For questions please refer to the page numbers of the paper
on my homepage.
Q--Will Bernstein-Bezier coeficients and control points be constructed
for the vertex type P which is the centroid of mesh face?
A --Yes. Centroids are treated like original points
This also answers your question
Q--about "determining coefficients by analogous
construction at the other subcell vertex of type P"
Q -- In Figure 2.3 on page 7, two of five layers of Bernstein-
Bezier coefficietns are shown. But I can't understand how the other
three layers of coefficients can be found. Benjamin ( Hong Kong )
A -- The construction is symmetric. Thus, if you understand how
to generate b_{ji} for j \in {0,1}, i\in {0,1,2,3,4}
you also have the formulas for b_{ij}; this
is explained for the case b_{12} and b_{21} on the page.
Now you are still missing b_{ij} for i,j >= 2.
However the construction is symmetric with respect to
the original point P and the centroid. That is you also have the
formulas for b_{ij} for i,j >= 2 (b_{22} is explicitely
set to C_i, but you can choose some average of surrounding
coefficients as well)
Q -- In page 7 of your paper, in finding both b12,i and b21,i+1,
there is 'j' there, I would like to ask what's the meaning oj it and
how it related to 'i'. c3801040@comp.polyu.edu.hk
A -- Look at Figure 2.3 directly below the formulas.
The $i$th neighbor of $P$, $M_i$, has the surrounding
panels labelled by $j$ as indicated.
Q --
$M_i$, has the surrounding panels labelled by $j$ as indicated in the
diagram, but how to determine which one of the four sections is Ci1,
...C14. It's because I can find the general relation between Cij and
Ci.
A-- Figure 2.3 indicates that C_i = C_{ij}; the labelling is relative
to this identity.
Q -- How to find b14,i? You seems to be not mention in thr paper.
A -- Well, b_{03,i} is defined. By symmetry of the labelling
you have the formula for b_{03,i-1} = b_{30,i} and by choosing
P to be the other vertex of the 4-sided sub-panel, b_{14,i}
is defined.
A --
1. c does not change meaning
2. boundaries are a whole chapter by themselves.
I usually interpolate the boundary polygon.
3. I use mathematically positive orientation -- but
any consistent orientation will do.
Q -- Also, in the paper, you said that 'there exist a projection of its
vertices into a plane such that none of the projected vertices
lies in the convex hull of the other project vertices'
Can you explain more on this?
A -- The statement is precise. If flat, the panels should be convex.