on the curvature of guided surfaces

Computer Aided Geometric Design 25 (2008) 69–79

www.elsevier.com/locate/cagd

On the curvature of guided surfaces

K. Karciauskas a, J. Peters b,∗

a Department of Mathematics and Informatics, Vilnius University, Lithuaniab Department C.I.S.E., University of Florida, CSW Building, Gainesville, FL, USA

Received 6 November 2006; received in revised form 26 May 2007; accepted 1 June 2007

Available online 13 June 2007

Abstract

Following [Karciauskas, K., Peters, J., 2007. Concentric tessellation maps and curvature continuous guided surfaces. ComputerAided Geometric Design 24 (2) 99–111], we analyze surfaces arising as an infinite sequence of guided C2 surface rings. However,here we focus on constructions of a too low degree to be curvature continuous also at the extraordinary point. To characterizeshape and smoothness of such surfaces, we track a sequence of quadratic functions anchored in a fixed coordinate system. These‘anchored osculating quadratics’ are easily computed in terms of determinants of surface derivatives. Convergence of the sequenceof quadratics certifies curvature continuity. Otherwise, the range of the curvatures of the limit quadratics gives a measure ofdeviation from curvature continuity.© 2007 Elsevier B.V. All rights reserved.

Keywords: Curvature continuity; Anchored osculating quadratic; Guided surface

1. Introduction and motivation

The paper (Karciauskas and Peters, 2007) introduces a framework for the construction of C2 surfaces from surfacerings xm, m = 0,1, . . . as illustrated in Fig. 1. To set the stage, we quickly review the key concepts and notations. Eachsurface ring xm joins smoothly and without overlap with its predecessor xm−1 if m > 0 and with existing boundarydata if m = 0. The sequence of rings contracts towards a central point x∞. The ring structure is inherited from a ct-map(concentric tessellation map) ρ :Σ × {1, . . . , n} → R

2. That is, a ct-map embeds n copies of a domain piece Σ intothe plane. Σ is for example a square or a square with a cutout. The scaled copies of ρ, λm−1ρ and λmρ for 0 < λ < 1,join with prescribed continuity and tessellate the neighborhood of the origin by a sequence of annuli. Table 1 listsseveral ct-maps defined in Karciauskas and Peters (2007) and their combinations with operators H explained below.The shape of a guided surface is inherited from a guide map g : R2 → R

3 by ‘sampling’ the composition of g withthe scaled copies of the ct-map. The guide g is a surface piece that outlines the intended shape, but need not fit therequirements of a modeling pipeline or match smoothly with the existing data. Specifically,

xm := H(g ◦ λmρ). (1)

* Corresponding author.E-mail addresses: [email protected] (K. Karciauskas), [email protected] (J. Peters).URL: http://www.cise.ufl.edu/~jorg.

0167-8396/$ – see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.cagd.2007.06.001

70 K. Karciauskas, J. Peters / Computer Aided Geometric Design 25 (2008) 69–79

Fig. 1. Sequence of guided rings xm and guided surface (Karciauskas and Peters, 2007).

Table 1Some combinations of ct-maps ρ and guided surface rings x

ct-map degree ρ layout ring degree x reproduces

ρC (3,3) �-sprocket xC6 (6,6) 2(e.g. Catmull–Clark) xC5 (5,5) 1

xC3 (3,3) 1ρL 4 �-sprocket xL8 8 2

(e.g. Loop) xL4 4 1ρp (3,1) polar xp6 (6,5) 2

xp63 (6,3) 2xp3 (3,3) 1

ρg (2,1) polar xg4 (4,3) 2

See Fig. 2 for an explanation of the entry ‘layout’. The first subscript of x indicates the ct-map ρ, the second the degree of the patches generated byH . The ct-map ρg results in G2 surface rings, all other ct-maps in C2 rings. The numbers k ∈ {1,2} in the last column indicate that xm reproducesgk ◦ ρ where gk is the kth order expansion of g at the central point.

Here H samples jets, i.e., a collection of derivatives, at fixed parameter values of its argument, and constructs xm sothat the resulting rings join smoothly and are internally smooth.

