Can U-splines deal with extraordinary vertices? If so, how? I understood that U-splines subsumes T-splines, which I understand to actually mean T-NURCCS (with extraordinary vertices).
I am experimenting on implementing U-splines, and don’t have access to a reference implementation, so perhaps someone could share what the result of the following toy example is on Coreform’s implementation.
Consider five four-sided faces which form a cube without a bottom face. The corners on the top face are then extraordinary vertices (each vertex has 3 shared faces). Suppose I require that these faces connect across edges with C1 continuity. If I run through my implementation with 2x2 patches, what I get is a cone: there are four control points which control the shape of the round curve at the bottom, and the top face is shrunk into a single control point which is the tip of the cone. Hence there is a sharp corner at the tip of the cone. This solution, I believe, is valid in the sense that the functions do connect with continuous derivatives; it’s just that the derivative goes to zero at the tip of the cone.
If I then increase the degrees of the faces to 4x4, then I get rid of the cone; i.e. the top face no longer collapses to a point, and I get a smooth surface except for the extraordinary vertices, where sharp corners remain. Again, I think it is differentiable with zero derivatives, but no well-defined tangent plane.
What I’d expect instead, as with T-NURCCS, is that the resulting surface had a well-defined tangent plane even at the extraordinary vertices. Is that possible?