APPROXIMATE G

CUBIC SURFACES FOR DATA

APPROXIMATION

Yingbin Liu and Stephen Mann

University of Waterloo, Waterloo, Ontario, Canada

Keywords:

CAGD, Triangular B

ezier patches, Continuity.

Abstract:

This paper presents a piecewise cubic approximation method with approximate G

continuity. For a given

triangular mesh of points with arbitrary topology, one cubic triangular B

ezier patch surface is constructed.

The resulting surfaces have G

continuity at the vertex points, but only requires approximate G

continuity

along the macro-patch boundaries so as to lower the patch degree. While our scheme cannot generate the

surfaces in as high quality as Loop’s sextic scheme, they are of half the polynomial degree, and of far better

shape quality than the results of interpolating split domain schemes.

1 INTRODUCTION

In computer graphics applications, a surface is usually

constructed from a given data set that represents more

or less precisely the surface shape. Due to their el-

egant geometric properties, triangular B

ezier patches

are often used to interpolate or approximate a triangu-

lated data mesh. The main difﬁculty to interpolate or

approximate a given data mesh is how to satisfy the

restrictions imposed on the control points to obtain

surface patches that meet with a given order of conti-

nuity. In most cases, we want the surface patches to

be visually smooth and meet with G

continuity.

In past work, different G

schemes using domain

splits were devised to ﬁt multiple patches (three or

four) to each data triangle (Peters, 1990; Piper, 1987;

Shirman and S

equin, 1991; Hahmann and Bonneau,

2003; Mann et al., 1992). Since it is impossible to

generated G

continuous surfaces on arbitrary data

sets using cubic patches (Piper, 1987; Peters, 1990),

these schemes use at least quartic patches to guar-

antee G

continuity along the macro-patch bound-

aries. It was later shown that the results from these

local interpolation schemes do not generate satisfac-

tory shapes (Mann et al., 1992). Loop designed a

scheme using sextic triangular B

ezier patches in one-

to-one correspondence with the data triangles (Loop,

1994). In Loop’s scheme, an approximation method

using transformed data vertices was also presented to

create smooth surfaces with high quality. In all these

schemes, control points are set to meet the constraints

required by G

continuity along the patch boundaries.

The price to fulﬁl the G

constraints is that we have

to use high degree (at least quartic) and/or multiple

patches and possibly sacriﬁce shape quality.

Approximate G

continuity is a relaxation of G

continuity where patches meet C

but the discontinu-

ity in the surface normals between adjacent patches

is small. Previous work on approximate G

construc-

tions includes parametric interpolation of positions,

normals, and second fundamental forms at the data

points (Mann, 1992); interpolations with cubic preci-

sion on positions and normals at the data points, both

in the functional case (Liu and Mann, 2005) and the

parametric case (Liu and Mann, 2007). In this paper,

the given data network is approximated by integrat-

ing techniques from Loop’s scheme to construct the

boundary curves (Loop, 1994). For each data triangle,

three cubic triangular patches are created to meet with

continuity. As for the macro-boundaries, i.e., the

boundaries between patches from different data trian-

gles, since we only require approximate G

continu-

ity, some discontinuity exists in the surface normals.

To decrease this normal discontinuity, two neighbour-

ing patches across the macro-patch boundaries are ad-

justed later to have equal normals at the middle point

Liu Y. and Mann S. (2008).

APPROXIMATE G1 CUBIC SURFACES FOR DATA APPROXIMATION.

In Proceedings of the Third International Conference on Computer Graphics Theory and Applications, pages 39-44

DOI: 10.5220/0001093400390044

 SciTePress

on the common boundary curve.

The remaining parts of this paper are arranged as

follows: Section 2 reviews the G

continuity con-

straints; the details of our approximation is introduced

in Section 3; in Section 4, we show some resulting

surfaces, which we compare to surfaces generated by

other G

schemes; conclusions and possible future

work are presented in Section 5.

2 CONTINUITY CONSTRAINTS

A triangular B

ezier patch of degree n is deﬁned as

S(u, v,w) =

∑

i+ j+k=n

i jk

(u,v,w).

