Reflection Lines

Question

I am analyzing the problem of G^1 continuity between patches. I have found the statement: if the reflection lines on a surface are C^0 then the surface will be G^1. I would like to know the proof of this statement; if someone has a reference on this and also on the reflecion lines as a surface interrogation method it would be useful. Thanks.

Answer 1

There are several surface interrogation methods that have similar properties. For example, so-called iso-photes and iso-clines have the same property you cited, and are much easier to compute than reflection lines. A fairly up-to-date paper can be found here. It includes numerous references to the earlier works.

Actually, the continuity condition is typically used in the opposite direction: if there are any reflection lines that are not $C_0$, then the surface join is not $G_1$. In other words, gaps in the reflection lines are used to detect discontinuities in the surface normal. In this direction, the condition is fairly obvious, I think: a "jump" of the surface normal is clearly going to create a "jump" in the reflection line.

If you want to get conclusions in the opposite direction, you have to be a bit careful. Specifically, which reflection lines (and how many) have to be $C_0$ in order to guarantee that the surface join is $G_1$.

There are analogous results for higher order continuity. See the paper referenced above. Again, conclusions in one direction are fairly straightforward differential geometry, but results about the other direction are more difficult, and in fact it looks like some are even unknown.