Snell's Law, Part IV: Arbitrary N

             {<−− Part 3}

The angle measure between vectors remains unchanged if the systems of equations which define them are translated, scaled or rotated.  Might as well check.

In the image below, N is no longer aligned with the z-axis.  The surface has been rotated by an arbitrary, positive angle, φ  (phi).

Phi, fie, fo fum. 

Ummmmmmmmm.... 

Oh.  Right.  The math.

We have, again in two parts which may be done in any order:

I. Calculate T for any η₁, η₂,  N and I

The dot products:


And so, as before:

The sign of the square root is taken positive, as defined in the last section.  I makes a positive angle with N less than 90° (π/2 radians).

II.  Give T as the sum of two vectors

               αI         +       βN         =        T,               α, β constant:


Using Cramer's Rule:


Solve for alpha:


Solve for beta:


Radness.  And at last:

THE RESULT

...being the same equation as Parts I and II.  So don't do it this way, because it's harder.
(It's over now.)

{CodeCogs you are the warp and weft of my Latex heart}

{<−− Part III}                                                                        {Part V upcoming}