IN AN ANAMORPHIC PICTURE THE IMAGE of what is represented is distorted so that it can be perceived only by viewing it in an unusual way. Some paintings and photographs of this kind must be viewed almost from the edge or by means of a reflecting cylinder or cone. Such pictures were an amusement in the Renaissance and remain largely a curiosity today. The methods of creating and viewing them are interesting as part of the larger subjects of the nature of perspective and of visual perception.

Recently I received several examples of anamorphic photography made by David G. Stork, a graduate student of visual perception at the University of Maryland at College Park. He employs a black-and-white negative or a color slide to make a print that is anamorphic. Normally a print is made by placing the negative in an enlarger. Light directed through the negative passes through a lens and falls on the print paper, which is developed for the final photograph. Stork mounts a shiny cone in the path of the light so that it reflects the rays to parts of the print paper where they would not normally go. At the center of such a print, of course, is an empty area corresponding to the circular base of the cone.

Usually the resulting print is unrecognizable, but the original perspective can be regained with the aid of the same cone. It is placed at the center of the print and the observer looks down from an appropriate height, usually with one eye. The sides of the cone show a reflection of the print with the perspective of the original scene restored.

The image from the negative is distorted in the print in two ways. The parts of the image near the center of the negative appear near the outside of the print. Conversely, the parts near the outside of the negative appear near the base of the cone in the print. This radial inversion of the center and the perimeter gives rise to the curious pattern in the print.

The other type of distortion of the image affects its angular width around the center of the print. The parts of the image cast near the outside boundary of the print are spread in a circle around the center of the print more than the parts cast near the base of the cone. As a result most of the normal clues to perspective are lost.

The equipment needed for Stork's anamorphic photography can be quite simple. Stork works with a heavy metal cone machined to size. You could substitute a sheet of reflecting Mylar formed into a cone. (These sheets are available at hobby shops and kite . stores.) In addition to the standard f-stop diaphragm of the enlarger you need a means of controlling the width of the light beam above the cone. Another f-stop diaphragm mounted on a ring stand will do. A substitute can be a series of cardboard sheets each with a hole of different diameter in it. By changing sheets appropriately the photographer can control the exposure almost as well as he could with an f-stop diaphragm.

The cone must be positioned properly over the print paper while the photographer is working in the dark. Stork made a cardboard mask to fit into the holder for the paper. At the center of the mask is a hole of the same diameter as the base of the cone. By feel Stork can position the cone in the hole and then lift the mask off without disturbing the cone.

The procedure for exposing the paper is-as follows. Put the negative into the enlarger, turn off the lights, remove a piece of print paper from its box and put it in the holder of the enlarger. Lay the mask for the cone over the paper. After you have positioned the cone in the hole lift the mask off the paper without moving the cone. Turn on the light in the enlarger to expose the paper.

Several exposures are needed for one print because the light falling on the paper near the base of the cone is more

concentrated than the light that strikes the outer boundary of the paper. The time of exposure for the paper near the base is normal; for the paper near the outer boundary the time should be between 15 and 20 times longer. Stork manages to achieve approximately the right exposures by making the first one with a relatively large aperture that he has mounted on the ring stand. The aperture is either the variable f-stop diaphragm or a piece of cardboard with a large hole in it. The last exposure is made with a comparatively small aperture that confines the light to the apex of the cone. Since that region sends light toward the outer boundary of the print paper, only the perimeter of the paper is exposed.

Starting with the large aperture, Stork steadily reduces the aperture size so that less of the cone is illuminated. (Either the variable f-stop diaphragm is narrowed or the cardboard is replaced with another piece having a smaller hole.) He continues until the exposed region is confined to the perimeter of the paper. It is important that in the first exposure the circle of light not extend beyond the base of the cone. If the circle is larger light can reach the area on the paper around the cone without having been reflected from the cone. The print would be ruined, since it would not be a true image resulting solely from reflection.

