Subdivision depth computation for n-ary subdivision curves/surfaces

Vis Comput (2010) 26: 841–851DOI 10.1007/s00371-010-0496-0

O R I G I NA L A RT I C L E

Subdivision depth computation for n-ary subdivisioncurves/surfaces

Ghulam Mustafa · Muhammad Sadiq Hashmi

Published online: 14 April 2010© Springer-Verlag 2010

Abstract This paper deals with subdivision depth compu-tation technique for n-ary subdivision curves/surfaces. Thistechnique also includes error bound evaluation technique forn-ary subdivision curves/surfaces with their control poly-gon. Both techniques provide error control tools in subdi-vision schemes.

Keywords Subdivision curve · Subdivision surfaces ·Subdivision depth · Error bound · Control polygon ·Forward differences

1 Introduction

Subdivision is a simple and elegant method to describesmooth curves and surfaces. The approach of subdivisionschemes is simple and efficient due to its mathematical for-mulation. Its application ranges from industrial design andanimation to scientific visualization and simulation. Due tothis, subdivision method is becoming a standard techniqueand now well understood by both academic and industrialcommunities. It is an algorithm to generate smooth curvesand surfaces as a sequence of successively refined controlpolygons. At each subdivision level, the subdivision schemedescribe the source grid maps to the subdivided grid, whichresults in increase in the number of points. The number of

This work is supported by the Indigenous Ph.D. Scholarship Schemeof Higher Education Commission (HEC) Pakistan.

G. Mustafa (�) · M.S. HashmiDepartment of Mathematics, The Islamia University ofBahawalpur, Bahawalpur, Pakistane-mail: [email protected]

M.S. Hashmie-mail: [email protected]

points inserted at level k+1 between two consecutive pointsfrom level k is called arity of the scheme. In the case whennumber of points inserted are 2,3, . . . , n the subdivisionschemes are called binary, ternary, . . . , n-ary, respectively.For more details on n-ary subdivision schemes, we may re-fer to [8], Jian-ao Lian [6, 7], thesis of Aspert [1], Ko [5],and Najma [12].

Although subdivision schemes have become importantin recent years because they provide a precise and efficientway to describe smooth curves/surfaces, however, little hasbeen done in the area of error control for n-ary subdivisioncurves/surfaces. The investigation of error control raises twoquestions [4]:

• How well the control polygon approximates the limitcurve?

• How many subdivision steps are needed to satisfy a user-specified error tolerance?

For given error tolerance, the subdivision levels performedon the initial control polygon, so that the error/distance be-tween the resulting control polygon and the limit curve/ sur-face would be less than the error tolerance is called subdivi-sion depth.

A subdivision depth and error bound based on forwarddifferences of control points have been presented by [2–4,9–11, 17], while the methods [13–16] are based on eigen-analysis. But nothing in this area has been done for moregeneral n-ary subdivision curves/surfaces yet. In this paper,we will answer the above said questions and present a sub-division depth computation technique based on error boundsfor n-ary subdivision curves/surfaces.

It is notified that the increase in arity offers greater free-dom than offered by low arity subdivision curve/surface interms of coefficients. Higher arity curves/surfaces allow arange of different behaviors than the lower arity curves/

mailto:[email protected]

mailto:[email protected]

842 G. Mustafa, M.S. Hashmi

surfaces. Ko [5] notified that subdivision curves/surfaceswith higher arity results in higher smoothness and ap-proximation order but smaller in support, which makes itmore practical in use. It is also noticed that higher aritycurves/surfaces have slightly lower computational cost thanlower arity curves/ surfaces. This discussion motivates usto calculate error bound and depth for higher arity sub-division curves/surfaces, i.e., in general for n-ary subdivi-sion curves/surfaces. Our method is the generalization ofMustafa et al. [3, 9, 10]. The paper is arranged as follows.

Section 2 is devoted for basic definitions and notations.In Sects. 3 and 4, we have computed subdivision depth forn-ary subdivision curves and n-ary subdivision surfaces, re-spectively. Section 5 presents applications of our results forn-ary subdivision curve and surfaces. Conclusion and futureresearch directions are given in Sect. 6. The typical mathe-matical proofs and tables are placed in Appendices A, B, C,and D for a transparent presentation of the paper.