The point of guided constructions is to decouple the definition of the shape from the final, more restrictive rep-resentation. The shape is outlined by the guide and the final representation of xm is in terms of low-degree Bézieror B-splines. Joining an infinite sequence of spline rings yields a surface x with subdivision structure. Only the useof a guide surface in place of a guide (subdivision) matrix sets guided subdivision apart from the usual procedure ofrefining meshes (see Section 2). Theorem 1 of Karciauskas and Peters (2007) showed that the limit surfaces x of thisprocess are C2, provided (1) the guide surface g is twice continuously differentiable at (0,0) and (2) the operator H

generates rings xm of sufficient degree to reproduce g2 ◦ ρ, where g2 is the piecewise quadratic expansion of g at thecentral point. This theorem applies to the guided surfaces with entry ‘2’ in the last column of Table 1.

The present paper focuses on the new constructions with entry ‘1’ in the last column of Table 1. These constructionsare analogous to the C2 constructions in Karciauskas and Peters (2007) and consist of a sequence of C2 rings joiningto form a C2 surface with an ever smaller central hole. However, their degree is lower than is needed to reproducethe composition g2 ◦ ρ. The surfaces are therefore C2 except in the limit extraordinary point, x∞ where they are C1.We will see that the surfaces have bounded curvature and are typically visually indistinguishable from their curvaturecontinuous siblings.

To quantify a surface’s deviation from curvature continuity, we turn to a classical notion of curvature, the conver-gence of a sequence of quadratic functions. These quadratics are defined in a fixed coordinate system, correspondingto the tangent plane and normal at x∞, rather than in a local coordinate system. They are therefore called ‘anchored’quadratics.

Overview: Section 2 contrasts standard subdivision with guided subdivision. Section 3 introduces the new con-structions, for �-sprocket, �-sprocket and polar layout, and establishes C1 continuity of the limit surfaces. Section4 defines curvature continuity via anchored osculating quadratics and provides curvature estimates and bounds forguided surfaces. Section 5 presents and discusses results of implementations of all three layouts.

K. Karciauskas, J. Peters / Computer Aided Geometric Design 25 (2008) 69–79 71

Fig. 2. Three types of ct-maps: ρC (�-sprocket), ρL (�-sprocket), ρp (polar). The rays emanating from the center demarcate the segment bound-aries of ρ and are mapped to the segment boundaries of g.

2. Subdivision and guides

Subdivision has both a discrete structure, the refinement of a control net, and a continuous structure, by associatingrefinable generating splines with the control nodes.

The early literature on generalized subdivision (now abbreviated to subdivision) (Doo and Sabin, 1978; Catmulland Clark, 1978; Ball and Storry, 1986), focused exclusively on the discrete structure, namely the refinement Amc ofthe control net c guided by a subdivision matrix A (Fig. 3). Alternatively, one can emphasize the generating systemassociated with the mesh, i.e., the spline rings that form the surface. In the early literature, these spline rings were tac-itly understood to be structurally simple spline functions so that the ‘tuning’ or improvement of properties of schemesfocused on adjusting the spectrum of the subdivision matrix. To obtain C2 continuity guided by a matrix, Prautzschand Umlauf (1998) induces zero curvature at extraordinary points; Xue et al. (2006) associate multiple values andfunctions with each node, achieving C2 continuity, but only for valence three; Levin and Levin (2003), Schaefer andWarren (2005) focus on reproduction of quadratics to combine subdivision schemes C2 except at extraordinary pointsand Zulti et al. (2006) adjusts subdivision functions to yield one C2 extraordinary point by applying special rules fortransitions across edges of the control net. Of the many C1 curvature-bounded subdivision schemes, we implementedthe constructions of Loop (2002) and Augsdörfer et al. (2006) for comparison (see Section 5).

The focus of guided subdivision is the continuous structure of the generating system. This is grounded in theanalytic point of view (Reif, 1995; Reif and Peters, 2005) that characterizes subdivision as a process of piecing asurface together as nested spline rings converging to an extraordinary point (see Fig. 3, right). In guided subdivision,this sequence is locally guided by a map g◦λmρ. If the guide surface were a single, global polynomial and we droppedthe operator H then guided surfacing would reduce to the approach in Prautzsch (1997), Reif (1998). However, suchglobal polynomial mappings, as well as low degree piecewise polynomial mappings, are known to create surfaceswith curvature flaws (see, e.g., Fig. 3 of Karciauskas and Peters (2007)). By sampling, guided subdivision decouplesthe need for a low degree standardized representation from the degree of the guide. Most recently, a scheme was

Fig. 3. Generalized subdivision and guided subdivision both generate a sequence of nested surface rings: generalized subdivision surface rings areguided by a subdivision matrix A, guided subdivision surface rings are guided by an explicit map g.


