On the curvature of guided surfaces
K. KarCiauskas and J. Peters
Department ofMathematics and Informatics Vilnius University
Dept CI.S.E., University of Florida
Abstract
Following [KP07], we analyze surfaces arising as an infinite sequence of guided C2 sur
face rings. However, here we focus on constructions of too low a degree to be curvature
continuous also at the extraordinary point. To characterize shape and smoothness of such
surfaces, we track a sequence of local quadratic functions. Convergence of the sequence
certifies curvature continuity. Otherwise, the range of the curvatures of the limit quadratics
gives a measure of deviation from curvature continuity. We observe additionally that, by
increasing the sampling density at each step, we can build guided surfaces with subdivision
structure that are C2 and of degree (3,3).
Key words:
1 Introduction and Motivation
The paper [KP07] introduces a framework
for the construction of C2 surfaces from
surface rings x", m = 0, 1,... as illus
trated in Figure 1. To set the stage, we
quickly review the key concepts and nota
tions. Each surface ring x" joins smoothly
and without overlap with its predecessor
x"1 if m > 0 and with existing bound
ary data if m 0 The sequence of
rings contracts towards a central point x".
The ring structure is inherited from a ct
map (concentric tesselation map) p : E x
I,
Fig. 1. Sequence of guided rings xm
and guided surface [KP07].
Email address: jorg@cise .ufl.edu, kestutis. karciauskas@mif .vu.lt
(K. Karciauskas and J. Peters).
URL: http: //www. cise. uf 1. edu/~j org (K. Karciauskas and J. Peters).
Preprint submitted to CISE, UF, Technical Report
17 April 2007
{1,..., n} + R2. That is, a ctmap embeds n copies of a domain piece E into the
plane. E is for example a square or a square with a cutout. The scaled copies of p,
A" 1p and A"p for 0 < A < 1, join with prescribed continuity and tesselate the
neighborhood of the origin by a sequence of annuli. Table 1 lists several ctmaps
defined in [KP07] and their combinations with operators H explained below. The
shape of a guided surface is inherited from a guide map g : R2  R3 by 'sam
pling' the composition of g with the scaled copies of the ctmap. The guide g is a
surface piece that outlines the intended shape, but need not fit the requirements of
a modeling pipeline or match smoothly with the existing data. Specifically,
xm : H(g o Amp). (1)
Here H samples jets, i.e. a collection of derivatives, at fixed parameter values of
its argument, and constructs x" so that the resulting rings join smoothly and are
internally smooth.
ct degree layout ring degree repro
map p x duces
pc (3,3) osprocket XC6 (6,6) 2
(eg CatmullClark) xc5 (5,5) 1
XC3 (3,3) 1
PL 4 Asprocket XL8 8 2
(eg Loop) XL4 4 1
pp (3,1) polar Xp6 (6,5) 2
xp3 (3,3) 1
pg (2,1) polar Xg4 (4,3) 2
Table 1
Some combinations of ctmaps p and guided surface rings x. See Figure 2 for an expla
nation of the entry 'layout'. The first subscript of x indicates the ctmap p, the second
the degree of the patches generated by H. The ctmap pg results in G2 surface rings, all
other ctmaps in C2 rings. The numbers k e {1, 2} in the last column indicate that x"
reproduces gk o p where gk is the kth order expansion of g at the central point.
The point of guided constructions is to decouple the definition of the shape from
the final, more restrictive representation. The shape is outlined by the guide and the
final representation of x" is in terms of lowdegree Bezier or Bsplines. Joining
an infinite sequence of spline rings yields a surface x with subdivision structure.
Only the use of a guide surface in place of a guide (subdivision) matrix sets guided
subdivision apart from the usual procedure of refining meshes (see Section 2). The
orem 1 of [KP07] showed that the limit surfaces x of this process are C2, provided
(1) the guide surface g is twice continuously differentiable at (0, 0) and (2) the op
erator H generates rings x" of sufficient degree to reproduce g2 o p, where g2 is
the piecewise quadratic expansion of g at the central point. This theorem applies to
the guided surfaces with entry '2' in the last column of Table 1.
Fig. 2. Three types of ctmaps: pc (osprocket), PL (Asprocket), pp (polar). The rays
emanating from the center demarcate the segment boundaries of p and are mapped to the
segment boundaries of g.
