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.