Fig. 4. h5,5 combines four 3 × 3 blocks of Bézier coefficients, representing partial derivatives up to second order at the four corners, into one patchof degree (5,5).

derived that combines the features of generalized subdivision and guided subdivision for polar layout (Karciauskas etal., 2006).

3. Guided surfaces

This section reviews the construction of guided rings xm in Karciauskas and Peters (2007). At first sight, guidedsurfacing appears to be more complex than subdivision based on the control mesh. However, Observation 1 of Karci-auskas and Peters (2007) shows that guided surfacing based on piecewise polynomial guides is numerically stable andstationary.

Table 1 lists ct-maps 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 Karciauskas and Peters (2007).

The operator h5,5 samples (2,2)-jets at each corner (Fig. 4) and combines 3n such sampled patches into a ring xC5of degree (5,5). h5,5 can be replaced by an operator h3,3 that substitutes, for each (5,5) patch, a 3 × 3 array of bicubicpatches. The formulas determining the Bézier coefficients are illustrated in Fig. 5. Alternatively, we could view the3 × 3 of patches as one bicubic surface piece in B-spline representation, with single knots in the interior and four-foldknots at the boundary.

Replacing the three layers of Bézier coefficients of xm abutting xm−1 by extrapolating xm−1 ensures C2 continuity;that is, the surface rings join C2 when xm−1 is represented in once-subdivided form to match the granularity. Theresulting type of surface ring is xC3. Similarly, replacing the degree (6,3) patches of xp6 by three bicubics yieldsxp3. A bicubic macro-patch construction corresponding to ρg has also been developed, but presents no additionalinsight (and a second curvature-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 dif-ferent, �-sprocket layout. The ring is defined in terms of the box-spline with directions

[ 1 0 1 1 0 10 1 1 0 1 1

]that also underlies

Loop subdivision. This box-spline is C2 and of degree 4. We will determine all box-spline coefficients of one ring

Fig. 5. Interpolation of a (2,2)-jet at the corners. (left) Stencil for Bézier coefficients to form a C2 composite curve. (right) xC3 construction: the3 × 3 jets in Bézier form, shown as •, are either obtained by sampling or, when adjacent to the previous ring, by subdividing the previous ring andextending it in a C2 fashion.


Fig. 6. (left) Locations of h4-sampling in consecutive steps m−1 and m. (middle) Representation in Bézier form and (right) representation in termsof box-spline coefficients.

Fig. 7. Construction of one segment of a ring in grey. (a) The four sample points of a segment. Two of the sample points are associated with theprevious, coarser layer m − 1 and depicted as larger circles. (b) Seven of the eight box-spline coefficients at level m, determined by h4-sampling atthe upper right sample point, are depicted as •. (c) Subdivision of the box-spline net at level m − 1, depicted as light grey disks, yields another setof coefficients at level m. The two topmost coefficients are borrowed from neighboring segments.

Fig. 8. Assembly of the double ring. (left) The box-spline coefficients from levels m−1 and m determining one of n segments. Coefficients markedas hollow circles ◦ stem from sampling at level m (cf. Fig. 7, left, middle). The remaining coefficients, marked as • were generated in level m − 1.(right) Partial surface ring xL4.

from sampled derivatives. By construction from box-spline coefficients, the rings are C2 and join C2 in the sense ofLoop subdivision.

The sampling operator h4 collects eight derivatives at a sample point:(∂

js ∂k

t f)

0�j+k�2, ∂2s ∂tf, ∂s∂

2t f, f := gi ◦ λmρi.

These eight derivatives define eight box-spline coefficients (Fig. 6, middle, right) and we select seven of these toassociate with the sample point (see Fig. 7(b)).

The construction of the box-spline coefficients of one guided ring is sketched in Fig. 7. At each iteration, h4

is applied n times to generate new box spline coefficients. Where these new box-spline coefficients overlap (seeFig. 7(b)), they are averaged. Subdivision of the box-spline net at the previous level (in step 0, of the input data)generates another set of box-spline coefficients at the same granularity as shown in Fig. 7(c). Where these coefficientsoverlap the newly created ones, they prevail and replace the new ones. Together (see Fig. 8, left), the box-splinecoefficients define a ring with 12n polynomial pieces arranged as a double layer (Fig. 8, right).


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 systemso that

g(0,0) = (0,0,0).