The present paper focuses on the new constructions with entry '1' in the last column
of Table 1. These constructions are analogous to the C2 constructions in [KP07]
and consist of a sequence of C2 rings joining to form a C2 surface with an ever
smaller central hole. However, their degree is lower than is needed to reproduce the
composition g2 o p. The surfaces are therefore C2 except in the limit extraordinary
point, x". We will see that the surfaces have bounded curvature and are typically
visually indistinguishable from their curvature continuous siblings.
To quantify the surfaces' deviation from curvature continuity, we turn to a classical
notion of curvature, the convergence of a sequence of quadratic functions. These
quadratics are defined in a fixed coordinate system, corresponding to the tangent
plane and normal at x", rather than in a local coordinate system. They are therefore
called 'anchored' quadratics.
Section 2 contrasts standard subdivision with guided subdivision. Section 3 intro
duces the new constructions, for Dsprocket, Asprocket and polar layout, and es
tablishes C1 continuity of the limit surfaces. Section 4 defines curvature continuity
via anchored osculating quadratics and provides curvature estimates and bounds
for guided surfaces. Section 5 leverages the results of the preceding section to show
that a new class of accelerated guided constructions generates C2 surfaces of degree
(3,3). Section 6 presents results of implementations of all three layouts.
2 Subdivision and guides
Subdivision has both a discrete structure, the refinement of a control net, and a
continuous structure, by associating refinable generating splines with the control
nodes.
The early literature on generalized subdivision (now abbreviated to subdivision)
[DS78,CC78,BS86], focused exclusively on the discrete structure, namely the re
finement Amc of the control net c guided by a subdivision matrix A (Figure 3).
Alternatively, one can emphasize the generating system associated with the mesh,
control net Amc
or
g o Amp generating system
refinement
surface ring x"
control net Am+lc 
or
g o rm+lp generating system nested rings
surface ring x"+1
Fig. 3. Generalized subdivision and guided subdivision both generate a sequence of nested
surface rings: generalized subdivision surface rings are guided by a subdivision matrix A,
guided subdivision surface rings are guided by an explicit map g.
i.e. the spline rings that form the surface. In the early literature, these spline rings
weres tacitly understood to be structurally simple spline functions so that the 'tun
ing' or improvement of properties of schemes focused on adjusting the spectrum
of the subdivision matrix. To obtain C2 continuity guided by a matrix, [PU98] in
duces zero curvature at extraordinary points; [XYD06] associate multiple values
and functions with each node, achieving C2 continuity, but only for valence three;
[LL03,SW05] focus on reproduction of quadratics to combine subdivision schemes
C2 except at extraordinary points and [ZLLTO6] adjusts subdivision functions to
yield one C2 extraordinary point by applying special rules for transitions across
edges of the control net. Of the many C1 curvaturebounded subdivision schemes,
we implemented the constructions of [Loo02] and [ADS06] for comparison (see
Section 6).
The focus of guided subdivision is the continuous structure of the generating sys
tem. This is grounded in the analytic point of view [Rei95,RP05] that characterizes
subdivision as a process of piecing a surface together as nested spline rings con
verging to an extraordinary point (see Figure 3, right). In guided subdivision, this
sequence is locally guided by a map g o ATp. If the guide surface were a single,
global polynomial and we dropped the operator H then guided surfacing would
reduce to the approach in [Pra97,Rei98]. However, such global polynomial map
pings, as well as low degree piecewise polynomial mappings, are known to create
surfaces with curvature flaws (see e.g. Figure 3 of [KP07]). By sampling, guided
subdivision decouples the need for a low degree standardized representation from
the degree of the guide. Most recently, a scheme was derived that combines the fea
tures of generalized subdivision and guided subdivision for polar layout [KMPO6].
3 Guided surfaces
This section reviews the construction of guided rings xm in [KP07]. At first sight,
guided surfacing appears to be more complex than subdivision based on the control
mesh. However, Observation 1 of [KP07] shows that guided surfacing based on
piecewise polynomial guides is numerically stable and stationary.
Table 1 lists ctmaps and the degree of rings obtained by applying operators H. The
operators corresponding to a '2' in the last column have been explained in detail in
[KP07].
0o0000
o o o 0 000
Fig. 4. h5'5 combines four 3 x 3 blocks of B6zier coefficients, representing partial deriva
tives up to second order at the four comers, into one patch of degree (5,5).
