# Which point on a diffuse reflecting plane is brightest to the viewer?

Given is the plane y = 0. A point light source is positioned at (1,10,1). The
plane reflects only diffuse. The viewer is positioned at (4,6,4) and sees all
points on the plane with 0 <= x <= 8 and 0 <= z <= 8. At which point on the
plane does the viewer perceive the highest reflection?


I think the answer is (1, 0, 1), the projection of the point light source onto the reflecting plane. Because a diffuse reflection scatters light equally in all directions so it doesn't matter where the reflection is relative to the viewer. All that matters is where the reflection is relative to the point light source, because the point light source shines a varying amount of intensity per unit area on different parts of the plane. But the highest concentration of intensity will be shined along the plane's normal vector aka the nearest point on the plane. Is this correct and if not how do you do this? Thank you

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I think that this problem is a bit more complicated, since a point on the diffuse surface will in its turn act as a light source which will also decrease in intensity the further you are away from it. So the distance from the viewer to the brightest point also affects the "location" of the brightest point. I am not 100% sure, but the brightest spot might be found by minimizing the total path length. But you would have to do some maths to be sure. –  fibonatic Jul 17 '13 at 13:15
Hmmm I see. No not quite this is practice exam. –  Aurast Jul 17 '13 at 13:26
I think @Gugg has it right with minimizing the product of the squares of the path length. Want to write it into an answer? If not I will. –  Kyle Jul 17 '13 at 13:32
@Gugg: Wasn't going to solve it, just give a hint. If Aurast wants to solve it fully in an answer of his own once he works it out definitively, that's great. Maybe I'll go the other way round... if he doesn't post a full answer, I'll put something together. –  Kyle Jul 17 '13 at 13:36
@Gugg Ummm... not sure where option 5 is coming from. If one path was << the other, then maybe, but they're going to be about of equal magnitude. –  Kyle Jul 17 '13 at 13:54

Yes, your analisys is right. Note that the question is not about which point the viewer receives the most light power from, but which one appears brightest. The "appear brightest" part factors out distance to the viewer. Distance will only make a object appear smaller, not different in brightness. Therefore, as you said, the part of the plane that appears brightest is the brightest part of the plane, and will appear to all observers to be the brightest part of the plane.

From the comments, it is clear people are having a problem with "observed brightness" as apposed to the amount of light reaching the observer. To be clear, I am talking about the brightness of the object focused on some image plane, like the film or sensor in a camera or your retina. The light power reaching the observer falls off with the square of the distance. However, the linear dimension of the focused object on the image plane falls off with distance, which means its area falls off with the square of the distance. Since the total light hitting the observer also falls off with the square of the distance, the focused projection onto the image plane remains the same brightness. No laws of conservation were violated since the image of the distant object is smaller, so the total light received is correct.

For a more intuitive feel for this, think of photographing a outdoor scene in daylight. You have a person 10 feet away and a mountain 10 miles away, both illuminated the same by the sun. Even though the mountain is 5000 time further away, the two objects produce about the same brightness on the film. If this weren't true, all those tourist pictures of someone ovelooking the Grand Canyon would be impossible.

I think the question is rather clearly worded to ask for this apparent visible brightness, not the total light power since that's not how we perceive brightness.

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Please define brightness. Explicitly describe how this is not related to the power one observes. –  Will Jul 17 '13 at 14:33
So you are saying that for example a big and very hot distant star would appear brighter than the moon, viewed from the night sky on the surface of earth? –  fibonatic Jul 17 '13 at 14:43
@Will: Observed brightness is the light power reaching the observer per solid angle. The same light bulb 1 m away and 3 m away has the same observed brightness. They both would expose film in a camera equally, for example. Distance causes the object to appear smaller instead of dimmer, so the total power reaching the observer falls off with the square of the distance. The brightness as imaged on film or a sensor or your retina stays the same. –  Olin Lathrop Jul 17 '13 at 15:41
@fibon: Your example is so extreme that other effects get significant. The sun seen from earth or Mars has the same brightness, but appears smaller on Mars. However, when the object on the image plane gets down to resolvable size due to limits of the optics, then it remains the same size and gets dimmer with distance, which is what happens with your distant star. I interpret the OPs question to imply large enough viewable size that this effect remains insignificant. –  Olin Lathrop Jul 17 '13 at 15:45
A TA got back to me and said this is correct. He referenced Lambert's cosine law and also said that an easy, non-mathematic way to think of it is that perceptual brightness is sort of an aspect of color, the difference between RGB (100,100,100) and RGB (200,200,200). For a diffuse surface you can think of the light sources as painting the surface with colors. No matter where the viewer is positioned the paint looks the same color. –  Aurast Jul 18 '13 at 8:28

