Neither of the above. Instead there are bright, narrow lanes in the form of a cross. The rest of the record is dim (it would be dark but for the light that is scattered to it from the walls and ceiling and objects in the room). One arm of the cross runs along the line between you and the lamp, and the other arm passes through the first at a right angle [see drawing at top right in Figure 1]. The figure assumes that the center of the record lacks the usual label and central hole and that it is covered with grooves just as the rest of the record is. If that were actually the case, you would see an especially bright spot at the center of the record.

If you shift the record to the left, the bright lane between you and the lamp begins to curve slightly, but the other lane remains straight and still extends horizontally through the center of the record, as is seen in the top left drawing. Provided that the record is not shifted too much, the especially bright spot lies on the straight lane, near the other lane. One end of the curved lane points toward the lamp (that end is labeled L in the illustration), and the other end (labeled O for "observer") points toward you. If instead you shift the record toward you from its original central location, it is the left-to-right lane that curves gently; the other lane remains straight, as in the bottom right drawing. The bright spot now lies on the straight lane on the opposite side of the record from you.

When the record is moved both toward the left and toward you, the lanes separate into shapes that resemble hyperbolas, as in the bottom left drawing. One lane still passes through the center of the record; if the record is not shifted too much, the bright spot is still on that lane. The second lane lies near an edge of the record; it enters at one point on the edge, heads for the center and then swerves toward another point on the edge. The pointing is now divided between the lanes: one end of the first lane points toward the lamp, and one end of the second lane points toward you. If the record is shifted farther, the second lane shrinks toward the edge and then finally disappears.

The width of the lanes depends on the size of the bulb and the nearness of the lamp. A large bulb in a lamp close by creates wide, rather indistinct lanes. You can improve the visibility of the patterns if you cover the opening of the lamp with a sheet of aluminum foil through which a small hole has been punched. Then the lanes are narrow and clear. (Take care that the lamp does not overheat.)

What accounts for these patterns? The answer was provided in 1963 in an impressive study by J. B. Lott, who had just graduated from the Felsted School in Essex, England. It was clear, of course, that the patterns are reflections from the grooves on the record; the problem was to determine which parts of which grooves have the proper orientation to reflect light to an observer. Lott's solution was to imagine that the lamp and the observer are inside an ellipsoid: a dirigible-like structure generated when an ellipse is rotated about one of its axes. Wherever part of a groove happens to coincide with the side of an ellipsoid, light reflects to the observer. I have found that the analysis also explains a more common observation: the bright streak produced by a streetlamp seen at night from a car through a windshield that is being swept clear of rain or snow by the wipers.

To follow Lott's analysis, you first must understand how light reflects from a flat surface. Suppose that the record lacks grooves. Also suppose that the bulb in the lamp is small enough to be considered a point. Light rays from the lamp spread over the record and reflect from the surface. A line that is perpendicular to the surface at a point of reflection is called a normal, and the angle of the ray from the lamp is measured relative to it [Figure 3]. The ray is called the incident ray, and its angle is the angle of incidence. The ray that leaves the surface is called the reflected ray, and its angle with the normal is the angle of reflection.

The reflection obeys two simple rules. The normal, the incident ray and the reflected ray must lie in a plane, and the angle of reflection must equal the angle of incidence. If the center of a smooth record lies halfway between you and the lamp, you intercept a ray that reflects from the center. All the other rays reflect from the record surface but not in the direction of your eye. What you perceive is a single spot of light at the center of the record, as if the spot, rather than the lamp, were the source of the light.

I shall call the spot the "mirror image" of the lamp, because in producing it the record functions as a mirror. When your eye is at the height of the lamp, the mirror image is always halfway between you and the lamp. You can control its location on the record by shifting the record over the table or (equivalently) by moving your position. For example, if you shift the record to the left and also toward you, the mirror image appears in the upper right quadrant of the record. (The mirror image is, of course, the especially bright spot that, as I mentioned above, is seen in the pattern of reflections from a grooved record.)

The rules of reflection also apply to a surface that is smoothly curved, but in this case the normal is perpendicular to a line that is tangent to the surface at the point of reflection, as is shown in Figure 2. Lott assumed that the grooves on a record are a series of smoothly varying hills and valleys that form circles around the center of the record. If a radial cross section is taken through any one of the hills, and if normals are erected at points along the hillsides in the section, the hill bristles with normals-somewhat like the quills on an angry porcupine. The normal at the top of the hill is vertical. All the others are tilted radially from the vertical, either toward the center f, of the record or away from it.

