Why for perfect reflections a surface must have G2 continuity (class A surface)?
I would like a mathematical answer.
Why for perfect reflections a surface must have G2 continuity (class A surface)?
I would like a mathematical answer.
What you see on reflects is the n-continuity of normals, which are the derivative of positions. -> a G1-only surface would have G0-only normal field, i.e., with sudden change of gradient in the normals (and thus, the reflects), that the eyes can notice. G2 surfaces have G1 normals fields, which is smooth enough for your eyes.
The G2 requirement does not mean that the surface is good quality. Just means that without this the surface is not going to have a continuous reflection flow so humans can see the difference. That may or may not be a good thing depends on what you want.
Mathematically the surface normal is:
$f(u,v) \frac{\partial}{\partial u} \times f(u,v) \frac{\partial}{\partial v}$
Since both sides are derived that means that the function field of the surface normal has one degree less than the original surface. So for the reflection to be first degree continuous it has to have a second degree continuity.
So far we have established the relationship between the continuity of the surface and the continuity of the reflection. Nothing thus far proves that the surface reflection needs to be first degree continuous. To understand why we must exit the realm of mathematics and enter the realm of biology.
The eye is equipped with an edge detection algorithm on a structural level right on the retina. This edge detection algorithm in essence works as a discrete derivative of the input signal. So, if your surface is not G2 continuous then the human edge detection kicks in and shows itself up. For references read on Mach Bands and so forth.
Since the edge detection is discrete G2 continuity is not enough. The change has to not only be locally satisfied but also satisfied on the retina. So the change should still be shallow enough not to cause problems.