Is Snell's Law valid even for on any curved surfaces? Proof of Snell's Law can also done by using Fermat's principle of least time on plane surfaces. Can we prove the same result for any curved surfaces ?
 A: Yes! (I believe) Whichever "point" your light ray strikes at the curved interface (surface) just draw a tangent along that point of the surface.
Then do the same thing that is done for the plane surfaces with the  angential surface created, you can go ahead and draw the normal, etc. Hence, Snell's law of ray optics is valid for all the points on a curved surface (interface between the two optical media). And yes, the fermat's principle of least time on plane surfaces can now be used easily.
Assuming that the point is macroscopic and does not require us to delve into the microscopic characteristics of the interface. Since ray or geometrical optics is only valid when the obstacle is of a larger order than the wavelength of the electromagnetic wave. 
A: Yes. By drawing a tangent to the surface and considering that region of the to be flat Snell's law can be applied. A small proof is given below. 
Consider the action 
$$S=\int{dt}$$
In a certain medium one can write 
$$S=\int{\frac{\sqrt{1+y'²}}{v}dx}$$
Here $y'$ represents the derivative of $y$ with respect to $x$. Solving the Euler Lagrange equations
$$\frac{d}{dx} \bigg(\frac{y'}{v\sqrt{1+y'²}}\bigg)=0$$
Hence 
$$ \frac{y'}{v\sqrt{1+y'²}}=c_1$$
Where $c_1$ is an integration constant. If light is moving at an angle of $\theta$ with the $x$-axis then we can write $y'=\tan\theta$. Substituting this in the above equation
$$ \boxed{\frac{\sin\theta}{v} =c_1}$$
This is Snell's law. In this derivation no surface was considered at all. For any two mediums one can write the relation (using appropriate coordinates)
$$ \frac{\sin\theta_1}{v_1}= \frac{\sin\theta_2}{v_2}$$
in general. 