How light reflects to an observer . from the grooves can be described in terms of the normals. If a ray is to reach the height of the observer's eye, the normal at the point of reflection must have a particular tilt. Lott argued that this requirement is easily met by every groove. Pick a spot around the circle of a groove, and then consider the normals along the sides of a cross h section through the hill. At least one of the normals will have the proper tilt for the light reflecting at the foot of the normal to be sent to the height of the eye. The requirement that a ray be sent to a certain height, then, is not responsible for the patterns seen on a record.

Lott reasoned that another, more restrictive requirement is responsible: the reflected ray must be directed properly, left or right, toward the eye. Only certain spots on certain grooves meet that requirement. The argument greatly simplifies the analysis, because it reveals that only the horizontal orientation of the normals to the circles formed by the grooves is important. The analysis is further simplified by the fact that horizontally the normals are radial: they can be extended through the center of the record. In what follows I shall ignore the vertical aspect of the normals and concentrate on their radial aspect.

To find which segments of which circles produce the patterns, Lott relied on a special property of an ellipsoid. If a light ray originates at one of the two focal points in an ellipsoid and reflects from the interior, it must pass through the other focal point. Lott imagined that when an observer receives a reflection from a record illuminated by a lamp, the lamp is effectively at one focal point of an ellipsoid and the observer at the other focal point. The record cuts horizontally through the lower part of the ellipsoid, and in the slice that is cut out, the wall of the ellipsoid forms an ellipse [see illustration below].

If the light is to reflect from the ellipsoid, it must reflect from the part of the wall that forms the ellipse. Because it must also reflect from the circle of a groove, the circle and the ellipse must coincide at the point of reflection. That in turn means that the circle and ellipse share a common normal there. Because the normal to any part of a circle is radial to the center of the record, the point on the ellipse that reflects light to an observer is a point with a radial normal.

To account for a full pattern of reflections on a record, one needs to consider a family of ellipsoids of different sizes. The record's slice through some of the ellipsoids leaves a large ellipse, and its slice through others leaves a small ellipse. In each case the points on an ellipse that are responsible for the reflection patterns are points with radial normals. These conditions were sufficient for Lott to solve equations for ellipses, circles and radial normals in order to find the points on a record that reflect light to an observer's eye. He restricted the situation to the one in which the lamp and the eye are at the same height above a horizontal record. The composite of the reflection points forms the lanes seen on the record.

I found that the general shape of the lanes could be predicted with a series of sketches without the need of equations. An example is shown in the illustration above, in which the record has been shifted from its initial central location directly to the observer's left. The top figure in the illustration is an overhead view of the ellipses that are left by ellipsoids in the plane of 11 the record. The y axis runs to the observer's left and right, and the x axis is perpendicular to the y. The bottom figure is a vertical cross section taken along the y axis. Included in it are the lowest parts of cross sections through two of the ellipsoids.

To construct the overhead sketch, you begin with the mirror image, which is the especially bright spot in the reflection patterns. Recall that when the observer's eye and the lamp are at the same height the mirror image is halfway between them. In the present case, that condition puts it on the y axis to the right of the record's center. The image is associated with an ellipsoid that just barely touches the top surface of the record, and so it leaves an ellipse that is as tiny as a point in the plane of the record. A segment of that ellipsoid is shown in the bottom figure in the illustration. Draw that point on an overhead sketch.

Around the point draw a small ellipse centered on the point and with its long axis parallel to the x axis. The ellipse is associated with an ellipsoid that is larger than the first one and extends through the record; this ellipsoid and its ellipse have the same h orientation. Now consider the normals at points around the ellipse. (Keep in mind the fact that a normal is perpendicular to a tangent to the ellipse.) Mark those points that have a normal that is radial-that is, one that can be extended through the center of the record. There are four such points: two are on the y axis (one on each side of the mirror image), and two are near the sharply curved ends of the ellipse.

Now repeat the procedure with similar but progressively larger ellipses derived from progressively larger ellipsoids. Each ellipse has four points with radial normals. After you have located a number of the points, connect them with lines; the lines replicate the bright lanes in a reflection pattern. If you tilt the record or bend it gently, the reflections are more difficult to explain, but the shapes of the patterns are similar.

When I came across Lott's paper and first took notice of the reflection patterns on a record, I suddenly felt I had seen one of the patterns in a different setting. Then I realized that whenever I drive through rain or snow at night and look at a streetlamp or the headlamp of an oncoming car through a wiper-cleared area of the front windshield the lamp has a streak running through it [see above right]. blight that streak be related to one of the bright lanes on a record?

The streak of light in a windshield was discussed in 1954 in a brief note by Paul Kirkpatrick of Stanford University. He attributed it to the fact that when the rubber blade of a windshield wiper rubs across the glass it scrapes furrows in gummy road debris that tends to adhere to the glass. Because the wiper moves in an arc, the furrows form circles around the pivot point of the wiper; the furrows may persist long after the windshield is dry.