A fair amount of practice is needed to master varying the size of the aperture and making the exposures. Beginners should work with black-and-white print paper so that experimenting with the exposures does not become too expensive. Stork recommends that when you are ready to make color prints from color slides, you work with Ilford Cibachrome color print paper.

According to Stork, almost any scene will do for a color slide. A good goal is to achieve an anamorphic print that resembles something totally different from the original scene. Better prints result if the slide has a relatively dark central area. Light from that area of the slide falls: mainly on the apex of the cone. No cone, not even a machined one, will have a perfect point. Hence light reflected from the apex will be distorted in an uncontrolled way.

If you could place the cone on the finished photograph exactly as it stood on the print paper during the exposure, you could undo this distortion. Such precise repositioning is arduous and not worth the trouble. Another difficulty with the apex is that the light reflected from it to the print paper is comparatively faint. You can avoid both problems by having a slide with a relatively dark central area so that what is reflected from the apex makes less differences in the print.

Stork suggests that if you are unable to machine a heavy metal cone or get one machined, you should mount Mylar on a solid wood cone. You can add weight to the cone by drilling a hole into the base and inserting a short length of metal rod into it. It is obviously important that the cone not move in the course of the repeated exposures.

A recent article in this magazine described similar distortions produced by reflections from plane and curved mirrors [see "Mirror Images," by David Emil Thomas; SCIENTIFIC AMERICAN, December, 1980]. Thomas presented a classification of the distortions that depends on the type of mirror employed. He defined two perpendicular axes on the surface of a mirror. The type of reflection from the surface can be classified by the type of curvature the axes have. For example, the side of a reflecting cylinder has one axis along the straight section of the surface (the axis runs along the side from one base to the other) and another axis (perpendicular to the first one) that travels around the circumference of the cylinder. The surface is a combination of a plane mirror (the straight axis) and a convex mirror (the curved axis). The reflections along the straight axis behave as they would with a plane mirror, and the ones along the curved axis are the same as those from a convex mirror.

In Stork's setup the light is reflected from the side of a shiny cone, which can also be considered a combination of a plane mirror and a convex one. The axis for the plane mirror lies along the side, running from the base to the apex. The axis for the convex mirror runs around the cone, perpendicular to the other axis and parallel to the base of the cone.

The cone introduces a complication, however, which is that the curvature of the convex axis changes along the side. Toward the apex the axis is highly curved but toward the base it is less curved. Still, the reflections leaving any small area on the side of the cone can be thought of as coming from a combination of a plane mirror and a convex one. The axis for the plane mirror is responsible for the exchange of the center and the perimeter of the slide when the print is made. The axis for the convex mirror is responsible for the spread of the image into a circle around the center.

Thomas demonstrated how a mirror can invert and reverse an image with respect to the object casting the image.

Inversion refers to an exchange of placement between the object and its image. The exchange can be of far and near, left and right or top and bottom. Plane and convex mirrors invert only far and near.

Suppose you hold a meter stick perpendicular to a plane mirror in such a way that the end with the zero mark touches the mirror. When you stand at the other end of the stick, the end closest to you in the mirror image is the one with the zero mark. The reflection from the plane mirror inverts far and near. A convex mirror does the same.

A plane mirror does not invert left and right or top and bottom. You might think left and right are inverted, but they are not. Face the mirror and hold a meter stick with your left hand on the zero mark and your right hand on the one-meter mark. Look at the reflection of the stick. The image of the zero mark is also on your left (yours, not the image person's) and the one-meter mark is on your right. Nothing has been exchanged.

Left and right are exchanged in the image person. The zero mark is held by the right hand, the one-meter mark by the left. Thomas calls such an exchange a reversal. Plane and convex mirrors cause a reversal and one inversion (of far and near). Any mirror that gives rise to an odd number of inversions reverses handedness.

I applied Thomas' analysis to the setup by Stork. In place of the slide I visualized a large letter F held by a small person. The variations in handedness could be followed by noting how the image of the F appears in the image of the person For this purpose I darkened a square in one corner of the F. In my imaginary demonstration light rays traveled from the F and the person through the lens of the enlarger and were then reflected from the surface of the cone onto the print paper.

