Snell's Law, Part III

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Given the two angles, incident and transmitted, for a ray passing from one medium to another, to find the Index of Refraction by geometric construction.


For simplicity, I have again assumed the surface is horizontal in the plane.

Along any line parallel to the surface, the distance from the normal to the points of intersection with lines  η₁ and η₂ are in the ratio η₁/η₂ .

Now, picture the horizontal black line labeled IOR sliding up and down, causing the lines η₁ and η₂ to follow.  The line η₂ will pull the vector i along with it, and by way of an auxiliary point of intersection of perpendiculars from i and t' to orthogonal radii of the circle, t' and t will move as well.  I say, as the bar is moved, the relationship between i and t will, for all positions, obey Snell's Law.  That is, the construction is complete, beginning with an arbitrary angle.

As an example, I have drawn the line IOR so that the distance from normal to η₁ is the same as the radius of the circle, which is known exactly by construction.  There is no need to appeal to sines or cosines.  The ratio of η₁/η₂  can then be worked out by segment subdivision with a compass until the desired accuracy is reached.
 In this (hypothetical) experiment, the IOR of the second medium is just slightly higher than 1.5x the first.  If the first is air, the IOR of the second is about 1.51  (η₁ leans a little too far to the left, so η₂ appears too close to it, and the subdivision marker).

This number should agree with trigonometric calculation (cheating).

If you mouse over the diagram, the red line is the continuation of θ₂  from t'.  For small angles, x is an excellent approximation of sinx.  Working directly with angles, we might use this approximation, and thus the ratio  θ₁/θ₂, instead of sinθ₁ / sinθ₂.  In the present case, that would require measuring the ratio along the arc, which is more difficult than using the correct ratio.
We could also use this value by mistake. (Not that anyone has ever done that.*  Nosirree.)


{<−− Part II}                                                          {Part IV −−>}

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*Ptolemy.