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In order to proceed with the discussion we have to define two terms. A wave front is the surface of constant phase. In a plane wave these are planes andin a spherical wave these are spheres. A ray travels perpendicular to the fronts.
Huygens postulated that as a wave propagates through a medium each point on the advancing wavefront acts as a new point source of the wave. This iscorrect physics for the water waves but not for light waves. However the Helmholtz equation for diffraction of EM waves gives a solution identical tothat give by Huygens' principle.
Look at the figure which shows a wavefront AB coming to a surface and is reflected creating the front CD. The point A hits the surface first. The pointB hits a time $vt$ later. During that time a spherical wave is emitted from A and travels a distance $vt$ . In fact this happens for every point along the wavefront. The next figureattempts to show how a number of waves line up along the line DC and that this is perpendicular to the line AD.
From this we see that $${\mathrm{sin}}{\theta}_{i}=\frac{vt}{AC}$$ and $${\mathrm{sin}}{\theta}_{r}=\frac{vt}{AC}$$ so ${\theta}_{i}={\theta}_{r}$
For refraction a similar thing happens. See figure (geometric optics / Huygens refraction.vsd )
In this case the velocities are different in the two media and so one obtains: $${\mathrm{sin}}{\theta}_{i}=\frac{{v}_{i}t}{AC}$$ and $${\mathrm{sin}}{\theta}_{t}=\frac{{v}_{t}t}{AC}$$ which then can be rearranged $$\frac{{\mathrm{sin}}{\theta}_{i}}{{v}_{i}t}=\frac{{\mathrm{sin}}{\theta}_{t}}{{v}_{t}t}$$ or rearranging some more $$\frac{{\mathrm{sin}}{\theta}_{i}}{{\mathrm{sin}}{\theta}_{t}}=\frac{{v}_{i}t}{{v}_{t}t}$$ or $$\frac{{\mathrm{sin}}{\theta}_{i}}{{\mathrm{sin}}{\theta}_{t}}=\frac{{n}_{t}}{{n}_{i}}$$ finally $${n}_{t}{\mathrm{sin}}{\theta}_{t}={n}_{i}{\mathrm{sin}}{\theta}_{i}$$ which is Snell's law. Now note that normally one uses rays, in which case the anglesare measured w.r.t. the normal to the surface.
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