We denote by ghk , k = 0, . . . , d , the homogeneous part of degree k of the guide surface cap g. That is gh

k is a linearcombination of monomials of total degree k in the expansion of g with respect to the parameters (u, v) and gh

k (λ·) =λkgh

k . Then

for m > 0, xmk := H

(ghk ◦ λmρ

) = λmkH(ghk ◦ ρ

). (2)

(Note that xmk is typically not homogeneous.) This yields the decomposition

xm =d∑

k=0

λ(m−1)kx1k =:

d∑k=0

lk(xhk , yh

k , zhk

). (3)

Retracing the proof of curvature continuity in Theorem 1 of Karciauskas and Peters (2007), but replacing reproductionof g2 ◦ λmρ, where g2 := gh

0 + gh1 + gh

2 by reproduction only of (gh0 + gh

1) ◦ λmρ, we see that x∞ = g(0,0), a uniquelimit of the tangent planes of the surface rings xm exists as m → ∞ and equals the tangent plane of the guide surfaceat x∞, and the guided subdivision surface x is C1 at x∞.

4. Curvature continuity and shape characterization

We now use a classical parametrization-independent notion of curvature continuity. The idea is to remove theinfluence of the surface parameterization by fixing the local coordinate system at (0,0) and isolate a quadratic mapso that surface rings are a higher-order perturbation of this quadratic. This approach can also measure deviation fromcurvature continuity. To make the idea precise, we denote piecewise polynomial terms of total degree greater than k

in the variables s1, . . . , si and with the coefficients depending on t1, . . . , tj by

π>k(s1, . . . , si; t1, . . . , tj ).If the coefficients are constant, we write π>k(s1, . . . , si).

Definition 1 (Curvature continuity). Let f : R2 → R3 be tangent plane continuous at f0 := f(0,0). We choose the

coordinate system so that f0 = (0,0,0) and the tangent plane at f0 is {z = 0}. Further assume for (s, t) �= (0,0) that(x, y, z) := f(s + σ, t + τ) satisfies the equation

z = q(s, t) · [x2, xy, y2, x, y,1]t + π>2(σ, τ ; s, t), q := [q0, . . . , q5], (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 f0 if the anchored osculating quadratics have a unique limit at f0.

Fig. 9. Illustration in one variable: (left) anchored osculating quadratics (thin stroke parabolas) are defined with respect to a fixed coordinate systemdefined by position and normal of g (thick curve) at the central point. (right) Quadratics in local coordinates.


Direct symbolic computation gives concrete formulas for anchored osculating quadratics. We use subscripts toindicate partial derivatives, e.g., xs := ∂x

∂s.

Lemma 1. If f(s, t) := (x(s, t), y(s, t), z(s, t)) then the maps qj : R2 → R are

q0 = [yt ys ]Q

[yt

ys

], q1 = −2 [xt xs ]Q

[yt

ys

], q2 = [xt xs ]Q

[xt

xs

], (5)

q3 = −2q0x − q1y − jyz/jxy, q4 = −q1x − 2q2y − jzx/jxy, (6)

q5 = q0x2 + q1xy + q2y

2 + (jyzx + jzxy + jxyz

)/jxy, (7)

where | · | denotes a determinant and

Q := 1

2(jxy)3

[L −M

−M N

], jxy :=

∣∣∣∣xs ys

xt yt

∣∣∣∣ ,L :=

∣∣∣∣∣∣xss yss zss

xs ys zs

xt yt zt

∣∣∣∣∣∣ , M :=∣∣∣∣∣∣xst yst zst

xs ys zs

xt yt zt

∣∣∣∣∣∣ , N :=∣∣∣∣∣∣xtt ytt zvt

xs ys zs

xt yt zt

∣∣∣∣∣∣ .

Let (s(s, t ), t (s, t )) be a different parametrization of (x, y, z). Then ∂sx(s, t) = xsss + xt ts , etc. By the multilinear-ity of the determinants, a lengthy but straightforward computation proves the terms qj to be unchanged under changeof variables and hence the following lemma.

Lemma 2. The anchored osculating quadratic at a point is a geometric invariant, i.e., independent of the parame-trization.

Similarly, multilinearity of determinants and elementary operations establish the following lemma where we writeqj (x, y, z) in place of qj (x(s, t), y(s, t), z(s, t)).

Lemma 3. If f = (x, y, z) := (lx1 + l2x, ly1 + l2y, l2z2 + l3z) then

liml→0

qj (x, y, z) = qj (x1, y1, z2). (8)

For the “height function” at a point, the anchored osculating quadratic simplifies even further.