The orientation of the F is altered by the lens. (I shall continue to use for the lens the terminology employed by Thomas for mirrors.) The lens inverts left with right and top with bottom. The result is an image of the F rotated 180 degrees from its normal orientation. (I am not being fully precise here. The image is not really present in the space between the lens and the cone because the rays of light passing through the lens have not yet crossed one another to create a focused image. If the image were in focus on a card inserted into the light, you would see what I have described.) The image also has a reversal of handedness. The darkened square is held by the left hand of the person in the initial arrangement, but it is held by the right hand in the image of the person cast by the lens.

The reflection of the F by the cone is essentially the same as the reflection from a combination of a plane mirror and a convex one except in one important respect: the reflection falls on a flat surface before the image reaches you. The rules governing inversion and reversal of an image of yourself in a mirror must be modified. The final image is still inverted top with bottom compared with the original orientation. The left-right inversion caused by the lens, however, is canceled by the reflection from the mirror onto the photographic paper, as is the reversal of handedness. The final image has the person holding the darkened square with the left hand just as in the initial setup. In sum, the image has had one inversion (top-bottom) and no reversal.

When the final print is viewed by means of a reflection in the side of the cone, the reflection is in effect from a combination of a plane mirror and a convex one. The rules governing the reflection, however, are not the same as those that hold when you see a reflection of yourself with such a combination of plane and convex mirrors. The reflected image is a virtual one, since the rays that are reflected do not cross to form a real image. Instead they diverge when they leave the side of the cone.

The observer's eye collects the diverging rays and focuses them onto the retina. The virtual image created by the cone does not have the same orientation as the image in the photographic print. Left and right are inverted. The reflection also reverses handedness. In the print the person is holding the darkened square with his left hand. When the image is reflected in the side of the cone, the square is held by the image person's right hand.

Stork has made anamorphic prints with two other setups. In one he reflects the light from the enlarger with a plane front-

surface mirror so that the rays fall obliquely across the print paper, which is placed off to the side rather than directly under the enlarger. The result is a distorted print, similar to some anamorphic paintings, that must be viewed almost from the edge in order to re-create the proper perspective. For example, the hot-air balloons in Figure 1 are clearer if the photograph is viewed (with one eye) from a slant angle or reflected in a plane mirror held tilted at one edge of the photograph. The sequence of inversion and reversal of an image in the exposure of the photograph is the same as it is with the cone.

To make the proper exposures on the anamorphic print made with a plane mirror you must adjust their duration across the width of the print paper. The areas farther from the mirror require more exposure than the areas closer to it. Stork adjusts the exposure by inserting a sheet of cardboard between the mirror and the paper. He begins the exposure with the cardboard fully blocking the light. Then he gradually withdraws the cardboard until finally the edge of the paper nearest the mirror is exposed. In this way the farther regions are exposed longer than the nearer ones.

Stork cautions that much experimentation and patience are required with both a reflecting cone and a plane mirror to make a satisfactory print. The focusing is never quite right. If one part of the paper is in focus, another part will not be. Stork generally adjusts the focus for the middle of the anamorphic print or for any particularly important feature of the print.

He also adjusts the f-stop diaphragm in the enlarger. A small aperture (a larger f number) gives more depth of focus. The smallness of the aperture means that longer exposures are needed because less light is reaching the print paper, but the improvement in focus is worth the extra effort.

Stork also employs a shiny cylinder to reflect the light rays to the print paper. The light from the enlarger is reflected from a plane mirror and then from the cylinder before it falls on the paper. To make the light from the plane mirror fall only on the cylinder and not directly on the paper, Stork blocks off part of the light so that only the cylinder is illuminated.

