By Grant Ocean

# Case B: Image in a Portion of Mirror and Window

What portion of a plan mirror must be used in order to view an image is a popular topic in optical physics. As depicted in the diagram below, if you stand in front of the mirror and E is your eye level, is the portion of the mirror required to view both the tip of your head (I) and your feet (I’). So is your entire image. After countless experiments and measurements done by the physics lab around the world, the conclusion is that you will need an amount of mirror equal to one-half of your height to view the entire image of yourself in a plane mirror. No matter where you stand in front of a mirror, you always need a half mirror to view yourself since dom = dim.

The result is consistent with geometry. For a right triangle such as in the above diagram, the middle line  is exactly one-half of the end line . This seems to be a proof that our perception of mirror images follows a geometrical pattern and can be explained by geometry. However, it will be demonstrated that this geometrical rule cannot really explain why you only need a half mirror to view your full image.

Let’s assume that you are 160cm tall and stand 100cm from a mirror. You set a ruler 10cm from your eye to measure the height of your mirror image. The result for your perceived image height (Pih) will be

Let’s assume that you need the height of the mirror (Hm) that is one-half of your own height, i.e., Hm = ½ H, to see your full height. When you stand the same distance from the mirror (dom = 100cm) and measure the perceived mirror height (Pmh) at Mp of 10cm, we get

Thus, Pih = Pmh when Hm = ½H. The results will be the same no matter how far away you stand in front of the mirror. Varying the values of dom and Mp will not affect the balance. The fact that one-half of a mirror is required to view your full image can be summarized as

Pih = Pmh

or

However, if you set Mp at 101cm, which is slightly more than the distance between you and the mirror which is 100cm, then your perceived image height is

On the other hand, the maximum perceived mirror height (which is determined by the maximum Mp that is 100cm) is

In this case Pih > Pmh. Of course, you cannot measure the mirror or the image in the mirror at a site beyond the mirror, which is physically impossible. But one can still argue that even though the maximum Mp, the measure-point, for the mirror is 100cm, which is the distance of the mirror from the observer, you can focus on a spot beyond the mirror since Mp is also the focal distance. This is the standpoint of optical geometry and our common sense. We think that we can look into the mirror and see the image in there. As shown in the above diagram, our eye located at E position is supposed to be able to see the full image II’ along the lines of and

The above calculations tell us that if you could look at your image beyond the mirror, you would need a mirror taller than half of your full height to view your entire body. Obviously, this is in conflict with the observations and experiments. It is a popular physics lab activity around the world to find out what portion of a mirror is required to view your image. The results are always the same: one-half of the mirror is all you need for the purpose.

Therefore, we have to reach the conclusion that we cannot look at an image beyond the mirror, and that the maximum Mp is dom. In accordance, we cannot see the full sized II’, which is the image’s height (Hi), unlike what is believed traditionally by the optical scientists and the general public. We can only see the perceived image height (Pih) on the mirror, which is half of II’. When we look at images in the mirror without a ruler to measure them or an object in between to focus on, the mirror itself is acting as the measure-point. The mirror is the maximum limit of Mp. In this case, the perceived mirror height equals the actual mirror height, i.e., Pmh = Hm.

The reason why we need only a half portion of the mirror to view our full image is because of this equation: Pih = Pmh, not because of the geometrical rule depicted in the diagram above. The fact that the portion of a mirror required for full view happens to follow such a geometrical rule is also a geometrical coincidence. The geometry can only be utilized to “save the appearances”, but cannot explain the phenomenon.

As we look into the mirror images, we have a tendency to think that it is similar to looking through a window at the objects outside. Actually, mirror images are similar to window images. If we look at an object outside a window, we can use Ph = Pwh (where Pwh is the perceived window height and it equals the actual window height when our eyes focus on the window) to determine what portion of a window is needed to view the entire object. We will be able to see the full object through a window that is one-half of the object’s height so long as the object is the same distance away from the window as we are away from the same window inside. The window is the maximum Mp just like the mirror.