Here (u,v,w) are barycentric coordinates for the para-

meter; B

i jk

(u,v,w) is the generalized Bernstein poly-

nomial; P

i jk

are the control points of the patch. For

two triangular patches to join each other with G

con-

tinuity, i.e., the tangent planes from the two adjacent

patches are coplanar at any point along the boundary

curve, certain constraints must be fulﬁlled.

i−1

Figure 1: G

continuity at a vertex.

For a vertex V with valence n in our mesh, we num-

ber the surrounding patches F

, i = 0, . ..,n − 1, in

anti-clockwise order. Figure 1, right, shows a pair

of neighbouring patches F

i−1

and F

with common

boundary H

, with their domain triangles appearing

on the left of the ﬁgure. The parametric direction for

the partial derivative along the common edge is de-

ﬁned from V to V

in the domain triangles.

In this paper, we focus on cubic B

ezier patches to

explore the G

continuity conditions. Two adjacent

cubic patches F

and F

i−1

with their control points are

shown in Figure 2. We deﬁne the following vectors:

= H

j+1

− H

, ~v

= F

− H

, ~w

= F

i−1

− H

with j = 0, 1,2. Then for any given point along the

boundary curve, the tangent planes calculated from

the two patches are deﬁned by vectors ~u, ~v and ~w, all

of which are quadratic B

ezier functions:

~u =

∑

j=0

(t), ~v =

∑

j=0

(t), ~w =

∑

j=0

(t).

i−1

Figure 2: Adjacent cubic patches.

If patches F

and F

i−1

meet with G

continuity, they

must share the control points of the common bound-

ary H

, j = 0,1, 2, 3. Moreover, the two tangent

planes at any boundary point should be coplanar, i.e.,

there exist scalar valued functions φ(t), µ(t) and ν(t)

such that for any parameter value t, t ∈ [0,1], we have

φ(t)~u = µ(t)~v + ν(t)~w. (1)

To guarantee the proper orientation of the tangents,

assume µ(t)ν(t) ≥ 0. Since it is not guaranteed to

create surface using cubic patches to have G

con-

tinuity (Piper, 1987; Peters, 1990), all the existing

schemes use at least quartic patches (Shirman and

equin, 1991; Mann et al., 1992)

For all the patches to meet with G

continuity at

vertex V , the G

constraints of Equation 1 are applied

to all boundaries originating from vertex V . Differen-

tiating these equations and evaluate at vertex V results

in a linear system. Solving this system is called the

“twist compatibility” or “vertex consistency” prob-

lem, and it has been shown that this linear system may

not have a solution (Watkins, 1988; Sarraga, 1988;

Loop, 1994). Even if there exists a G

solution by

solving the twist compatibility problem, it is not clear

that the resulting surface has a satisfactory shape.

In the past work, one approach to construct G

continuous surfaces is to perform a Clough-Tocher

like domain split on each patch F

surrounding vertex

V (Clough and Tocher, 1965). At least quartic degree

patches are used in a domain split method (Shirman

and S

equin, 1991). Alternatively, Loop invented a so-

lution to solve the twist compatibility using one sextic

patch for each data triangle (Loop, 1994), with details

introduced in Section 3.1.

Instead of using high degree patches to fulﬁl the

constraints, we focus on using low degree (cu-

bic) patches to create surfaces with approximate G

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

continuity. With the approximate G

method, we al-

low a small amount of normal discontinuity across the

boundary curves. However, we still require G

conti-

nuity at the vertex points to keep the surface disconti-

nuity going too high. Using the freedom provided by

this relaxation, the goals of our method include:

• Use of cubic patches,

• Better shaped surfaces.

3 CUBIC APPROXIMATION

The goal of our work is to approximate the shape rep-

resented by a given data mesh as accurately as pos-

sible, but using lower degree patches than the G

schemes. We achieve this reduction in degree by

only requiring approximate G

continuity along patch

boundaries. In the cubic scheme we describe in this

paper, we construct the surface by creating boundary

curves for each edge of a data triangle, and then per-

form a Clough-Tocher like domain split on the cubic

patch. After the interior of the patches is ﬁlled, we

then do some adjustments to lower the normal dis-

continuity along the boundaries.