2 Definitions and notations

2.1 n-ary subdivision curve

Given a sequence of control points pki ∈ RN , i ∈ Z, N > 1,

where the upper index k > 0 indicates the subdivision level.n-ary subdivision curve [1] is defined by

pk+1ni+α =

m∑

j=0

aα,jpki+j , α = 0,1, . . . , n − 1, (1)

where m > 0 and

m∑

j=0

aα,j = 1, α = 0,1, . . . , n − 1. (2)

The set of coefficients {aα,j , α = 0,1, . . . , n − 1}mj=0 is

called subdivision mask. Given initial values p0i ∈ RN, i ∈

Z. Then in the limit k → ∞, the process (1) defines an in-finite set of points in RN . The sequence of control points{pk

i } is related, in a natural way, with the diadic meshpoints tki = i/nk, i ∈ Z. The process then defines a schemewhereby pk+1

ni and pk+1ni+n replaces the values pk

i and pki+1

at the mesh points tk+1ni = tki and tk+1

ni+n = tki+1, respectively,

while pk+1ni+α is inserted at the new mesh points tk+1

ni+α =1n((n − α)tki + αtki+1) for α = 1,2, . . . , n − 1. Labeling of

old and new points is shown in Fig. 1 which illustrates sub-division scheme (1).

2.2 n-ary subdivision surface

Given a sequence of control points pki,j ∈ RN , i, j ∈ Z,

N ≥ 2, where the upper index k ≥ 0 indicates the subdivi-sion level. n-ary subdivision surface is tensor product of (1)defined by

pk+1ni+α,nj+β =

m∑

r=0

m∑

s=0

aα,raβ,spki+r,j+s ,

α,β = 0,1, . . . , n − 1, (3)

where aα,r satisfies (2). Given initial values p0i,j ∈ RN, i, j ∈

Z, then in the limit k → ∞, the process (3) defines an in-finite set of points in RN . The sequence of values {pk

i,j }is related, in a natural way, with the diadic mesh points( ink ,

j

nk ), i, j ∈ Z. The process then defines a scheme

whereby pk+1ni+α,nj+β replaces the value pk

i+α/n,j+β/n at

the mesh point (i+α/n

nk ,j+β/n

nk ) for α,β ∈ {0, n}, while the

values pk+1ni+α,nj+β are inserted at the new mesh points

( ni+α

nk+1 ,nj+β

nk+1 ) for α,β = 0,1, . . . , n − 1 (where α and β arenot zero at the same time). Labeling of old and new pointsis shown in Fig. 2 which illustrates subdivision scheme (3).

2.3 Subdivision depth

Given control polygon of n-ary subdivision curve/surfaceand an error tolerance ε, if we subdivide control polygon k

times so that the error between resulting polygon and subdi-vision curve/surface is smaller than ε, then k is called subdi-vision depth of subdivision curve/surface with respect to ε.

2.4 Notations

Here, we settle some notations for fair reading of this paper.Assume

χ = maxi

∥∥p0i+1 − p0

i

∥∥, (4)

δ1 = maxβ

{∣∣∣∣∣

m∑

j=0

bβ,j

∣∣∣∣∣, β = 0,1, . . . , n − 1

}, (5)

δ2 = maxα,β

{∣∣∣∣∣

m∑

s=0

aα,s

m∑

r=0

bβ,r

∣∣∣∣∣, α,β = 0,1, . . . , n − 1

}, (6)

Fig. 1 Solid lines show coarsepolygons whereas dotted linesare refined polygons.(a)–(c) represent binary, ternary,and quaternary refinement ofcoarse polygons using (1) forn = 2,3,4, respectively

Subdivision depth computation for n-ary subdivision curves/surfaces 843

Fig. 2 Solid lines show oneface of coarse polygons whereasdotted lines are refinedpolygons. (a)–(c) can beobtained by subdividing oneface into four, nine, and sixteennew faces by using (3) forn = 2,3,4, respectively

where{

bβ,j = ∑j

t=0(aβ,t − aβ+1,t ), β = 0,1, . . . , n − 2,

bn−1,j = a0,j − ∑n−2β=0 bβ,j .

(7)

Also,

γ = maxα

{∣∣∣∣∣

m−1∑

j=0

aα,j

∣∣∣∣∣, α = 0,1, . . . , n − 1

}, (8)

η1α,β =

∣∣∣∣∣aβ,0

m∑

t=1

aα,t − α(n − β)

n2

∣∣∣∣∣ +∣∣∣∣∣aβ,0

m−1∑

s=1

aα,s

∣∣∣∣∣, (9)

η2α,β =

∣∣∣∣∣

m∑

t=1

aβ,t − β

n

∣∣∣∣∣ +∣∣∣∣∣

m∑

r=0

aα,r

m−1∑

s=1

aβ,s

∣∣∣∣∣, (10)