The procedure for exposing the paper is similar to that for making an exposure with a cone. In darkness Stork mounts the paper in its holder and lays on the paper a placement mask that has a semicircular cutout into which he positions the cylinder. Then he lifts the mask off the paper before he illuminates the cylinder. If the mask is shaped so that it does not block any of the light reflected from the cylinder, it can be left on the paper during the exposure.

The geometry of the cone setup is simple. Consider a cone that has an apex with a known angle. I shall call the half angle of the apex phi. It should be less than 45 degrees; a half angle of 30 degrees works well. Next consider a ray of light that is initially vertical. It strikes the side of the cone at an angle to the surface that is also phi. A light ray always reflects from a surface at the angle of its approach. Therefore in the cone setup the angle between the reflected ray and the surface is also phi.

Suppose the center of the cone's base is considered as being the center of a coordinate system. If the cone had not been in the way, the vertical ray would have struck the paper at a distance r from the center. With the cone in place the reflection redirects the ray so that it now strikes the paper at a distance greater than r from the center. The new distance is given by the equation in Figure 5.

One can predict the appearance of the anamorphic print by plotting the new positions of each section of the original scene in the slide. With the cone displaced focus the slide on a sheet of ordinary paper. Choose a small area of the image and measure its distance r from the center point (where the center of the cone's base will be once it is positioned on the paper). Then use the equation to determine the new distance r' for that section of the scene. A simple procedure for measuring the half angle of the cone's apex is to-take the inverse sine of the radius of the base divided by the distance along the side of the cone (from the apex to the base).

To map the original scene into the anamorphic one completely position a circular grid on the paper. Choose a section of the original scene that is at a distance r from the center and lies within a certain angle around the center. That area of the picture should be reproduced at a new distance r' and within the same angular extent around the center. If the new location is adjacent to the base of the cone, the new area will resemble the original area. If the new location is away from the base, the new section will appear to be stretched out along a section of a circle around the center. This procedure for mapping normal drawings into anamorphic ones has been described in these pages by Martin Gardner [see "Mathematical Games," SCIENTIFIC AMERICAN, January, 1975].

A faster way is to have a computer plot the anamorphic pattern. The input must include the coordinate positions of all the sections of the original scene: the initial radius R and the angular extent of the section, say one degree. The computer will compute the new positions and (if a plotter is attached) plot the results within one degree at the new radius.

Waldo Tobler of the University of California at Santa Barbara created an anamorphic computer-graphical picture from a drawing of a woman Stork had made on graph paper. Patricia S. Irle, one of Tobler's students, entered 1,450 data points for the drawing into a Tektronix 4051 computer. The original drawing was printed out by the computer. Then the program effectively reflected the image from a cone by applying the transformation equation to determine the new distance of each point from the center of the picture. Finally the distorted picture was printed out by the computer. When Stork positions his cone at the center of the printout and peers down into the reflecting surfaces of the cone, he sees an image of the original drawing. Without the aid of the cone he would see only the anamorphic version.

The same kind of transformation can be done with a home computer. If you do not have a printer, the transformation can be displayed directly on the monitor screen. Write a program that "draws" a design close to the center of the screen. The pattern will actually be composed of many small rectangular elements that are turned on by the computer. Since the positions of these elements will have to be remembered by the computer, you must store the positions of the rectangles.

For each rectangle have the program apply the equation for the transformation by a cone. New rectangles, farther from the center of the screen, will be turned on. This new set constitutes the anamorphic picture. Place a cone at the center of the screen and look into its sides. In the reflections you will see a likeness of the original drawing.

The reproduction is not complete because the monitor lights up rectangles, not small dots. Some of the details of a curve are lost. The rectangularity of the elements also means that the angular extent of part of the design will not be mapped faithfully at the new radius. Still, the speed of the computer makes it possible to analyze many patterns. Other anamorphic pictures can be studied if the equation for the transformation is altered. Examples can be found in the article by Andy A. Zucker listed in the bibliography for this issue.