3.1 Loop’s Scheme

In Loop’s scheme, boundary curves are constructed

so that the solutions for the twist terms are guaran-

teed (Loop, 1994). Loop’s sextic scheme has the fol-

lowing steps:

1. Construct quartic boundary curves.

2. Solve the twist terms.

3. Construct tangent ﬁelds along the boundaries.

4. Elevate the degree of the boundary curves to sex-

tic.

5. Set the second row of control points in the sextic

patch to interpolate the tangent ﬁelds.

6. Set the remaining center control point by averag-

ing the control points from above steps.

When the valences of the three vertices of a data trian-

gle are equal, the patch constructed by Loop’s scheme

is quintic; if the three valences are all six, then the

patch degree reduces to quartic (Loop, 1994). Our

cubic scheme integrated the boundary tangents con-

struction from the ﬁrst step in Loop’s scheme, but

generating cubic boundary curves.

i+1

i−1

i+1

i−1

Figure 3: Construction of boundary curves.

3.2 Boundary Construction

For the edge between the vertex V and its neighbour

, we use some ideas of Loop (Loop, 1994) to con-

struct a cubic boundary curve H

, as illustrated in Fig-

ure 3. Since the curve is cubic, there are four control

points: H

, H

to be constructed. For the ver-

tex V , the control points H

and H

are constructed as

= αV + (1 − α)A = αV +

1 − α

n−1

∑

k=0

, (2)

= H

n−1

∑

k=0



cos



2(k − i)π



. (3)

Here n is the valence of the vertex V ; point A is the

average of all neighbours of vertex V ; α and β are two

shape parameters. When α = 1, the resulting surface

interpolates the data vertices, shown as V

in Figure 3.

For α to be other values, the resulting surface approx-

imates the given data mesh. Parameter β deﬁnes the

length of the tangent vector at H

. In our cubic ap-

proximation scheme, the α and β are set with an ex-

perimental value the same as in Loop’s scheme:

α =



4 + cos(

2π

)



β =



1 + cos(

2π

)



. (4)

With this α value, our scheme does not interpolate the

given data vertices, but generating an surface approx-

imating the shape of the given data mesh.

The construction of control points H

and H

performs a ﬁrst order Fourier transformation on all

the neighbouring vertices of V (Loop, 1994; Davis,

1979). If all the control points H

are connected,

they form a regular n-gon with H

being the centre,

APPROXIMATE G1 CUBIC SURFACES FOR DATA APPROXIMATION

as shown in the Figure 3. Point H

is the afﬁne com-

bination of the data vertex V and the average center of

V ’s neighbours, and all the boundary curves connect-

ing V share the same position of point H

. The other

two control points H

and H

are calculated similarly

with Equation 3, using the information of V

’s neigh-

bours.

In Loop’s scheme, quartic boundary curves have

to be used to fulﬁl the strict G

continuity conditions,

and later the patch degree is elevated to sextic for the

same reason.

3.3 Patch Construction

Past work shows that it can be impossible to gener-

ate G

continuous surfaces using cubic patches (Piper,

1987; Peters, 1990). When using approximate G

continuity, we found there was insufﬁcient degrees of

freedom if we only used one cubic patch per data tri-

angle. In particular, with one cubic patch per face,

since the center control point of this patch affects the

normals along all three boundaries, it is difﬁculty to

place this control point to yield surfaces with accept-

able smoothness. To improve surface smoothness and

still use cubics, we perform a Clough-Tocher like do-

main split (Clough and Tocher, 1965) to construct

three micro-patches for each data triangle.

Figure 4: Macro patch split.

We ﬁrst construct one cubic patch for each data trian-

gle, as shown on the left in Figure 4. The boundaries

of this initial patch are set with the methods described

in Section 3.2. We set the center control point to be

the weighted average of the other nine control points,

using a method similar to the way introduced in one

author’s previous paper (Mann, 1999).

