMON

REFERENCES

1. W. Simon, A method of producing a three-dimensional cathode ray tube display, Behavioral Res. Methods Instrumentation 1, 1969, 179.

2. A. C. Traul, A new 3-dimensional display technique, MITRE Rept. M68-4, 1968 (AD684252). 3. E. G. Rawson, 3-d computer generated movies using a varifocal mirror, Appl. Opt. 7, 1968,

1505-1511. 4. E. G. Rawson, Vibrating varifocal mirror for 3-d imaging, Spectrum 6, 1969, 37-43.