Unlike the plane mirror, a movable window can be used to illustrate some points because it can be moved back and forth without changing the distance between the observer and the object. Let’s assume that you are looking at an object of 160cm in height at a distance of 200cm from you outside the window. If you put the window right in the middle between yourself and the object, you can look through an 80cm high window (as shown earlier for the plane mirror) and see the whole object. And if you place the window at 40cm from you, we can calculate and get

As a result, you can look through a 32cm high window and see the object. If you bring the window to a spot that is only 10cm away from you, you can look through an 8cm high window (as calculated earlier) and see the entire object. Thus, without changing the distance between you and the object which is still the same size, the closer the window is from you, the smaller portion of it is needed to view the full object, and vice versa. (By the way, all the calculations in Case B have been confirmed by the follow-up experiments.)

There is another traditional standpoint on our size perception, which is called the “size-distance invariance hypothesis” (Gregory, 1998) or SDIH for short. This SDIH states that retinal size (R) is transformed into perceived size (S) after taking apparent distance (D) into account. The hypothesis can be expressed in an equation form

S = R · D

It is worth noting that the perceived size S and the apparent distance D are subjective, not the measured values as Ps and d in the new perceived size equation. It is because our knowledge of distance is traditionally believed to be forever veiled from the mind. All we have about an object’s size is the information that can reach our retinal surface. The distance of an object has to be inferred or estimated by using our experience, “educated” knowledge, and depth cues. As a consequence, the apparent distance of an object would be various from one individual to another due to their differential experiences and knowledge about depth. Since S and D are supposed to maintain their relative values in relation to each other, the perceived size S will co-vary with the differently estimated distance D to keep the above equation balanced. This is why the perceived size S in SDIH is subjective and cannot be measured precisely at all. But, the fact that a precise portion of the mirror or window is needed to view an image illustrates that there are no such things as subjective S and D. The SDIH is not congruous with the observations and experiments.

The perceived size S in SDIH is the actual linear size of an object, e.g., H and Hi. It means that we can always perceive the actual object’s size as long as we have the distance more or less figured out. As we have already known, it is impossible to perceive the actual size of an object on a mirror or through a window because mirrors and windows set a limit on how far away we can perceive the objects. At most, we can merely perceive the images of objects on the mirror or window. There is absolutely no way that we could see images beyond the mirror or window.

When you stand in front of a window and look at an object outside, apparent distance between you and the object should not change dramatically after the window has been moved closer to you. You should have the knowledge that the distance between you and the object in principal stays the same so long as you and the object stay put on the same spots, even though the window has moved. In addition, the moving window should not have the slightest effect on the visual angle θ, which determines the size of the retinal image. So the perceived size S in the equation S = R · D should remain the same no matter where the window is. Therefore, the SDIH cannot explain why the object appears smaller when the window has been moved closer to the observer, and why it seems getting larger when the window has been moved farther away from the observer.

The visual angle θ can be calculated by tan θ. The size of the retinal image can be obtained by the formula: R = tan θ · n, where n is the distance between the retina and the nodal point which is about 0.17cm. Let’s calculate the retinal size by using these values: H = 160cm and d = 200cm. Now we have

R = tan θ · n = 0.8 rad × 0.17cm = 0.136cm

It is very unlikely that we perceive the image as small as 1.36mm when we look at a 160cm object from two meters away. Even if we measure the object at only 1cm away from the eye, the object still looks as tall as 8mm. The convergence micropsia (Hollins, 1976) is probably the case in which the most reduced size of an object could be perceived. Nonetheless, the perceived size in the condition of micropsia is still much bigger than the calculated retinal size. As a result, we may not perceive the retinal size R at all.

# References

Gregory, R.L. (1998). Eye and brain (5th ed.). Oxford: Oxford University Press.

Hollins, M. (1976). Does accommodative micropsia exist? American Journal of Physiology, 89, 443-454.

Kaneko, H, & Uchikawa, K. (1997). Perceived angular size and linear size: The role of binocular disparity and visual surround. Perception 26 (1), 17–27.

Kosslyn, S.M. (1975). Information representation in visual images. Cognitive Psychology, 7, 341-370.

Zygmunt, P. (2001). Perception viewed as an inverse problem. Vision Research, 41(24), 3145-3161.

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