The operator h5,5 samples (2,2)jets at each corner (Figure 4) and combines 3n
such sampled patches into a ring XC5 of degree (5,5). h5,5 can be replaced by an
operator h3'3 that substitutes, for each (5,5) patch, a 3 x 3 array of bicubic patches.
The formulas determining the Bezier coefficients are illustrated in Figure 5. Alter
natively, we could view the 3 x 3 of patches as one bicubic surface piece in Bspline
representation, with single knots in the interior and fourfold knots at the boundary.
Replacing the three layers of Bezier coefficients of x" abutting x"1 by extrapo
lating xm1 ensures C2 continuity; that is, the surface rings join C2 when x"1 is
represented in oncesubdivided form to match the granularity. The resulting type
of surface ring is XC3. Similarly, replacing the degree (6,3) patches of xp6 by three
bicubics yields xp3. A bicubic macropatch construction corresponding to pg has
also been developed, but presents no additional insight (and a second curvature
2 4 2 1
33 3 3 6 see
t**~*
1 7 1 1
36 36
* ***
* * *
Fig. 5. Interpolation of a (2,2)jet at the covers. (left) Stencil for B6zier coefficients to form
a C2 composite curve. (right) XC3 construction: the 3 x 3 jets in B6zier form, shown as *,
are either obtained by sampling or, when adjacent to the previous ring, by subdividing the
previous ring and extending it in a C2 fashion.
bounded construction of degree (3,3) does not seem worth reporting on when the
G2 construction Xg4 uses patches of only degree (4,3).)
3.1 C2 guided rings of total degree 4
The construction of XL4 is conceptually alike the previous constructions, but merits
an extra section due to the different, Asprocket layout. The ring is defined in terms
of the boxspline with directions [ 01 1 1 ] that also underlies Loop subdivision.
This boxspline is C2 and of degree 4. We will determine all boxspline coefficients
of one ring from sampled derivatives. By construction from boxspline coefficients,
the rings are C2 and join C2 in the sense of Loop subdivision.
The sampling operator h4 collects eight derivatives at a sample point:
(OaOtf)O
These eight derivatives define eight boxspline coefficients (Figure 6, middle,right)
and we select seven of these associated with the sample point (see Figure 7, (b)).
00
Fig. 6. (left) Locations of h4sampling in consecutive steps m 1 and m. (middle) repre
sentation in B6zier form and (right) representation in terms of boxspline coefficients.
0 0 /"
S 0 0 06 ,' ,
0 ******
(a) (b) (c)
Fig. 7. Construction of one segment of a ring in grey. (a) The four sample points of a seg
ment. Two of the sample points are associated with the previous, coarser layer m 1 and
depicted as larger circles. (b) Seven of the eight boxspline coefficients at level m, deter
mined by h4sampling at the upper right sample point, are depicted as *. (c) Subdivision of
the boxspline net at level m 1, depicted as light grey disks, yields another set of coeffi
cients at level m. The two topmost coefficients are borrowed from neighboring segments.
The construction of the boxspline coefficients of one guided ring is sketched in
Figure 7. At each iteration, h4 is applied n times to generate new box spline coeffi
cients. Where these new boxspline coefficients overlap (see Figure 7 (b)), they are
averaged. Subdivision of the boxspline net at the previous level (in step 0, of the
input data) generates another set of boxspline coefficients at the same granularity
as shown in Figure 7 (c). Where these coefficients overlap the newly created ones,
they prevail and replace the new ones. Together (see Figure 8, left), the boxspline
coefficients define a ring with 12n polynomial pieces arranged as a double layer
(Figure 8, right).
Fig. 8. Assembly of the double ring. (left) The boxspline coefficients from levels m 1
and m determining one of n segments. Coefficients marked as hollow circles o stem from
sampling at level m (cf Figure 7, left,middle). The remaining coefficients, marked as *
were generated in level m 1. (right) Partial surface ring XL4.
3.2 C1 continuity
To analyze the surfaces in the limit, we assume that g is C2 at the center point, and
choose the coordinate system so that
g(0, 0) (0,0,0).