Lemma 4. If f(s, t) := (s, t, h(s, t)) then

q0 = 1

2hss, q1 = hst , q2 = 1

2htt . (9)

The idea now is to insert f := xm into (4) and to bound the deviation from the quadratic as m → ∞.

Theorem 1. Let x be a guided surface of a type listed in Table 1. Then the curvature of x at the central point isbounded. If the entry in last column is ‘2’ then the surface at x∞ is curvature continuous and its anchored osculatingquadratic agrees with that of the curvature continuous guide g.

Proof. We fix the coordinate system so that the tangent plane of g at x∞ = (0,0,0) is {z = 0} and g has the expansion

g = (u + π>1(u, v), v + π>1(u, v), au2 + buv + cv2 + π>2(u, v)

). (10)

In this coordinate system, applying formulas (7), (6) and (5) to xm = (x, y, z) expanded according to (3)

xm =(

lxh1 + l2

(d∑

lk−2xhk

), lyh

1 + l2

(d∑

lk−2yhk

), lzh

2 + l3

(d∑

lk−3zhk

)), (11)

k=2 k=2 k=3


we obtain from Lemma 3

liml→0

qj (x, y, z) = qj

(xh

1 , yh1 , zh

2

), (12)

and, since the arguments are homogeneous, we can read off

liml→0

qj = 0, j = 3,4,5. (13)

Now, we define

q∞j := lim

l→0qj (x, y, z) = qj

(xh

1 , yh1 , zh

2

), for j = 0,1,2. (14)

For type 1 schemes, i.e., schemes that do not reproduce quadratics composed with the ct-map, q∞j depend on s and

t and are polynomials divided by the corresponding polynomial piece of jxy on each of its kn domain pieces (unittriangles or squares). For example, for the scheme of line 2 of Table 1, k = 3 and for line 3, k = 27. Since |jxy | > 0by injectivity of ct-maps and each of the kn domain pieces of a q∞

j is compact, each q∞j is bounded.

Since type 2 schemes reproduce quadratics,

zh2 = a

(xh

1

)2 + b(xh

1 yh1

) + c(yh

1

)2 (15)

allows us to reparameterize q∞j = qj (s, t, as2 + bst + ct2) with the coefficients of the anchored osculating quadratic

of g at x∞. Indeed, by Lemma 4, q∞0 = a, q∞

0 = b and q∞0 = c. With these well defined limits at x∞, we can apply

the mean value theorem to conclude that the surface is C2 at x∞. �The theorem yields an alternative proof of curvature continuity of the surfaces considered in Karciauskas and Peters

(2007). Moreover, it gives a concrete way to compute bounds.

4.1. Computing curvature bounds

Specifically, we compute as follows.

(i) Bound the coefficients q0, q1, q2 for the functions f1 := u2, f2 := uv, f3 := v2. This yields nine intervals thatallow computing the bounds for any function f = γ1u

2 + γ2uv + γ3v2.

(ii) If (x, y, z) is an orthogonal coordinate system in R3 with x∞ = (0,0,0), tangent plane {z = 0} at x∞ and

g = (e11u + e12v + π>1(u, v), e21u + e22v + π>1(u, v), au2 + buv + cv2 + π>2(u, v)). Then the bounds of thecoefficients q ′

k of the osculating quadratic in such system are calculated from the previous bounds on coefficientsqk as

q ′0 = q0e

20 + q1e0e1 + q2e

21, q ′

2 = q0e22 + q1e2e3 + q2e

23,

q ′1 = 2q0e0e2 + q1(e0e3 + e1e2) + 2q2e1e3, (16)

where( e0 e2

e1 e3

) := ( e11 e12e21 e22

)−1.

(iii) To bound Gaussian and mean curvatures, we observe that for (x, y, q ′0x

2 + q ′1xy + q ′

2y2) the mean curvature H

and the Gaussian curvature K at the origin (0,0,0) are

H = q ′0 + q ′

2, K = 4q ′0q

′2 − q ′2

1 .

Substituting formulas (16), we get

H = (e2

0 + e2)2

q0 + (e0e1 + e2e3

)q1 + (e2

1 + e23)q2,

K = (e0e3 − e1e2)2(4q0q2 − q2

1

). (17)

For Gaussian curvature K , we get a tighter estimate if, in formula (17), the part 4q0q2 − q21 is precomputed with

respect basis functions f1, f2, f3. Consistent with Peters and Umlauf (2001), there are six precomputed intervalsfor k and nine for H .