Some types of anamorphic distortion are particularly interesting. For example, Stork sent me the results of three simple drawings that had been distorted anamorphically in a cone. In one he had drawn a heart (in Valentine style). He colored the heart red and the area around it white. After the transformation what he saw was an inverted white heart with a red surround. The color and orientation were reversed because the cone exchanged the center of the sketch for the perimeter and vice versa.

For another example Stork drew a "happy face": a circle with two dots for eyes and a semicircle for the mouth. The transformation results in another happy face of the same orientation. I noted that if the mouth was originally almost a straight line, the transformation to the anamorphic version made the face happier. Can you think of any design for which transformation by a cone results in a recognizable design with a transformed meaning?

Last November I described a visual illusion that arises when you scan a television screen horizontally. When you scan the screen properly, several images of the picture on it can be seen floating in the air off to the side of the set. The illusion depends primarily on two things: the persistence of vision and the rapid creation and disappearance of the picture on the screen.

Brian Glassner, a student at Cleveland State University, pointed out that the same illusion can be seen in the electronic displays on some types of calculator and on many of the newer pinball machines and other electronic games. If the display consists of light-emitting diodes, it is likely to be pulsed. The pulsation is too fast for the eye to detect, but it does mean the display is constantly appearing and disappearing.

If you sweep your eyes rapidly across your field of view, the flashing light will excite a series of areas on your retina. After the sweep each area in the series provides a separate image of the display. As a result you perceive the images as floating to the side of the actual display.

I described another visual illusion in this department for March, 1978. If a swinging pendulum is viewed while one eye is covered with a dark but not opaque filter, the pendulum appears to move in an ellipse. This illusion, known as the Pulfrich illusion, has been attributed by some investigators to the delay of perception by the covered eye. Apparently the reduced intensity of the light reaching that eye results in a delay of the signal transmitted to the brain. The resulting discrepancy of information from the two eyes forces the observer to interpret the pendulum as being closer or farther away than it actually is. The difference between the apparent distance and the true distance varies with the speed of the pendulum, being less when the pendulum slows as it reaches the end of a swing.

Recently Jerry Harris of Worthington, Ohio, described a related illusion that can be produced with a display of a Lissajous figure on an oscilloscope. He connected a signal generator to the vertical input of the oscilloscope and another signal generator to the horizontal input. Sine-wave signals from the generators swept the oscilloscope trace vertically and horizontally.

When the frequencies of the signals were adjusted properly, the trace on the screen formed a Lissajous figure. If the frequencies were slightly displaced from the optimum values, the figure still appeared, but it gradually varied. An observer will often interpret the varying display as a three-dimensional object revolving slowly about a fixed axis. The perspective of the seemingly three-dimensional object appears to vary.

When the frequencies match, the figure might be a circle. With a slight mismatch the circle appears to revolve about an axis on the screen. Sometimes the full plane of the circle is seen. At other times the circle is seen on edge. The direction of the rotation is arbitrary; some observers can switch rapidly from one interpretation of the direction to the other.

Harris modified this classic demonstration by holding a dark filter (a pair of partly crossed polarizing filters) over one eye to delay that eye's perception of the oscilloscope display. The delay introduced depth into his interpretation of the object, presumably as it does in the Pulfrich illusion. The appearance of depth removed the ambiguity of the rotation of the apparent three-dimensional object, which was seen to rotate about a vertical axis in the plane of the screen. When Harris tilted his head so that his eyes were along a vertical axis, the rotational axis of the object appeared to be horizontal. If he shut either eye, the three-dimensional effect disappeared.

Bibliography

HIDDEN IMAGES: GAMES OF PERCEPTION, ANAMORPHIC ART ILLUSION. Fred Leemann. Harry N. Abrams, Inc., 1976.

ANAMORPHIC ART. Andy A. Zucker in Creative Computing, Vol. 3, No. 4, pages 137-140; July/August, 1977.

ANAMORPHOSCOPES: A VISUAL AID FOR CIRCLE INVERSION. Philip W. Kuchel in The Mathematical Gazette, Vol. 63, No. 424, pages 82-89; June, 1979.

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