We denote by gh, k 0,..., d, the homogeneous part of degree k of the guide
surface cap g. That is g) is a linear combination of monomials of total degree k in
the expansion of g with respect to the parameters (u, v) and g (A.) Akg. Then
for m > 0, x := H(gh o Atp) XAkH(g o p). (2)
(Note that x' is typically not homogeneous.) This yields the decomposition
d
Xrm S (ml)kx (3)
k=0
Retracing the proof of curvature continuity in Theorem 1 of [KP07], but replacing
reproduction of g2 o Amp, where g2 := g0 + g + g2 by reproduction only of
(g g + g) o Amp, we see that x" g(0, 0), a unique limit of the tangent planes of
the surface rings x" exists as m  o and equals the tangent plane of the guide
surface at x', and the guided subdivision surface x is C1 at x'.
4 Curvature Continuity and Shape Characterization
We now use a classical parametrizationindependent notion of curvature continuity
that allows quantifying deviation from curvature continuity. We denote piecewise
polynomial terms of total degree greater than k in the variables si,..., si and with
the coefficients depending on t, ... tj by
71T>k(S1, . Si; tl, . ,tj)
If the coefficients are constant, we write 7>k (sl,.. si).
Oc
Fig. 9. Illustration in one variable: (left) anchored osculating quadratics (thin stroke parabo
las) are defined with respect to a fixed coordinate system defined by position and normal of
g (thick curve) at the central point. (right) Quadratics in local coordinates.
Definition 1 (curvature continuity) Let f : R2 + R3 be tangentplane continuous
at fo := f(O, 0). We choose the coordinate system so that fo = (0, 0, 0) and the
tangentplane at fo is {z = 0}. Further assumefor (s, t) / (0, 0) that (x, y,z :
f(s + a, t + 7) satisfies the equation
z= (x2,xy, Y2,x,y, 1)q(s,t) + T>2(7, ; s, t), q := (qo,..., q)T (4)
where q is a continuous function of(s, t). Then
(i) q(s, t) is called an anchored osculating quadratic and
(ii) f is curvature continuous at fo if the anchored osculating quadratics have a
unique limit at fo.
The idea now is to insert f := xm into (4) and then to bound the deviation from the
quadratic as m + oc.
Theorem 1 (Curvature Boundedness) Let x be a guided surface of a type listed
in Table 1 and q the unique osculating quadratic of its curvature continuous guide
g. Then the curvature of x at the central point is bounded.
Proof We fix a coordinate system so that, near x" = (0, 0, 0),
g =(u + 7>l(u, v), v + 7r>(u, v), a2 + buv + cv22 + (>U,' ))
In the following, we consider only one sector i. With (s + o, t + 7) a point near
(s, t) in the compact domain of one of the n segments of the C2 surface ring, we
expand the ith segment (u, v) of p to second degree at (s, t):
(u(s + a,t + ), v(s + a,t + ))
(U2(oa, T; s, t) + 7>2(o, T; 2. s, t) V, 1 t) + 7t>2(7, T; S, t)),
u2(o, T; s, t) : uo + hoa + h27 + h4a2 + h6.7 + hs72,
v2(, T; t) : vo + hlo + h3T + h5o2 + h7oT + h972.
Here uo, vo and hi are piecewise polynomials in s and t. By expansion (3), choice
of coordinates and reproduction of gi o p and with I := AX,
xT =(( 1 + 12E1, 2 + 12E2, 12 + 13E3) + 7>2(7,7; ,t,1)
z2 := ho + h1a + h27 + h3a2 + h47r + h_572, : (5)
where Ek are (piecewise) polynomials in (o, 7) with coefficients depending on s,
t, 1. Substituting (x, y, z) = x into Equation (4) and collecting the coefficients of
o, 2 2, a, , 1 yields a system of six linear equations with entries depending on
(s, t) and in the six unknowns qj(s, t):
Mq = r. (6)
The first three columns of M have a factor 12, the next two a factor I and the last no
factor of 1. The right hand side r has a factor of 12. Away from the origin, M is of
full rank since detM "((ho h3 hh2)4 + > (1; s, t), hoh3 h2 > const > 0
and all p from Table 1 are injective on their domain. It follows that
lim qj 0 for j 3, 4, 5.
