UF TR 0003: Higher Order Smooth Patching of
Refined Triangulations
Jorg Peters*
September 6, 2000
Abstract
This technical report supplements the paper "Smooth Patching of Refined Tri
angulations" [3]. That paper gives formulas for smoothly filling nsided holes in
a 3direction boxspline (Loop) surface at extraordinary mesh nodes with poly
nomial pieces of degree 4. If n A 6 and n is even then alternating sums of the
radial neighbors of the extraordinary mesh node have to vanish. This technical
report gives simple constructions of degree 5 and of degree 6 that do not require
the alternating sums to vanish.
1 Context
Please read the sections 'Motivation' and 'Threedirectionboxspline and B1zier patches'
as well as the first subsection of 'From Mesh to Surface at Extraordinary Nodes' in
"Smooth Patching of Refined Triangulations" [3]. This report sketches three alter
native constructions of degree 5, 5 and 6, respectively. The first construction uses the
same linear reparametrization / = 2c(1 t) + t as the degree 4 scheme in [3] but
quartic boundary curves. The construction of boundary curves by the threedirection
box spline guarantees that the even alternation requirement holds automatically. The
drawback of the scheme is that the boundary to the edgeadjacent patch is modified and
increased to degree 5. The other two constructions use the quadratic reparametrization
a(t) = 2c(1 t)2 + 2(1 t)t + t2 to isolate the boundary to the edge adjacent patch
from change. However, for degree 5, a cubic boundary curve is constructed and hence,
even though the even alternation requirement for the first ring Pi holds automatically,
this scheme must still perturb the boundary to the edgeadjacent patch. Only the de
gree 6 construction does not require change in the neighboring patches. It is a simpler
version of the construction in [2].
In either case, the degree 5 as well as the degree 6 algorithm we define the n by n
*supported by NSF NYI CCR9457806
Figure 1: Subscripts of the mesh nodes Pk (i) in the neighborhood of an extraordinary
mesh node Po.
004 013
103 112
022 031 040 005 014 023 032 041 050
121 130 4 113 122 131 14
'I i' 211 20
301 310
41111
20 212 221 20
302 311 20
401 410
500
Figure 2: Coefficient labels of a degree 4 patch (left) and of a degree 5 patch (right).
\
matrices An and B, with rows i = 1,... n and columns j = 1,... n and entries
2a 2(
An(i, j) = cos( (i j)) a = 1(default) and
S(1)"nj ifn is odd,
B(i,j) = (1) 2. j(1)Ji/n if n is even,
nij = mod (n+i j,n).
For example, B6(3,3) = 1. We also use as default choice of the extraordinary
point and the normal at the extraordinary point is the limit position and normal of the
extraordinary mesh node under subdivision with Loop's rule. According to ([1], p42)
this limit position is
n 24
Q4oo = aPo + (1 a) Pi,i/n, a = 2
55 12c 4c2
i= 1
I prefer a = (4 + c)/9. The polynomial surface has the limit normal of the Loop
subdivision surface if
Q31o = Po + AnP1/4.
The implicit scaling of the tangent vectors Q31o(i) Q400 is chosen to agree with the
regular Sabin conversion rules for n = 6 [4].
2 Degree 5, linear reparametrization
The degree 5, linear reparametrization algorithm is characterized by and enforces
(2c(l t) + t) l .t) = 'I. t) + (t, ).
Sv i aui ovi+1
Since *'0(, t) is of degree three and the right hand side of degree 4, there are five
coefficients to be set equal. As with the degree 4 scheme, the first of the equations
will hold due to the application of An and the second, for the mixed derivatives at
t = 0, due to the choice of Bn in computing Q211: the answer is a solution since
=1 (1)iQ220(i) = 0 for n even always holds for a quartic boundary curve deter
mined by Sabin's rules [4].
The third equation will be enforced by perturbation of Q221 and Q212, the fourth
by perturbation of Q113 and Q131 and the final equation at the t = 1 holds due to
the unchanged C1 transition of the box spline. The C1 continuity with the edge
adjacent patch is enforced by moving the boundary coefficients Q023, Q032 and the
coefficient Q122. This, unfortunately means increasing the degree of the boundary of
edgeadjacent patch from 4 to 5.
