Stereographics Projections by Digital Computer

A Michael Noll

May 1965

Computers and Automation

The human eye is the receptory organ of an extremely complex vision system. This system has the ability to perceive a brightness range from 10,000 millilamberts to 0.00001 millilambert - a ratio of one billion to one.[1] The images focused upon the retina of each eye are each slightly different, and the brain, by some presently unknown method, translates these differences into an effect which we call depth.

Our depth-perceptive abilities yield much information about our three-dimensional environment. When added to photographs, the illusion of depth becomes a source of considerable realism and excitement; so exciting are these prospects that LOOK magazine sponsored 13 years of research to produce one such picture in mass quantities.[2] However, the illusion of depth also has important applications in the visualization of scientific data. A physics text-book, for example, has used two perspective drawings side-by-side so that a three-dimensional effect is obtained when viewed properly, and anaglyphs have been used in a Hungarian descriptive-geometry text. [3][4]

A few years ago, psychological research into depth perception was initiated using random patterns that depicted surfaces when viewed stereoptically.[5][6] These patterns were produced by a digital computer programmed to calculate and automatically plot the stereoscopic projections. The technique depicted only surfaces however, and so was not applicable to the presentation of scientific curves and figures.

The obvious next step was to use the digital computer to calculate and plot stereographic projections of general-purpose scientific data. A computer has been so programmed, and the results are reported in this article. The aviation industry has also been interested in computer stereographic techniques, and the results of its efforts have been described.[7][8]

Stereographic Projection

The basic technique for producing a three-dimensional drawing is the technique of stereographic projection. This technique consists of producing two perspective drawings corresponding to the images seen by the left and right eyes. Usually the drawing of such perspectives is quite tedious, and in practice various approximations such as isometric, one vanishing point, and two vanishing point projections are used. However, the digital computer is so adept at performing tedious calculations that straight-forward methods for producing a perspective can be utilized.

To produce a perspective drawing of an object, it is first necessary to choose some point (representing the eye) from which the object is viewed (see Fig 1). In descriptive geometry terminology, this point is called a station point.[11] A plane, more specifically called a picture plane, is inserted between the object and the station point. Projection lines (actually visual rays) are then drawn from the object to the station point, and their points of intersection with the picture plane are connected to complete the perspective drawing.

Since two perspectives are required to produce a stereographic drawing, two station points (one for each eye) and two picture planes must be chosen. The object can be viewed from any angle if an angle of inclination and an angle of rotation of the station points are introduced. Assuming that the object is specified in a rectangular coordinate system, the stereographic scheme can be depicted as in Fig 2. The left and right picture planes, the left and right station points, and the angles of inclination and rotation are shown. If an object were to be projected stereographically, lines would be drawn from it to the station points. The intersections of these lines with the picture planes produce two slightly different perspectives, corresponding to the left and right-eye images. When viewed stereoscopically, these two perspectives create the illusion of depth. Of course, the computer does not have the ability to physically draw lines from the object to the station points, and so an analytic treatment of stereographic techniques is required.

The derivation of the projection formulas is straightforward, and the formulas will be supplied upon request to the author.

Stereographic Projection by Computer

If the rectangular coordinates of some point are known, then the corresponding left and right perspectives can be easily computed. The introduction of angles of inclination and rotation of the viewing point makes the computations only slightly more complex. The projection technique is thus reduced to equations that can be evaluated by a digital computer. It is only necessary to represent the object to be projected by straight lines connecting points. These points are given to the computer, along with parameters, and the computer then computes the corresponding coordinates of the points in the left and right picture planes.

The remaining problem is to plot the projected points and to then connect lines between them thereby producing the left and right perspectives. This is a job far too tedious to do by hand; fortunately, an elaborate device manufactured by the Stromberg Carlson division of General Dynamics is available for plotting digital data.

The Stromberg-Carlson SC-4020 microfilm plotter consists primarily of a cathode ray tube and a 35-mm camera for taking pictures of the information displayed on the face of the tube. Instructions for the SC-4020 are written on magnetic tape; the tape is then decoded by the SC-4020 and used to generate commands for opening and closing the shutter of the camera, for advancing the film, and for deflecting the beam of the cathode ray tube. Development of the film produces a 35-mm microfilm transparency which consists of lines connecting points, drawn, in effect, directly under the control of a digital computer. In this manner, the perspective points computed by an IBM 7094 digital computer are used as the input to an off-line SC-4020 microfilm plotter through an intermediate magnetic-tape storage. After photographic development, the microfilm can then be viewed directly in a stereoscope, and the final result is an illusion of depth created by a completely computerized technique as diagrammed in Fig 3.