η3α,β =

∣∣∣∣∣

m∑

t=1

aα,t

m∑

t=1

aβ,t − αβ

n2

∣∣∣∣∣ +∣∣∣∣∣

m∑

t=1

aβ,t

m−1∑

s=1

aα,s

∣∣∣∣∣, (11)

where⎧⎨

⎩

aα,0 = ∑mt=1 aα,t − α

n,

aα,j = ∑mt=j+1 aα,t , j ≥ 1,

α = 0,1, . . . , n − 1. (12)

Suppose further for α,β = 0,1, . . . , n − 1,

Nkα = max

i

∥∥∥∥pk+1ni+α − 1

n

((n − α)pk

i + αpki+1

)∥∥∥∥, (13)

Mkα,β = max

i,j

∥∥∥∥pk+1ni+α,nj+β − 1

n2

{(n − α)(n − β)pk

i,j

+ α(n − β)pki+1,j + (n − α)βpk

i,j+1

+ αβpki+1,j+1

}∥∥∥∥. (14)

Assume⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

Δki,j,1 = pk

i+1,j − pki,j ,

Δki,j,2 = pk

i,j+1 − pki,j ,

Δki,j,3 = pk

i+1,j+1 − pki,j+1,

χt = maxi,j‖Δ0i,j,t‖, t = 1,2,3.

(15)

Furthermore, suppose

ϑ = maxα,β

{3∑

t=1

(χt )(ηt

α,β

), α,β = 0,1, . . . , n − 1

}. (16)

3 Depth for n-ary subdivision curves

In this section, we find subdivision depth for n-ary subdi-vision curves. Moreover, we prove that error bounds for bi-nary, ternary, and quaternary subdivision curves [3, 9, 10]are special cases of our bounds.

Lemma 1 Given initial control polygon p0i = pi , i ∈ Z, let

the values pki , k ≥ 1 be defined recursively by subdivision

process (1) together with (2) then

Nkα ≤ γχ(δ1)

k, (17)

where χ , δ1, γ, and Nkα are defined by (4), (5), (8), and (13),

respectively.

Proof is given in Appendix A.

Lemma 2 Given initial control polygon p0i = pi , i ∈ Z, let

the values pki , k ≥ 1 be defined recursively by subdivision

process (1) together with (2). Suppose P k is the piecewiselinear interpolant to the values pk

i and P ∞ is the limit curveof the process (1). If δ1 < 1, then the error bound betweenlimit curve and its control polygon after k-fold subdivisionis

∥∥P k − P ∞∥∥∞ ≤ γχ

((δ1)

k

1 − δ1

), (18)

where χ , δ1 and γ are defined by (4), (5), and (8), respec-tively.

Proof Let ‖.‖∞ denote the maximum norm. Since the max-imum difference between P k+1 and P k is attained at a pointon the (k + 1)th mesh, we have

∥∥P k+1 − P k∥∥∞ ≤ max

α

{Nk

α, α = 0,1, . . . , n − 1}, (19)


where Nkα is defined by (13). From (17) and (19), we get

∥∥P k+1 − P k∥∥∞ ≤ γχ(δ1)

k,

where χ , δ1 and γ are defined by (4), (5), and (8), respec-tively.

Triangle inequality yields (18). This completes theproof. �

Remark 1 Theorem 1 in [10], Theorem 2.1 in [9], and The-orem 2.1 in [3] designed to estimate error bound for bi-nary, ternary, and quaternary subdivision curves, respec-tively (i.e., each edge is divided in 2, 3, and 4 subedges,respectively). But for the higher arity subdivision curves,such as for n = 5,6, . . . (when each edge is divided in5,6, . . . subedges), error estimates are not feasible by exist-ing results. Our Lemma 2 provides freedom to evaluate errorbound for all arities. Here, we also mention that Lemma 2for n = 2,3,4, reduces to [6, Theorem 1], [7, Theorem 2.1],and [5, Theorem 2.1], respectively.

Now we offer the computational formula of subdivisiondepth for n-ary subdivision curves.

Theorem 1 Let k be subdivision depth and let dk be theerror bound between n-ary subdivision curve P ∞ and itsk-level control polygon P k . For arbitrary ε > 0, if

k ≥ logδ1−1

(γχ

ε(1 − δ1)

),

then

dk ≤ ε.

Proof From (18), we have

dk = ∥∥P k − P ∞∥∥∞ ≤ γχ

((δ1)

k

1 − δ1

).

This implies, for arbitrary given ε > 0, when subdivisiondepth k satisfies the following inequality

k ≥ logδ1−1

(γχ

ε(1 − δ1)

),

then

dk ≤ ε.