10
Moreover, we can bring the system into the form
HO M2,2 q ( ) O, 3 ., 3 R3'3, H := 2hoh2 hoh3+h1h2 2h1h3
0 M2,2 ' R"H h2 h2h3 h2
where the entries in the 3 by 3 matrix O vanish as 1 0. Let D := detH
(hoh3 h1h2)3 and De the determinant of the matrix obtained by replacing the
(f + 1)th column of H by the limit of rl as 1 0. Since D > 0,
D
D, e 0,1,2 (7)
is welldefined and, since the domain of qe is compact, the restriction of q(s, t) to
the ith segment is bounded as claimed. III
If g2 o p is not reproduced then qe, = 0, 1, 2, varies with (s, t) and can be different
for each of the n sectors. If g2 op is reproduced, as is the case for the schemes listed
in Table 1 with rightmost entry '2', then there exists a unique central quadratic
(a, b, c, 0, 0, 0)t at the extraordinary point, defined by g2, and we can reformulate
(6) as
Mq= r M(a, b, c, 0, 0, 0)t. (8)
That is, q defines the deviation from the central quadratic. However, the right hand
side now has a factor of 13 so that q vanishes in the limit. This yields an alternative
proof that the surfaces considered in [KP07] are curvature continuous.
4.1 Computing Curvature Bounds
The practical calculation of the bounds on the coefficients go, qi and q2 of the oscu
lating quadratic is simplified since D and De share factors hoh3hlh2. Specifically,
we compute as follows.
(i) Bound the coefficients go, qi, q2 for the functions fl := u2, f2 := uv, f3 := 2.
This yields nine intervals that allow computing the bounds for any function f
71U2 + 2UV +y73V2.
(ii) If (x, y, z) is an orthogonal coordinate system in R3 with x' = (0, 0, 0), tan
gent plane {z 0} at x" and g (elu + el2v + 7r>i(u, v), e21 + e22 +
7r>1(u, v), au2 + buy + cv2 + 7>2 (u, v)), then the bounds of the coefficients q' of
the osculating quadratic in such system are calculated from the previous bounds
on coefficients qk as
9 qoeo2 + q1eoei + q2e1 q2 qge2 + qle263 + q2e6
02(9)
q1 =2qoeoe2 + q(e03 + e62) + 2q2e63 ,
where (o :2 ( I11 )12 1
Ce e 3 C 21 Ce22
(iii) To bound Gaussian andmean curvatures, we observe that for (x, y, q'x2 +q xy+
q'y2) the mean curvature H and the Gaussian curvature K at the origin (0, 0, 0)
are
2 1
H = q + q K l= 1'.. q2
Substituting formulas (9), we get
H (e + e2)2qo (eo60 + 6263)ql + (e + )2 ,
1 ;0 1 N2(10)
K =(e6e3 ee2)2(4qo2 q)
For Gaussian curvature K, we get a tighter estimate if, in formula (10), the part
4qoq2 q2 is precomputed with respect basis functions fl, f, f3. Consistent with
[PU01], there are six precomputed intervals for k and nine for H.
(iv) Defining fl : u2 + v2, f2 := uv, f3 := v2 gives an immediate impression of
how guided subdivision reproduces canonical elliptic, hyperbolic and parabolic
shapes. Table 4.1 lists bounds on the Gaussian curvature for these shapes when
n =8.
Table 2
Gauss curvature bounds on monomials for different x and n = 8.
guide K K(xc3) K(xc5) K(Xp3)
2 + v2 4 [3.87501, 4.22213] [3.96505, 4.02945] [3.98505, 4.03352]
uv 1 [1.06493, 0.96478] [1.00426, 0.99261] [1.05229, 0.97047]
v2 0 [0.15738, 0.15644] [0.02054, 0.02241] [0.09641, 0.05184]
5 Accelerated bicubic guided C2 schemes
We consider a sampling scheme that is no longer stationary, since, in each step,
we sample with increasing density. At level m, for the osprocket (CatmullClark)
layout, each quad is evenly subdivided into 4m subquads and h3,3 is applied on
each, creating 4" pieces of 3 x 3 bicubic patches that are joined C2. Prolongation,
as explained in Section 3, assures that rings join smoothly. For polar layout, there
are 2" subquads (see Figure 10, right). We call these schemes accelerated.
Fig. 10. Structure of a sector of accelerated bicubic subdivision. (left) osprocket (Cat
mullClark) layout and (right) polar layout.
To characterize shape and smoothness of such surfaces, we track a sequence of local
quadratic functions. Convergence of the sequence, using symbolic computation,
certifies curvature continuity.
Theorem 2 Accelerated bicubic guided subdivision surfaces x are curvature con
tinuous.