Edit 3: Yes, you are correct. For an excellent explanation, see the Wikipedia entry on Lambert's cosine law. You can safely ignore the remainder of this answer. NB: Thanks for the question!

Edit 1: It appears that in this answer I reasoned that we are looking for luminance (even though I was not familiar with the terminology at the time), and having now looked at Wikipedia, I still think that is correct. I assume that what I "derived" below comes fairly close or exactly down to the use of its definition. Olin's answer seems to refer to luminosity, and since I didn't "derive" that, I think that luminosity is "wrong" for this situation. :)

Edit 2: Also, I took your "a diffuse reflection scatters light equally in all directions" for granted and referring to points, although it seemed kind of weird. But since browsing the Wikipedia, I came across Lambert's cosine law, which indicates that I might have to read that differently. Correcting for a Lambertian surface, I fear that Olin might very well have been right all along. (And the question then does become almost tautological, because the cosines cancel out by definition, which is why his answer is so short.)

I'm going to have a half-hearted shot at this. It's a bit messy, and I recommend not relying on it. It also shows how you can make a mess of something that perhaps is as easy as Olin makes it seem. Perhaps this is even plain wrong...

The length of the first leg from $(1,10,1)$ to $(x,0,z)$ equals $$d_1=\sqrt{(1-x)^2+10^2+(1-z)^2}.$$

The intensity of the incoming light at $(x,0,y)$ is proportional to $$\frac{1}{d_1^2}=\frac{1}{(1-x)^2+10^2+(1-z)^2}.$$ Note or simply assume* that nearby points $(x\pm\delta,0,z\pm\varepsilon)$ have virtually equal intensity of incoming light.

The length of the second leg from $(x,0,z)$ to $(4,6,4)$ equals $$d_2=\sqrt{(4-x)^2+6^2+(4-z)^2}.$$

The intensity of light received from this exact point is proportional to $$\frac{1}{d_1^2}\frac{1}{(4-x)^2+6^2+(4-z)^2}=\frac{1}{(1-x)^2+10^2+(1-z)^2}\frac{1}{(4-x)^2+6^2+(4-z)^2}.$$ (I think I am assuming something here, but I'm not sure what.)

But note that as $d_2$ increases, point ($x,0,z$) becomes "smaller". You might interpret this as having to "include more points" $(x\pm\delta,0,y\pm\varepsilon)$ in the immediate vicinity (neighbourhood). But: How much more points?

I suggest that that depends on the angles $\varphi\pm \gamma$ between the line from $(4,6,4)$ to $(x\pm\delta,0,z\pm\varepsilon)$ and the $xz$-plane (with $\delta$ and $\varepsilon$ possibly dependent, because they represent a "viewing cone").

So the amount of points to be included is proportional to the area consisting of all intersections of all lines from $(4,6,4)$ to some $(x\pm\delta,0,z\pm\varepsilon)$ and the $xz$-plane with angles $\varphi\pm\gamma$. Let's call this $A(x,0,z;\gamma)$, because I'm very bad at trigonometry.

So, the final intensity to be perceived is proportional to $$A(x,0,z;\gamma)\frac{1}{(1-x)^2+10^2+(1-z)^2}\frac{1}{(4-x)^2+6^2+(4-z)^2}.$$

Maximize that with respect to $(x,0,z)$. This should give you an answer in terms of $\gamma$.

Take a realistic $\gamma$ or take the limit $\gamma\downarrow0$. (Taking the limit will probably justify the earlier assumption*.)

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