When you look through the array of circles and toward a lamp, you intercept light that reflects from the sides of the furrows. In daytime the light is too dim to be visible, but at night it can be so apparent as to be annoying. The streak is often straight, with one end pointing toward the wiper's pivot. As your angle of view of the light source changes, the direct image of the lamp slides over the windshield, and the streak rotates around the image so that one end continues to point toward the pivot. Kirkpatrick mentioned that sometimes the streak is noticeably curved, with the lower end still pointing toward the pivot. Sensitized by Lott's analysis, I wondered whether the streak might be related to a bright lane on a record-the one on which the mirror image is seen.

After a little thought I realized that the situations were almost identical. The display I see in the windshield is essentially the one I see on the far side of a record when I move the record toward me and thus shift the mirror image to the far side, as in either of the bottom drawings in Figure 1. In both cases a bright line extends through the center of the circles, or at least points there, and the line can be straight or curved depending on the circumstances. Somewhere along the line there is an especially bright spot. With the record, the bright spot is a mirror image and the light source is on my side of the circular array. With the windshield, the bright spot is a direct image and the light source is on the opposite side of the array. I concluded that Lott's analysis applies also to a windshield streak.

Still, one detail nagged me. With the windshield, the lack of "near quadrants" (corresponding to the near side of a record) apparently eliminated the second bright lane or streak; any second streak (which would run left to right or would be strongly curved and separate from the first one) would presumably be below the pivot point of the wiper. Was there some way I could bring the second lane into sight? I tried and failed several times to make the streak appear. Then one day, with the setting sun seen through the curved section of the windshield on the passenger's side, I saw both bright streaks near the sun; they resembled the bottom left drawing in Figure 1. (If you watch for sun streaks, be careful about looking directly toward the sun for more than a second because of the obvious danger to your eyes.) I found that by altering my perspective I could make them join or separate at will-and in the process I almost drove off the road. I still do not know exactly why that second streak appeared.

During the past winter I enhanced the visibility of the streaks by allowing the salt that is spread to clear roads of snow and ice to build up on my windshield. The furrows ground out by the windshield wiper were then quite apparent. The streaks from them were probably due more to a complex scattering of the light than to simple reflections as envisioned by Kirkpatrick, but the results were similar. Once I noticed a novel feature of the streaks. As I drove in the direction of a low sun, the bare limbs of trees blocked part of the sun's image and radiated dark lines down through the sun streak.

When a car approaches you at night, the extension of the streaks from its headlamps toward the pivot point of your wiper forces the streaks to converge. Each streak also narrows as it nears the pivot, because the circles nearer the pivot are more strongly curved than the circles farther out.

Kirkpatrick mentioned an illusion of depth that can be perceived in the streak from a streetlamp. Because of the separation of your eyes, each eye sees its own streak. The separation between the streaks is greatest near the pivot and is least at the direct image of the lamp. You might perceive both streaks, but if your brain successfully merges them into a composite, their actual separation creates the illusion of depth: you perceive a single streak that appears to be a lit path extending from you to the lamp. If the streaks are gently curved, the path is curved too, as if it leads over an invisible valley and then up an invisible hill to the lamp. (Similar depth can sometimes be seen in a reflection pattern on a record.)

Most windshields are curved, particularly at the sides. The curvature alters the shape of the streaks just as the bending of a record changes a reflection pattern. The curvature also allows regions of gummy material to scatter light to you that would miss you if the glass were flat. It also momentarily traps some of the light from a lamp, as is shown in the illustration above, before the light is sent in your direction. During the entrapment the light reflects and scatters many times inside the glass, moving away from the direct image of the lamp. What you see is a bright streak that extends from the direct image. If you reach out the car window and block the direct image with a finger, the streak disappears. Similar streaks arise from corrective eyeglasses, but in a car they can be distinguished from the windshield streaks if you tilt your head: the streaks from the glasses rotate.

Here are a few more questions raised by reflections from circular arrays. If you shift a record enough in the light of a lamp, why does the bright lane through the center develop an end at which the reflection dims and then disappears? (A hint: recalling Lott's assumption that there is always a normal on a groove that sends a ray to the height of your eye, consider just how tilted a normal can be at the juncture of a hill and valley.) What sort of reflections are seen in a "laser disk"? Why are the reflections on both a record and a laser disk colorful when the lamplight is white?

Bibliography

A BINOCULAR ILLUSION. Paul Kirkpatrick in American Journal of Physics, Vol. 22, No. 7, page 493; October, 1954.

REFLECTIONS ON A GRAMOPHONE RECORD. J. B. Lott in the Mathematical Gazette, Vol. 47, No. 360, pages 113-118; May, 1963.

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