If parallel light rays come in (like from a far away source) then they will focus at a point that is a distance half the radius from the center. Sorry, I feel like I should derive this R/2 focal point – but maybe I will do that later.

So, the picture is for a 2 dimensional mirror. Normally, you use these 3-d mirrors, not just a flat curve. However, in this case it is an extended 2D circle instead of a real circular mirror (the building in no round). What does this mean? Let me call the direction perpendicular to the ground, the y-direction. The reflections in the y-direction are just like a flat plane mirror. The reflections in the x and z-direction are like a circular mirror. This makes it a little complicated to analyze.

How do I deal with a cylindrical curved mirror instead of a 2-d mirror. My first mistake was to think about the cylindrical mirror as a whole bunch of 2-D circular mirrors stack on top of each other, like this.

This seems like a logical way to approach the problem, but it isn’t. The Sun is not in the same 2-D plane as this slice. This means that I can’t use the same 2-D ideas and just add them up to make it 3-D. Here is a better way to slice it.

If you can’t tell from my image, in this representation the curved cylindrical mirror is a series of vertical plane mirrors. These are easier to deal with. Looking from the above the hotel, the image should look just like the 2-d circular mirror. Here is a sample. I found some curved mirrors and stacked them on top of each other. This image shows the reflected light when the curved mirror is facing the Sun.

Suppose I break my building into n vertical mirrors of width ? s such that the total length of all the mirrors is s, the length of the building. You can see that there would be a hotspot. The width of the spot would depend on the size of the vertical mirrors. There is no need to go to a continuously curved model because the actual hotel is probably made of a whole bunch of flat mirrors (I could be wrong here though). Let me consider light reflecting off of one of these vertical mirrors.

This if for light from the Sun at an angle ? above the horizon where the plane of the mirror is perpendicular to the light. How big is the area of reflected light? Well, if the width is ?s, then it would have an area of reflection of:

Why is this important? Intensity, that is why. If the intensity of the sunlight is I₀ (about 1000 Watts/m²), and the mirror is perfect, then this light would be distributed over the reflected area. For a mirror that has a height h, the intensity of sunlight in the reflected area would be:

A couple of notes:

• The angle, ?, is the angle the sunlight makes with respect to the horizontal. If the Sun is directly overhead (which wouldn’t happen in Las Vegas), then the reflected intensity would be infinite because the size of the reflected area goes to zero. It is still a finite amount of energy.
• I₀ is in there twice. This is because light from the Sun hits the reflected area as well as light reflected from the mirror.
• The position of the hotspot doesn’t really depend on how high the Sun is in the sky or how tall the building is. There will be a hot spot as long as the reflected areas are long enough to reach the hot spot area.
• Remember, this is just for one mirror – the center mirror with the Sun straight on. The other mirrors would have different reflections because the light would be coming from the side. But, I will just assume that these reflections are all about the same.
• This analysis is getting way longer than I had projected. Oh well, I can’t stop now.

What about the intensity from the whole thing? Suppose the building has a radius of curvature R and is s long (arc length) with n segments. Then the total intensity will be:

At first, I didn’t like this answer. It doesn’t depend on the size of the mirrors, just the number. If use smaller mirrors, the intensity increases. Actually, this is ok. As I use smaller mirrors, the size of the hot spot gets smaller too.

Data about the actual building

What do I need to know? Well, the radius of curvature would be nice – but that only tells me the location of the hot spot. I guess the only thing I really need to know is n and ?. From the Las Vegas Review article, it says the size of the hotspot is 10 x 15 feet. So perhaps the size of each window is about 10 feet. From google maps, I estimate that the length of the building is around 300 feet. This would mean that there should be about 30 of these mirrors (each 10 foot). What about ?? How about I use something like 60 degrees above the horizon?

Reflection efficiency is also important. The article mentions that the designers installed a film that would scatter (in other directions) 70% of the light.

Estimated intensity

If I use an intensity of sunlight around 1000 Watts/m², then the intensity of the light in the hot spot would be:

Ok, that is a little bit higher in intensity than I expected.

Melting Bags

I have no idea what that intensity of light would do. In particular, I want to consider what it would take to melt a bag (like in the article). So, I melted a bag. Here is a quick video of me melting a bag with a magnifying glass (diameter of 5 cm). Notice how the colored part of the bag melted fine, but the white part wouldn’t get hot enough.

What kind of intensity of light is that bright spot? If assume a the lens is maybe 80% efficient, has a diameter of 5 cm and a spot size of about 0.3 cm then what is the intensity?

So, that is way larger than the estimation from the building. However, that bag melted right away. I bet if I decreased the intensity by a factor of 10, it would still melt but take a lot longer (just like in the article).

I guess that is good enough.

Authors: Rhett Allain

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