Bilinear Coons patches Suppose that we have two bilinear Coons patches which share a common curve. Studying on Farin I find that, generally, these two surfaces join with C^0 continuity along that common curve, but I don't know why. Thanks.
 A: Look at the equation for a Coons patch, $P(s,t)$. If you set $s=0$, or $s=1$ or $t=0$ or $t=1$, the equation degenerates into one of the defining curve equations. In other words, the patch exactly interpolates all four of its defining curves. 
So, if two patches share a common curve, each of them has an edge that exactly matches this curve. Those two edges therefore match each other (in position) which means that the two patches meet with $C_0$ continuity.
To show that two patches do not join in a $C_1$ fashion, in general, all you need is a counterexample -- a simple example of two patches that share a common edge but are not $C_1$. A trivially simple example will work. Define each patch by four line segments that form a square. Then the patch will simply be the planar region in the interior of the square. If the two squares are not coplanar, then the join will not be $C_1$.
Another approach is to calculate partial derivatives, and show that they are not equal, in general, but that seems more difficult, to me.