This completes the proof. �

4 Depth for n-ary subdivision surfaces

In this paragraph, we compute subdivision depth for n-arysubdivision surfaces. Moreover, we show that results of er-ror bounds for binary, ternary, and quaternary subdivision

surfaces [3, 9, 10] are special cases of our result. Here, weneed following lemmas for Theorem 2. The proof of firsttwo lemmas are shown in Appendices B and C, respectively.

Lemma 3 Given initial control polygon p0i,j = pi,j , i, j ∈

Z, let the values pki,j , k ≥ 1 be defined recursively by subdi-

vision process (3) together with (2) then

maxi,j

∥∥Δki,j,t

∥∥ ≤ (δ2)k max

i,j

∥∥Δ0i,j,t

∥∥, (20)

where δ2, Δki,j,t , t = 1,2,3 are defined by (6) and (15),

respectively.



vision process (3) together with (2) then

Mkα,β ≤ (δ2)

k

3∑

t=1

(χt )(ηt

α,β

), (21)

where δ2, ηtα,β,Mk

α,β,χt , α,β = 0,1, . . . , n − 1 are definedby (6), (9)–(11), (14) and (15).



vision process (3) together with (2). Suppose P k is the piece-wise linear interpolant to the values pk

i,j and P ∞ is the limitsurface of the subdivision process (3). If δ2 < 1, then the er-ror bound between the limit surface and its control polygonafter k-fold subdivision is

∥∥P k − P ∞∥∥∞ ≤ ϑ

((δ2)

k

1 − δ2

), (22)

where δ2 and ϑ are defined by (6) and (16), respectively.

Proof Let ‖.‖∞ denote the uniform norm. Since the maxi-mum difference between P k+1 and P k is attained at a pointon the (k + 1)th mesh, we have∥∥P k+1 − P k

∥∥∞ ≤ maxα,β

{Mk

α,β,α,β = 0,1, . . . , n − 1}, (23)

where Mkα,β is defined by (14). Using (21) and (23), we get

∥∥P k+1 − P k∥∥∞ ≤ ϑ(δ2)

k,

where δ2 and ϑ are defined by (6) and (16), respectively.By triangle inequality, we get (22). This completes the

proof. �

Remark 2 Theorem 7 in [10], Theorem 3.2 in [9], and The-orem 3.3 in [3] designed to estimate error bound for binary,ternary, and quaternary subdivision surfaces, respectively


(i.e., each face is divided in 4, 9, and 16 subfaces, respec-tively). But for the higher arity subdivision surfaces, suchas for n = 5,6, . . . (when each face is divided in 52,62, . . .

subfaces), error estimates are not feasible by existing re-sults. Since estimation of error bound for n-ary subdivi-sion surfaces is quite necessary in these cases, therefore,our Lemma 5 gives error bound of all arities subdivisionschemes. Here, we also mention that [6, Theorem 7], [7,Theorem 3.2], and [5, Theorem 3.3] are special cases ofLemma 5 for n = 2,3,4, respectively.

Here, we suggest the computational formula of subdivisiondepth for n-ary subdivision surfaces.

Theorem 2 Let k be subdivision depth and let dk be theerror bound between n-ary subdivision surface P ∞ and itsk-level control polygon P k . For arbitrary ε > 0, if

k ≥ logδ2−1

(ϑ

ε(1 − δ2)

),

then

dk ≤ ε.

Proof From (22), we have

dk = ∥∥P k − P ∞∥∥∞ ≤ ϑ

((δ2)

k

1 − δ2

).

This implies, for arbitrary given ε > 0, when subdivisiondepth k satisfies the following inequality

k ≥ logδ2−1

(ϑ

ε(1 − δ2)

),

then

dk ≤ ε.


5 Applications

5.1 Error bound and depth of n-ary interpolatingsubdivision curves

In this section, we estimate the error bound and subdivisiondepth of following (2b + 2)-point n-ary interpolating subdi-vision curve [12]

pk+1ni = pk+1

i ,

(24)

pk+1ni+s =

b+1∑

t=−b

(∏b+1m=−b(s − nm)

Ast

)pk

t+i ,

Fig. 3 Here, dotted lines present control polygons and solid lines rep-resent: (a) A four tile triangular curve generated by quinary (i.e., n = 5)

interpolating subdivision curve; (b) Rectangular lattice generated byquinary approximating subdivision curve

where Ast = (s − nt)(−1)(b−1−t)(n)(2b+1)(b + t)!(b −t + 1)! and s = 1,2, . . . , n − 1, b = 1,2,3, . . . . and n standsfor n-ary interpolating subdivision curve, i.e., n = 2,3,4, . . .

stands for binary, ternary, quaternary, and so on, respec-tively.

