Q:
I did not quite understand the significance of the two pictures of
cyclides alone. One you just say is a cyclide, OK. But the other is
an order h^4 approximation to a cyclide? However, the Dupin cyclide
is algebraically a degree 4 surface anyway...or have I misunderstood
something? Perhaps these are surface splines of parametric order 4.
Maybe I should have read the text.
A: Surface splines have off-hand nothing to do with the implicit
box-spline-based representation of the paper. Ie. I am
separately working on parametric and implicit representations.
The approximation order is to be understood as follows:
p(x,y,z) = sum a_i M([x,y,z]-i) + O(h^4)
That is the defining polynomial of degree 4 is captured except for some terms
of total degree 4.
These terms are, however, approximated to within
a multiple of meshwidth^4 -- and the constant is so good in the
example that we cannot visually distinguish. Only the theory tells
us that we should be off by a bit. So my statement
concerning reproducibility is conservative.