Stereographic Projection Program

The preceding paragraphs have indicated that the computer requires only the coordinates of the end points of lines to compute the stereographic projections. The projected points are then used as the input to an off-line microfilm plotter which actually draws lines between them. The command structure of the microfilm plotter has been designed to draw either a single line between two points or a sequence of connected line segments between a set of points. Thus, if all the points are stored in one master array for programming convenience, when they are to be plotted, the proper sets must be unpacked from the projected master array. This can be done conveniently with two subroutines, one to store and pack the coordinates of the points of each set, and a second to actually compute the stereographic projections, unpack them, and instruct the plotter to draw on microfilm the left and right images. Thus, the first subroutine is called repeatedly until all the sets of points to be projected have been packed together.

The functions of the stereographic computing subroutines are indicated in Fig 4. ARRAY is called to store the coordinates of the points of each set. After all the sets of points have been called, a call to PLOT computes the stereographic projections, using the previously-derived equations. The argument of PLOT specifies the distance to the origin, the interfocus distance, and the angles of inclination and rotation.

Since the size of the microfilm frame is restricted, it is important that the perspectives be centered and scaled in size to adequately fill each frame. Accordingly, PLOT searches for the maximum and minimum of the arrays. The maximum and minimum are used to determine the shifting required to center each perspective in its frame. A scaling factor is also computed and used to scale the perspectives in size to assure that they are neither too big nor too little. PLOT then instructs the microfilm plotter to draw lines between the points specified in the shifted and scaled arrays.

Four Examples

Several examples of stereoscopic drawings produced by the computer are here given. In order to see them, as printed here, it is necessary to decouple one's eyes sufficiently to produce double images. The left and right perspectives are presented next to each other. The trick in looking at them is to decouple the eyes sufficiently to produce a third image centered between the left and right perspective. This task is made easier by first gazing beyond the page and then dropping the eyes to the page without refocusing; a piece of paper placed between and perpendicular to the two perspectives may also help to produce the third image. The third image, when and if obtained, is in depth but is at first blurred. If one continues to look at it for a while, it will become clear, and look remarkably solid.

One of the major disadvantages of stereographic projection of scientific data is that a stereoscope is usually necessary for viewing them. This reason plus the tedious drafting work required to prepare the projections are two reasons why stereographic presentations have not been used more frequently. However, considerable research is being devoted to solving the first problem; and the computer techniques described in this article almost eliminate the second.

Figure 5 shows a three-dimensional bundle of lines whose end points have been determined at random. This type of pattern is excellent for demonstrating depth, since each perspective by itself contains no monocular perspective clues. Speech spectrograms have been plotted in Figure 6. The frequency in cycles per second is plotted to the right; the vertical distance measures the log amplitude of each spectral component, and each spectral slice is separated in time by 15 milliseconds. The educational possibilities of stereographic projections by computer are exemplified by the flow diagram shown in Figure 7. An electrical engineering application is given by the transfer function plotted in Figure 8.

Discussion

The examples given in the preceding paragraphs are representative only of a few of the many provocative visual presentations using depth made possible by the computer technique described in this article. The most obvious use is in the presentation of curves and functions of three variables. When visualized in true depth, many important trends in data become quite evident, as, for example, the formant structure of the speech spectra shown previously.

Here is a method for presenting stress diagrams, the construction of beams and bridges, the structure of molecules, functions of a complex variable, and much more - all viewed from any angle and any distance. It is apparent that further applications are limited by only the imagination of the prospective user.

The three-dimensional examples were contributed by R M Golden, H M Kalish, and J C Noll, and are gratefully acknowledged.

References

1. Lawrence J. Fogel, Biotechnology: Concepts and Applications, Prentice-Hall, Inc., Englewood Cliffs, 1968, p. 93.

2. LOOK, February 25, 1964, pp. 102-105.

3. Philip M. Morse and Herman Feshbach, Methods of Theoretical Physics, McGraw-Hill Book Company, Inc., New York, 1953.

4. Imre Pal, Descriptive Geometry With Three-Dimensional Figures, Hungarian Technical Publishers, Budapest, 1962.

5. Bela Julesz, Binocular Depth Perception of ComputerGenerated Patterns, Bell System Technical Journal, Vol. 39, September, 1960, pp. 1125-1162.

6. Bela Julesz and Joan E. Miller, Automatic Stereoscopic Presentation of Functions of Two Variables, Bell System Technical Journal, Vol. 41, March 1962, pp. 668-676.

7. Gary A. McCue, Visualization of Functions by Stereographic Techniques, North American Aviation, Inc., SID63-170, January 20, 1963.

8. H. R. Puckett, Computer Method for Perspective Drawing, Journal of Spacecraft and Rockets, Vol. 1, No. 1, January, 1964, pp. 44-48.

9. E. G. Pare, R. 0. Loving and I. L. Hill, Descriptive Geometry. The MacMillan Company, New York, 1959.