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Snell's law from geometrical optics states that the ratio of the angles of incidence $\theta_1$ and of the angle of refraction $\theta_2$ as shown in figure1, is the same as the opposite ratio of the indices of refraction $n_1$ and $n_2$.

$$ \frac{\sin\theta_1}{\sin \theta_2} = \frac{n_2}{n_1} $$

figure 1

(figure originally from wikimedia)

Now let $P$ be a point in one medium (with refraction index $n_1$) and $Q$ a point in the other one as in the figure. My question is, is there is a nice geometrical construction (at best using only ruler and compass) to find the point $O$ in the figure such that Snell's law is satisfied. (Suppose you know the interface and $n_2/n_1$)?

Edit A long time ago user17762 announced to post a construction. However until now no simple construction was given by anybody. So, does anybody know how to do this?

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    @user3445: Do you know the interface? (I think if you know just the position of $P$, $Q$ and the ratio $\frac{n_2}{n_1}$, there can be many solutions.)2011-02-20
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    Yes, I know the interface.2011-02-20
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    Assume that the interface is the $x$ axis. The algebraic equations that characterize the point $O$ is a quartic polynomial with coefficients given by the coordinate values of $P$, $Q$, and $n_2/n_1$. So the ruler-and-compass constructibility can probably be [considered in the usual way](http://en.wikipedia.org/wiki/Constructible_number).2011-02-20
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    @WillieWong So the ruler-and-compass constructibility probably is impossible.2013-01-06
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    Solving a quartic equation like I've found as: $$x^2(B^2+(d-x)^2)=(n_2/n_1)^2(d-x)^2(A^2+x^2)$$ seems to be prohibitive; it's a mess even with a computer algebra system.2015-03-02
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    Do you know point $P$ and $Q$?2015-03-03
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    @student: Points P and Q are given and fixed. After that, how is the interface line given or defined? 1) Is its slope only given? 2) Is a point that it passes through given? Or 3) Are the slope *and* point passing through *both* are given?2015-03-05
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    @Narasimham: Just assume that the interface is the abscissa of the coordinate system. $P$ and $Q$ are given, the point $O$ should be constructed.2015-03-08
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    @student : I gave a solution, using only trigonometry and kinematics, hopefully you will like it.2015-10-23

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