A while ago I was wondering how we could use mathematics to increase the efficiency of solar panels. The kind of mathematics I was thinking about in particular was Dynamical Billiards. Though I think it is improbable that it is currently technologically feasible to create the solar panel I am thinking about, I guess the following question could be interesting from a mathematical point of view.

We first need some definitions:

An $(k+l)$-setting is a collection of two sets of points in $\mathbb{R^d}$, $\{n_1,...,n_k \}, \{m_1,...,m_l \}$ such that each of the points $m_i \in \{m_1,...,m_{l-1} \}$ is connected with the point $m_{i+1}$, by some continuous function (the continuous function may differ for each pair of points $(m_i,m_{i+1})$).

For example, this is an $(k+l)$-setting in $\mathbb{R^2}$:

enter image description here

I hope it's somewhat readable. In this case, we have $k=2$ and $l=5$.

Furthermore, we say that a $(k+l)$-setting is good, if it is possible create a circle, such that the points $m_1,...,m_l$ are in the circle, but the points $n_1,...,n_k$ are not in the circle. If a $(k+l)$-setting isn't good, it's bad. For example:

enter image description here

Please notice that the first example of a $(k+l)$-setting is bad. No matter how you draw the circle, the point $n_2$ is always contained in it.

Now, the point of these definitions is that I would like to launch a light rays from the the points $n_1,...,n_k$, that bounces against the continuous functions. These continuous functions act as a mirror, causing the light ray to reflect according to the laws of reflection. We assume that the light ray loses no energy with each reflection, thereby maintaining its intensity on it whole course. The continuous functions that act as a boundary of the "solar panel" aren't affected by a reflection either.

Question 1: Does there exist a good $(1+l)$-setting in $\mathbb{R^2}$, such that we can send a light ray from point $n_1$ into the circle that encloses the points $m_1,...,m_l$, in such a way that the light ray never leaves the circle?

If you can answer the question in the affirmative, I have a number of follow-up questions:

Question 2.1: If there exists such a setting, how is it visualised?

Question 2.2: What about the case $k>1$ ?

Question 2.3: What about the case $d>2$ ?

If you answer the question in the negative, I also have some follow-up questions:

Question 3.1: Why does such a setting not exist? Can you prove it cannot exist?

Question 3.2: What if $d>2$ ?

Please notice that, when $d=3$, the continuous functions between the points become continuous surfaces, and when $d=4$, they become volumes, etc.

By the way, it would be great if someone told me how I could center the images, or places them in the middle themselves.

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    $\begingroup$ If the function is pointwise linear instead of smooth, I believe the answer is no. I also think that taking the limit of the pointwise linear function will prove that the answer to your question is no as well. Intuitively, one can imagine that if the light came in, it must be able to come out, unless you can get it to bounce infinity inside. $\endgroup$ – nbubis Jul 9 '12 at 19:14
  • $\begingroup$ @nbubis Yes that's the idea, the light ray(s) should bounce an infinite amount of times in order for it/them to stay inside the circle. $\endgroup$ – Max Muller Jul 9 '12 at 19:17
  • $\begingroup$ I will replace these pictures by some prettier ones in some time, by the way. $\endgroup$ – Max Muller Jul 9 '12 at 19:18
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    $\begingroup$ I'd go with $(k+l)$-setting, based on your description, since $(1+m)$-setting (for example) makes no sense, otherwise. Then your question 1 is asking about $(1+l)$-settings, and your first example is a $(2+5)$-setting. $\endgroup$ – Cameron Buie Jul 9 '12 at 19:23
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    $\begingroup$ @MaxMuller If it's just one ray, then the answer is fairly easily yes. $\endgroup$ – Alex Becker Jul 9 '12 at 20:00

This has been studied, under various assumptions. For the case of a parallel beam of light rays, there is a nice description in Serge Tabachnikov's 2005 book Geometry and Billiards, pp. 52-53:

enter image description here
enter image description here

enter image description here

Going deeper is a paper by Daswon et al., "Light Traps" (PDF link). They show it is possible to trap a pencil of rays (rays through a point), but it is impossible to trap diffuse light. Their main negative theorem

states that if the initial light has nonzero measure in the phase space, it cannot be trapped.

  • $\begingroup$ Thank you! I may come back with more questions, based on the answers you have given. $\endgroup$ – Max Muller Jul 10 '12 at 17:05

OK - I believe the answer is yes:

Take an ellipse, make a small hole, and send in a ray through one of the ellipses focai. You can see that this leads to a series of rays ever converging towards the major axis.

enter image description here

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    $\begingroup$ Two remarks. First, the hole doesn't need to be small; two portions of ellipse near the major axis would have the same effect. Second, if the ray doesn't pass exactly through a focus, then it will leave the system, even if the hole is very small. $\endgroup$ – D. Thomine Jul 9 '12 at 21:29

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