A four tile triangular curve generated by (24) for n = 5is shown in Fig. 3(a). Its error bounds and subdivision depthare shown in Appendix D in Tables 1 and 2, respectively.In these tables, we have also shown the error bounds anddepth of different arity interpolating subdivision curves byusing Lemma 2 and Theorem 1 with χ = 0.1. Its error isalso explained by Fig. 5(a).

5.2 Error bound and depth of n-ary approximatingsubdivision curves

In this section, we estimate the error bound and subdivisiondepth of following (2b + 2)-point n-ary approximating sub-division curve [12]

f k+1ni+s =

b+1∑

t=−b

(∏b+1m=−b(2s + 1 − 2nm)

Bst

)f k

t+i , (25)

where Bst = (2s + 1 − 2nt)(−1)(b−1−t)(2n)(2b+1)(b +t)!(b − t + 1)! and s = 0,1,2, . . . , n − 1, b = 1,2,3, . . .

and n stands for n-ary approximating subdivision curve.The well-known rectangular lattice generated by (25) for

n = 5 is shown in Fig. 3(b). Its error bounds and subdivi-sion depth are shown in Appendix D in Tables 3 and 4,respectively. In these tables, we have also shown the er-ror bounds and depth of different arity approximating sub-division curves by using Lemma 2 and Theorem 1 withχ = 0.1.

5.3 Error bound and depth of n-ary interpolatingsubdivision surfaces

By taking the tensor product of (24), we get following (2b+2)-point n-ary interpolating subdivision surface


pk+1ni,nj = pk

i,j ,

pk+1ni+s1,nj

=b+1∑

t1=−b

(∏b+1m1=−b(s1 − nm1)

As1t1

)pk

t1+i,j ,

pk+1ni,nj+s2

=b+1∑

t2=−b

(∏b+1m2=−b(s2 − nm2)

As2t2

)pk

i,t2+j ,

(26)

pk+1ni+s1,nj+s2

=b+1∑

t1=−b

b+1∑

t2=−b

(∏b+1m1=−b(s1 − nm1)

As1t1

)

×(∏b+1

m2=−b(s2 − nm2)

As2t2

)pk

t1+i,t2+j ,

where s1, s2 = 1,2, . . . , n − 1, b = 1,2,3, . . . . and n standsfor n-ary interpolating subdivision surface.

Figure 4 is generated by (26) for n = 5, its error boundand depth are shown in Appendix D in Tables 5 and 6, re-

Fig. 4 Here, (a), (c) present control polygons and (b), (d) present 1stsubdivision level of control polygon using quinary (i.e., n = 5) inter-polating subdivision surface

spectively. By using Lemma 5 and Theorem 2 with χ = 0.1,we get the error bound and depth for other arity interpolatingsubdivision surfaces, also shown in these tables.

5.4 Error bound and depth of n-ary approximatingsubdivision surfaces

By taking the tensor product of (25), we get following (2b+2)-point n-ary approximating subdivision surface

pk+1ni+s1,nj+s2

=b+1∑

t1=−b

b+1∑

t2=−b

(∏b+1m1=−b(2s1 + 1 − 2nm1)

Bs1t1

)

×(∏b+1

m2=−b(2s2 + 1 − 2nm2)

Bs2t2

)pk

t1+i,t2+j , (27)

where s1, s2 = 0,1,2, . . . , n−1, b = 1,2,3, . . . and n standsfor n-ary approximating subdivision surface.

The error bounds and subdivision depth for approximat-ing surfaces (27) are shown in Appendix D in Tables 7 and 8,respectively. Its error is also explained by Fig. 5(b).

6 Conclusion and future work

We have computed subdivision depth based on error boundsfor more general n-ary subdivision schemes. Furthermore,we have shown that error bounds for binary, ternary, andquaternary subdivision schemes [3, 9, 10] are special casesof our bounds. The authors are looking, as a future work,to extend the computational techniques of subdivision depthfor n-ary subdivision schemes over volumetric models.