We then subdivide the patch into three cubic

patches (called micro patches), as shown in the right

of Figure 4. The subdivided control points are shown

as small black dots in Figure 5. The small white and

the big white points are constructed as the gravity cen-

tre of the adjacent three control points, as in Clough-

Tocher’s scheme (Clough and Tocher, 1965; Farin,

1986; Mann, 1999). After the split, the center control

point of each micro-patch, shown as big black dots in

Figure 5, only affects the continuity along the corre-

sponding macro-patch boundary. The initial positions

Figure 5: Control points construction.

i−1

Figure 6: Ratios of intersection.

for these three points are set by local averaging. As

shown in Figure 6, The two pairs of side panels are

set to be coplanar to have equal normals at end data

points H

and H

. Let F

i−1

intersect H

at P

i−1

intersect H

at P

, as shown in Figure 6.

Here the points F

and F

i−1

( j = 0, 1, 2) are the micro

patch control points resulting from the Clough-Tocher

like split. The ratios of the intersection are then de-

ﬁned as

− H

, γ

i−1

− P

i−1

− F

− H

, γ

i−1

− P

i−1

− F

We set the initial position for point F

and F

i−1

= H

+ (1 −

+ γ

)

− F

i−1

+ F

− F

i−1

+ η

− H

i−1

= H

−

+ γ

− F

i−1

+ F

− F

i−1

+ η

− H

). (5)

The two center control points across the macro-

patch boundary are then adjusted by adding an offset

GRAPP 2008 - International Conference on Computer Graphics Theory and Applications

vector that is calculated with the method introduced in

our previous paper. After the adjustments, the normal

vectors of two patches at the middle boundary point is

equal. Theoretically, we can adjust the center points

to have equal normals at two or three points along the

boundary, but that will make the scheme not stable. In

all the examples shown in Section 4, all the surfaces

by our scheme have equal normals only at one middle

boundary points.

This cubic scheme is approximately G

along the

macro-patch boundaries, although patches meet with

continuity at the vertex points; the patches meet

with C

continuity along the boundaries between two

micro-patches; and the three micro-patches meet with

at the split point, a result of the averaging in Step

3 of (Farin, 1986).

4 RESULTS

We compared the resulting surfaces of our cubic

scheme to those generated by Loop’s scheme, with

both the interpolation and the approximation results.

Since Loop’s scheme does not use the domain split

method, we also generated surfaces on the same

models using Shirman-S

equin’s quartic scheme to

make the comparison more comprehensive. Shirman-

equin’s scheme generates surfaces with data vertices

interpolated, and it also performs a domain split sim-

ilar to Clough-Tocher’s (Shirman and S

equin, 1991).

In Figure 7, we show four surfaces ﬁt to an icosa-

hedron. The top left is Loop’s interpolating sur-

face; the bottom left is Loop’s approximating surface;

the top right is the quartic, G

scheme by Shirman-

equin; the bottom right is our cubic, approximate G

scheme. Loop’s approximate scheme has much bet-

ter shape than his interpolating scheme; our cubic ap-

proximation scheme has similar shape to Loop’s. The

highest normal discontinuity is 0.48 degree on our cu-

bic surface.

In Figure 8, we show four surfaces ﬁtted to the

Stanford bunny model. The order of the schemes is

the same as in Figure 7. While the two interpolat-

ing schemes show numerous shape artifacts, both ap-

proximating schemes produce surfaces with far bet-

ter shape quality. In this model, the highest normal

discontinuity is 19.57 degree for our cubic surface.

However, boundaries with high normal discontinuity

angles (more than 10 degrees) account for about 0.8%

of all boundary curves. While some artifacts are visi-

ble in our approximate G

scheme, we see substantial

improvement over the similar interpolating scheme.

Figure 7: Icosahedron surfaces.

Figure 8: Bunny surfaces.

5 CONCLUSIONS

In this paper we presented a cubic approximating

scheme to generate surfaces characterizing the shape

of given data mesh with low price. Although the nor-

mal discontinuities on the resulting surfaces are visi-

ble on some test models, the overall surface shape is

good. Further, our scheme is only cubic and it will of-

fer great advantage for easy implementation and sta-

bility.

APPROXIMATE G1 CUBIC SURFACES FOR DATA APPROXIMATION

In the future, we will do further analysis on the

cross boundary discontinuity, and derive an upper

bound for it. We will also look for methods to guar-

antee that the normal discontinuity does not exceed

a speciﬁed bound. This will likely result in condi-

tions similar to the vertex consistency conditions, but

our expectation is that we will be able to trade patch

shape for lower normal discontinuity.

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