Table 2Gauss curvature bounds on monomials for different x and n = 8

guide K K(xC3) K(xC5) K(xp3)

u2 + 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]

(iv) Defining f1 := u2 +v2, f2 := uv, f3 := v2 gives an immediate impression of how guided subdivision reproducescanonical elliptic, hyperbolic and parabolic shapes. Table 2 lists bounds on the Gaussian curvature for theseshapes when n = 8.

5. Discussion

Traditional ‘tuning’ of subdivision algorithms focused on explicitly setting eigenvalues or on formulating anoptimization problem to approximate several spectral properties (e.g., Barthe and Kobbelt (2004)). More recently,subsubdominant eigenvectors have gained attention in the context of optimizing surfaces with respect to shape charts(Karciauskas et al., 2004; Augsdörfer et al., 2006; Ginkel and Umlauf, 2006). This allows enforcing, for example,

Fig. 10. Curvature-distribution-vs-degree trade-off. The guide for the examples is g := (x, y, x2 + y2). The left column corresponds to schemesreproducing g2 ◦ ρ. All figures are shaded according to Gauss curvature with the curvature values evenly spread over the red–green–blue range.(top row) xC6, xC5 and xC3; (middle row) xL8, xL4; (bottom row) xg4, xp3. The additional side views show (middle row) xL4 and (bottom row)xC6 (top) and xg4. (For interpretation of the refereces to color in this figure legend, the reader is referred to the web version of this article.)


Fig. 11. Constructions of degree (3,3). (top row, left) Input mesh: the characteristic control net of Catmull–Clark subdivision projected onto theparaboloid z = −x2 − 0.6y2; (middle) Tuned construction of Augsdörfer et al. (2006); (right) guided construction xC3. (bottom row) Gaussiancurvature shading: the C2 construction xC6 (for comparison) (Augsdörfer et al., 2006), xC3.

Fig. 12. Constructions of degree (3,3) for double-saddle data. (left) Tuned construction of (Augsdörfer et al., 2006): Gauss curvature is both negativeand positive (red, where segments meet). (right) guided construction xC3 is non-positive as it should be. (For interpretation of the refereces to colorin this figure legend, the reader is referred to the web version of this article.)

Fig. 13. Constructions of total degree 4. (top row, left) Input mesh: the characteristic control net of Loop subdivision projected onto the paraboloidz = −x2 − 0.6y2; (middle) surface rings 3,4,5 of (Loop, 2002); (right) surface rings 3,4,5 of xL4. (bottom row) Gaussian curvature shading: theC2 construction xL8 (for comparison), Loop (2002), xL4.

convexity. In guided subdivision we prescribe, via the guide surface, both the dominant eigenvalues and the domi-nant eigenvectors up to the order of continuity. Moreover, prescribing the guide surface yields a regional averagingas opposed to the very localized averaging of standard subdivision rules. To set the guided constructions in perspec-tive, Figs. 11, 12 and 13 compare the Gauss curvature to two recently published tuned curvature-bounded surfaceconstructions. Overall, the shape of guided surfaces appears to be the most predictable of the constructions.

In the 2005 version of this paper (Karciauskas and Peters, 2005), we observed that, by increasing the samplingdensity at each step, we can build guided surfaces with subdivision structure that are C2 and of degree (3,3). Thisapproach is theoretically interesting in that it sheds new light on the challenge to build C2 subdivision surfaces of thedegree of Catmull–Clark surfaces and on the lower bounds on the degree of curvature continuous subdivision surfaces(Reif, 1996; Prautzsch, 1998). The approach is however at present not practical since the patch size shrinks rapidlyand the number of patches grows too fast.


We have been able to implement and thus practically test all low-degree guided constructions presented in thispaper for all currently used subdivision layouts: �-sprocket, �-sprocket and polar. While the combination of degreeand patch layout is too low to be curvature continuous at the extraordinary point, the surfaces are C2 everywhereelse and their curvature is bounded. In fact, the surfaces are typically visually indistinguishable from their curvaturecontinuous siblings defined in Karciauskas and Peters (2007). In Fig. 10 and Table 2 we therefore resort to Gauss-curvature images to illustrate the trade-off and effect of choosing low-degree approximations. We note that the lowerthe degree, the more fractured the distribution of curvature.

While high-end engineering applications will require excellent curvature distribution, we expect that the low-degreeconstructions will be useful for high-end rendering where a good visual impression suffices.

Acknowledgement

This work was supported by NSF Grants CCF-0430891 and DMI-0400214.

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