The ig .. diii First, we create a degree 4 patch. The 9n B6zier coefficients
Qi,j,k(l) E R3 with i < 2 and i + j + k = 4, = 1,..., n as well as Q202 and Q220
(c.f. Figure 3) are determined by Sabin's formulas. Leaving out neighbors if n > 6 or
005 014 023 032 041 050 006 015 024 033 042 051 060
122 131 14
12 221 20
311 20
41 0
500
15 114 123 132 141 150
24 213 222 231 24
303 312 321 330
402 411 420
501 10
0
Figure 3: Indices of degree 5 and 6 B6zier patches.
wrapping around if n < 6 Sabin's rules are also be applied at the extraordinary point
to generate Q211. The remaining Qaol, Qa3o and Q400 may be chosen freely as in the
degree 4 scheme.
Second, we raise the degree of the patch from four to five. By symmetry it suffcies
to specify the rules
Q500 = Q400,
Q410 = (Q400 + 4Q310)/5,
Q320 = (2Q310 + 3Q220)/5,
Q311 = (Q310 + 3Q211 + Q301)/5,
Q221 = (2Q121 + Q22o + 2Q211)/5.
Third, with c = cos(27r/n) the tangent and twist coefficients are
Q401 Q50oo + AQ401,
6c 1
r311 = Q401 + (Q202 Q301) + (Q301 Q400),
10 10
Q311 Bn 311
and the remaining coefficients are updated by computing
1 2c
d212 (Qi03 Q202),
1 2c
dll3 = (Qo04 0103),
04 113
203 2
32
\
and adding
d212(i) to Q212i),Q22(i 1)
d113(i) to Qxi3(i),Qo23(i),Q32(i 1)
dii3(i) +d13(i 1) to Q122(i).
Finally, we adjust the edgeadjacent patch to match Q032 and Q023 by degreeraising
the boundary and overwriting the two middle coefficients.
3 Degree 5, quadratic reparametrization
The degree 5, quadratic reparametrization algorithm is characterized by and enforces
for a(t) = 2c( t)2 + 2(1 t)t + t2
a(t), t) = 0 t) + , ,).
*aui OVi+i
This construction also requires a perturbation of the edgeadjacent boundary and is
therefore not elaborated.
4 Degree 6, quadratic reparametrization
The degree 6 construction is straightforward and avoids the even alternation require
ment and changing the boundary of the edgeadjacent patch but comes at the cost of 28
coefficients per triangle. It is a simpler version of the scheme in [2].
The coefficients of the degree six patch are computed by degreeraising the degree
4 patch construction
Q600 Q400,
Q51o = (Q400 + 2Q31o)/3,
Q42 = (Q400 + 8Q310 + 6Q22o)/15,
Q330 = (Q310 + 3Q220 + Q13o)/5,
Q411 = (4Q31o + 6Q211 + 4Q3o1 + Q400)/15,
Q321 = (3Q121 + 3Q220 + 6Q211 + Q301 + 2Q310)/15.
followed by enforcing with a(t) = 2c(1 t)2 + 2(1 t)t + t2
a ,
a(t) .0,t)= t)+ ,(t,0)
.Oi Ov+1
This yields the formulas
Q501 * Q600 + AQ501,
6c 1
411 = Q401 + (Q202 Q301) + (Q301 Q400),
15 15
Q411 Bnr411
and the updates
1 2c
d312 (03 202)
1 2c
d213 (Qo004 Q103)
10
with
to be added to
to be added to
Q312(i), Q321(i 1),
Q213(i), Q231(i 1).
5 Summary
This technical report presents alternative constructions for smoothly filling an nsided
hole in a regular triangulation. The formulas are simple and explicit and yield smoothly
connected, standard B6zier patches of degree 5 and 6 respectively.
References
[1] Charles T. Loop. Smooth subdivision surfaces based on triangles, 1987. Master's
Thesis, Department of Mathematics, University of Utah.
[2] Charles T. Loop. A G' triangular spline surface of arbitrary topological type.
Computer Aided Geometric Design, 11(3):303330, 1994.
[3] Jorg Peters. Smooth patching of refined triangulations, 2000. submitted.
[4] Malcolm Sabin. The use ofpiecewise shapes for the numerical representation of
shape. PhD thesis, Dissertation, Computer and Automization Institute, Hungarian
Academy of Science, Budapest, 1976.
Jorg Peters
Dept C.I.S.E., CSE Bldg
University of Florida
Gainesville, FL 326116120
jorg@cise.ufl.edu
tel (352) 3921226
fax (352) 3921220
d312(i)
d213 (i)