Fig. 5 (a) Presents comparisonamong the error bounds ofdifferent arity subdivisioncurves (24), i.e., forn = 2,3,4,5,6; (b) presentscomparison among the errorbounds of different aritysubdivision surfaces (27), i.e.,for n = 2,3,4,5,6. Here ,kpresents subdivision level, n

presents the arity and ε presentsthe user-specified error tolerance


Appendix A: Proof of Lemma 1

Proof From (1) and (2) for α = 0,1, . . . , n − 1, we obtain

pk+1ni+α − 1

n

((n − α)pk

i + αpki+1

)

=m−1∑

j=0

aα,j

(pk

i+j+α+1 − pki+j+α

), (28)

where aα,j is defined by (12).By (1), (2), and induction on m for β = 0,1, . . . , n − 1,

we get

pkni+β+1 − pk

ni+β =m∑

j=0

bβ,j

(pk−1

i+j+1 − pk−1i+j

), (29)

where bβ,j is defined by (7). It follows from (12), (13),and (28) that

Nkα ≤ max

α

∣∣∣∣∣

m−1∑

j=0

aα,j

∣∣∣∣∣maxi

∥∥pki+1 − pk

i

∥∥. (30)

Using (29) recursively gives

maxi

∥∥pki+1 − pk

i

∥∥

≤(

maxβ

∣∣∣∣∣

m∑

j=0

bβ,j

∣∣∣∣∣

)k

maxi

∥∥p0i+1 − p0

i

∥∥. (31)

By (30) and (31) we get (17). This completes the proof. �

Appendix B: Proof of Lemma 3

Proof From (2), (3), and using similar approach as we didfor (29), we obtain

pkni+α+1,nj+β − pk

ni+α,nj+β

=m∑

s=0

aβ,s

(m∑

r=0

bα,r

(pk−1

i+r+1,j+s − pk−1i+r,j+s

))

, (32)

pkni+α+1,nj+n − pk

ni+α,nj+n

=m∑

s=0

a0,s

(m∑

r=0

bα,r

(pk−1

i+r+1,j+s+1 − pk−1i+r,j+s+1

))

, (33)

pkni+α,nj+β+1 − pk

ni+α,nj+β

=m∑

r=0

aα,r

(m∑

s=0

bβ,s

(pk−1

i+r,j+s+1 − pk−1i+r,j+s

))

, (34)

pkni+n,nj+β+1 − pk

ni+n,nj+β

=m∑

r=0

a0,r

(m∑

s=0

bβ,s

(pk−1

i+r+1,j+s+1 − pk−1i+r+1,j+s

))

, (35)

where bβ,r is defined by (7) and α,β = 0,1, . . . , n − 1.Now using (32) recursively together with notations de-

fined by (15), we get

maxi,j

∥∥Δki,j,1

∥∥ ≤(

maxα,β

∣∣∣∣∣

m∑

s=0

aα,s

m∑

r=0

bβ,r

∣∣∣∣∣

)k

maxi,j

∥∥Δ0i,j,1

∥∥.

From (6) and the above inequality, we get

maxi,j

∥∥Δki,j,1

∥∥ ≤ (δ2)k max

i,j

∥∥Δ0i,j,1

∥∥.

Again using (33) recursively and by utilizing (6) and (15),we have

maxi,j

∥∥Δki,j,3

∥∥ ≤ (δ2)k max

i,j

∥∥Δ0i,j,3

∥∥.

Similarly, using (34) and (35) recursively together with (6)and (15),

maxi,j

∥∥Δki,j,2

∥∥ ≤ (δ2)k max

i,j

∥∥Δ0i,j,2

∥∥.


Appendix C: Proof of Lemma 4

Proof From (2) and (3), we get

pk+1ni,nj − pk

i,j =m∑

r=0

a0,r

(m∑

s=0

a0,s

(pk

i+r,j+s − pki,j

))

. (36)

Since

m∑

s=0

a0,s

(pk

i+r,j+s − pki,j

)

= a0,0(pk

i+r,j − pki,j

) + a0,1(pk

i+r,j+1 − pki,j

)

+ a0,2(pk

i+r,j+2 − pki+r,j+1 + pk

i+r,j+1 − pki,j

)

+ a0,3(pk

i+r,j+3 − pki+r,j+2 + pk

i+r,j+2 − pki+r,j+1

+ pki+r,j+1 − pk

i,j

) + · · ·+ a0,m

(pk

i+r,j+m − pki+r,j+m−1

+ pki+r,j+m−1 − · · · − pk

i+r,j+1 + pki+r,j+1 − pk

i,j

),


therefore,

m∑

s=0

a0,s(pki+r,j+s − pk

i,j )

= a0,0(pk

i+r,j − pki,j

)

+m∑

t=1

a0,t

(pk

i+r,j+1 − pki,j

)

+m−1∑

s=1

a0,s

(pk

i+r,j+s+1 − pki+r,j+s

),

where a0,s is defined by (12). Taking summation on bothside of above equation we get

m∑

r=0

a0,r

(m∑

s=0

a0,s

(pk

i+r,j+s − pki,j

))

= a0,0

m∑

r=0

a0,r

(pk

i+r,j − pki,j

)

+m∑

t=1

a0,t

(m∑

r=0

a0,r

(pk

i+r,j+1 − pki,j

))

+m∑

r=0

a0,r

(m−1∑

s=1

a0,s

(pk


))

.