Sketch of Proof Since accelerated subdivision reproduces g, o p and inherits
the tangent plane of the guide surface at x', we can follow the proof of Theo
rem 1. According to the construction, the piecewise polynomial g o p : [0..1]2 R
is sampled by h3'3 at the comers of the 4m subquads of of each osprocket seg
ment xT or at the comers of the 2m subquads of of each polar segment x'. Then
m can be chosen so large that the value and partial derivatives of g2 o Amp dif
fer from those of x' everywhere by less than any prescribed constant and the
functions hi and hj in (6) deviate from their value at the covers by less than a
fixed e. Let (a, b, c, 0,0, O)t be the unique osculating quadratic of g at (0, 0). Then
limmi o qo = a limmi o ql = b limmi m q2 = c and we have a unique limit
quadratic. III
Since the limit of the anchored osculating quadratics of x coincides with that of
guide surface g, accelerated subdivision surfaces are C2 in the sense of differential
geometry. This sheds new light on the challenge to build C2 subdivision surfaces
of the degree of CatmullClark surfaces and on the lower bounds on the degree of
curvature continuous subdivision surfaces derived in [Rei96,Pra98].
6 Discussion
Traditional 'tuning' of subdivision algorithms focused on explicitly setting eigen
values or on formulating an optimization problem to approximate several spec
tral properties (e.g. [BK04]). More recently, subsubdominant eigenvectors have
gained attention in the context of optimizing surfaces with respect to shape charts
[KPR04,ADS06,GU06]. This allows enforcing, for example, convexity. In guided
subdivision we prescribe, via the guide surface, both the dominant eigenvalues and
the dominant eigenvectors up to the order of continuity. Moreover, prescribing the
guide surface yields a regional averaging as opposed to the very localized averaging
of standard subdivision rules. To set the guided constructions in perspective, Fig
ures 12, 13 and 14 compare the Gauss curvature to two recently published tuned
curvaturebounded surface constructions. Overall, the shape of guided surfaces ap
pears to be the most predictable of the constructions.
We have been able to implement and thus practically test lowdegree guided con
structions for all currently used subdivision layouts: Dsprocket, Asprocket and
polar. While the degree chosen is too low to be curvature continuous at the extraor
dinary point, the surfaces are C2 everywhere else and their curvature is bounded. In
fact, the surfaces are typically visually indistinguishable from their curvature con
tinuous siblings defined in [KP07]. In Figure 11 and Table 4.1 we therefore resort
to Gausscurvature images to illustrate the tradeoff and effect of choosing low
degree approximations. We note that the lower the degree, the more fractured the
distribution of curvature.
We expect that the lowdegree constructions will be useful for highend render
ing where a good visual impression suffices, as opposed to highend engineering
applications where the curvature distribution is important.
Acknowledgement: This work was supported by NSF Grants CCF0430891 and
DMI0400214.
0 _4
 4
%P 10
Fig. 11. Curvaturedistributionvsdegree tradeoff. The guide for the examples is
g : (x, y, x2 + y2). The left column corresponds to schemes reproducing g2 o p. All
figures are shaded according to Gauss curvature with the curvature values evenly spread
over the redgreenblue range. (top row) XC6, XC5 and XC3; (middle row) XL8, XL4; (bot
tom row) Xg4, Xp3. The additional side views show (middle row) XL4 and (bottom row) XC6
(top) and xg4.
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Fig. 12. Constructions of degree (3,3). (top row, left) Input mesh: the characteristic control
net of CatmullClark subdivision projected onto the paraboloid z x2 0 1/12, (middle)
Tuned construction of [ADS06]; (right) guided construction xc3. (bottom row) Gaussian
curvature shading: the C2 construction xc6 (for comparison), [ADS06], xc3.
Fig. 13. Constructions of degree (3,3) for doublesaddle data. (left) Tuned construction of
[ADS06]: Gauss curvature is both negative and positive (red, where segments meet). (right)
guided construction xc3 is nonpositive as it should be.
40*
Fig. 14. Constructions of total degree 4. (top row, left) Input mesh: the characteristic control
net of Loop subdivision projected onto the paraboloid z 2 2 il,2, (middle) surface
rings 3,4,5 of [Loo02]; (right) surface rings 3,4,5 of xL4. (bottom row) Gaussian curvature
shading: the C2 construction xLs (for comparison), [Loo02], XL4.