Since

m∑

r=0

a0,r

(pk

i+r,j − pki,j

)

= a0,1(pk

i+1,j − pki,j

) + a0,2(pk

i+2,j

− pki+1,j + pk

i+1,j − pki,j

)

+ a0,3(pk

i+3,j − pki+2,j + pk

i+2,j

− pki+1,j + pk

i+1,j − pki,j

) + · · ·+ a0,m

(pk

i+m,j − pki+m−1,j + pk

i+m−1,j − · · ·+ pk

i+2,j − pki+1,j + pk

i+1,j − pki,j

),

therefore,

m∑

r=0

a0,r

(pk

i+r,j − pki,j

)

=m∑

t=1

a0,t

(pk

i+1,j − pki,j

)

+m−1∑

s=1

a0,s

(pk

i+s+1,j − pki+s,j

).

Similarly,m∑

r=0

a0,r

(pk

i+r,j+1 − pki,j

)

= a0,0(pk

i,j+1 − pki,j

)

+m∑

t=1

a0,t

(pk

i+1,j+1 − pki,j

)

+m−1∑

s=1

a0,s

(pk

i+s+1,j+1 − pki+s,j+1

).

Substituting these summations into (36) then by (14), weobtain

Mk0,0 =

(a0,0

m∑

t=1

a0,t

)(pk

i+1,j − pki,j

)

+(

m∑

t=1

a0,t

)2(pk

i+1,j+1 − pki,j+1

)

+m∑

t=1

a0,t

(pk

i,j+1 − pki,j

)

+ a0,0

m−1∑

s=1

a0,s

(pk

i+s+1,j − pki+s,j

)

+m∑

t=1

a0,t

m−1∑

s=1

a0,s

(pk

i+s+1,j+1 − pki+s,j+1

)

+m∑

r=0

a0,r

(m−1∑

s=1

a0,s

(pk


))

.

(37)

Similarly from (2), (3), and (14) for α,β = 0,1, . . . , n − 1(where α and β are not zero at the same time), we obtain

Mkα,β =

(aβ,0

m∑

t=1


n2

)(pk

i+1,j − pki,j

)

+(

m∑

t=1

aα,t

m∑

t=1

aβ,t − αβ

n2

)(pk

i+1,j+1 − pki,j+1

)

+(

m∑

t=1

aβ,t − β

n

)(pk

i,j+1 − pki,j

)

+ aβ,0

m−1∑

s=1

aα,s

(pk

i+s+1,j − pki+s,j

)

+m∑

t=1

aβ,t

m−1∑

s=1

aα,s

(pk

i+s+1,j+1 − pki+s,j+1

)

+m∑

r=0

aα,r

(m−1∑

s=1

aβ,s

(pk


))

,

(38)

where aα,s is defined by (12).


Using (6), (20), (37), and (38) for α,β = 0,1, . . . , n − 1,

we get

Mkα,β

≤ (δ2)k

{(∣∣∣∣∣aβ,0

m∑

t=1


n2

∣∣∣∣∣

+∣∣∣∣∣aβ,0

m−1∑

s=1

aα,s

∣∣∣∣∣

)χ1

+(∣∣∣∣∣

m∑

t=1

aβ,t − β

n

∣∣∣∣∣ +∣∣∣∣∣

m∑

r=0

aα,r

m−1∑

s=1

aβ,s

∣∣∣∣∣

)χ2

+(∣∣∣∣∣

m∑

t=1

aα,t

m∑

t=1

aβ,t − αβ

n2

∣∣∣∣∣ +∣∣∣∣∣

m∑

t=1

aβ,t

m−1∑

s=1

aα,s

∣∣∣∣∣

)χ3

}.

Utilizing notations (9)–(11), (14), and (15), we get (21).This completes the proof. �

Appendix D: Tables

Table 1 Error bounds of n-aryinterpolating subdivision curves:Here, n presents the arity ofsubdivision curve and k presentsthe subdivision level

n\k 1 2 3 4 5 6 7

2 0.100000 0.050000 0.025000 0.012500 0.006250 0.003125 0.001562

3 0.050000 0.016667 0.005556 0.001852 0.000617 0.000206 0.000069

4 0.033333 0.008333 0.002083 0.000521 0.000130 0.000033 0.000008

5 0.025000 0.005000 0.001000 0.000200 0.000040 0.000008 0.000002

6 0.020000 0.003333 0.000556 0.000093 0.000015 0.000003 0.000000

Table 2 Subdivision depth ofn-ary interpolating subdivisioncurves: Here, n presents thearity of subdivision curve and ε

presents error tolerance

n\ε 0.100000 0.050000 0.025000 0.012500 0.006250 0.003125 0.001562

2 1 2 3 4 5 6 7

3 1 1 2 3 3 4 5

4 1 1 2 2 3 3 4

5 1 1 1 2 2 3 3

6 1 1 1 2 2 3 3

Table 3 Error bounds of n-aryapproximating subdivisioncurves: Here n presents the arityof curve and k presents thesubdivision level

n\k 1 2 3 4 5 6 7

2 0.125000 0.062500 0.031250 0.015625 0.007812 0.003906 0.001953

3 0.058333 0.019444 0.006481 0.002160 0.000720 0.000240 0.000080

4 0.037500 0.009375 0.002344 0.000586 0.000146 0.000037 0.000009

5 0.027500 0.005500 0.001100 0.000220 0.000044 0.000009 0.000002

6 0.021667 0.003611 0.000602 0.000100 0.000017 0.000003 0.000000

Table 4 Subdivision depth ofn-ary approximatingsubdivision curves: Here n

presents the arity of curve and ε

presents error tolerance

n\ε 0.100000 0.050000 0.025000 0.012500 0.006250 0.003125 0.001562

2 2 3 4 5 6 7 8

3 1 2 2 3 4 4 5

4 1 1 2 2 3 3 4

5 1 1 2 2 2 3 3

6 1 1 1 2 2 3 3


Table 5 Error bounds of n-aryinterpolating subdivisionsurfaces: Here n presents thearity of surface and k presentsthe subdivision level

n\k 1 2 3 4 5 6 7

2 0.300000 0.150000 0.075000 0.037500 0.018750 0.009375 0.004688

3 0.166667 0.055556 0.018519 0.006173 0.002058 0.000686 0.000229

4 0.116667 0.029167 0.007292 0.001823 0.000456 0.000114 0.000028

5 0.090000 0.018000 0.003600 0.000720 0.000144 0.000029 0.000006

6 0.073333 0.012222 0.002037 0.000340 0.000057 0.000009 0.000002

Table 6 Subdivision depth ofn-ary interpolating subdivisionsurfaces: Here n presents thearity of surface and ε presentserror tolerance

n\ε 0.100000 0.050000 0.025000 0.012500 0.006250 0.003125 0.001562

2 3 4 5 6 7 8 9

3 2 3 3 4 4 5 6

4 2 2 3 3 4 4 5

5 1 2 2 3 3 4 4

6 1 2 2 2 3 3 4

Table 7 Error bounds of n-aryapproximating subdivisionsurfaces: Here n presents thearity of surface and k presentsthe subdivision level

n\k 1 2 3 4 5 6 7

2 0.369141 0.184570 0.092285 0.046143 0.023071 0.011536 0.005768

3 0.191114 0.063705 0.021235 0.007078 0.002359 0.000786 0.000262

4 0.129272 0.032318 0.008080 0.002020 0.000505 0.000126 0.000032

5 0.097708 0.019542 0.003908 0.000782 0.000156 0.000031 0.000006

6 0.078537 0.013090 0.002182 0.000364 0.000061 0.000010 0.000002

Table 8 Subdivision depth ofn-ary approximatingsubdivision surfaces: Here n

presents the arity of surface andε presents error tolerance

n\ε 0.100000 0.050000 0.025000 0.012500 0.006250 0.003125 0.001562

2 3 4 5 6 7 8 9

3 2 3 3 4 5 5 6

4 2 2 3 3 4 4 5

5 1 2 2 3 3 4 4

6 1 2 2 3 3 3 4

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Ghulam Mustafa received a Ph.D.(2004) in Mathematics from theUniversity of Science and Technol-ogy of China, Peoples Republic ofChina. He is an Assistant Profes-sor in the Department of Mathemat-ics, The Islamia University of Ba-hawalpur, Pakistan. His research in-terests include computer aided geo-metric design and applied approxi-mation theory.

Muhammad Sadiq Hashmi re-ceived a Masters Degree (2003) inMathematics from the Islamia Uni-versity of Bahawalpur, Pakistan.Currently, he is doing his Ph.D. inMathematics from The Islamia Uni-versity of Bahawalpur. He is alsoworking as lecturer at the Govern-ment Degree College Yazman. Hisresearch interests include computeraided geometric design and appliedapproximation theory.

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Subdivision depth computation for n-ary subdivision curves/surfaces