Chiew-LanTai ,Shi-MinHu *,Qi-XingHuang ApproximatemergingofB-splinecurvesviaknotadjustmentandconstrainedoptimization

Approximate merging of B-spline curves via knot adjustment and constrained optimization

Chiew-Lan Tai

, Shi-Min Hu

*, Qi-Xing Huang

aDepartment of Computer Science, The Hong Kong University of Science and Technology, Hong Kong, People’s Republic of China

bDepartment of Computer Science and Technology, Tsinghua University, Beijing 100084, People’s Republic of China Received 5 March 2002; received in revised form 18 September 2002; accepted 30 September 2002

Abstract

This paper addresses the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. The basic idea of the approach is to find the conditions for precise merging of two B-spline curves, and perturb the control points of the curves by constrained optimization subject to satisfying these conditions. To obtain a merged curve without superfluous knots, we present a new knot adjustment algorithm for adjusting the endkknots of akth order B-spline curve without changing its shape. The more general problem of merging curves to pass through some target points is also discussed.

q2003 Elsevier Science Ltd. All rights reserved.

Keywords:B-spline curves; Merging; Knot adjustment; Constrained optimization

1. Introduction

Approximate conversion is an important issue in data communication between different CAD systems [1]. As mentioned by Hoschek [2], approximate conversion includes the following two problems:

† Degree reduction: finding a parametric curve of degreen that approximates the given curve of degreemðn,mÞ:

† Merging: merging as many curve segments of degreenas possible to get one curve segment of degreemðn#mÞ: Degree reduction methods for Be´zier, Ball, and B-spline curves and surfaces have been extensively investigated [3 – 13]. Merging is one of the main methods for data reduction; by merging as many curve segments as possible into one curve, the amount of geometric data needed for communication can be reduced. InRef. [14], we presented a method for approximate merging of a pair of Be´zier curves using constrained optimization method. The objective of this paper is to consider approximate merging of B-spline curves.

The approximate merging problem we address is as follows. Given two adjacent orderkB-spline curvesP(t) and

R(s) with knot vectorsT¼{t₀;t₁;…;t_k;…;t_n;…;t_nþk} and S¼{s₀;s₁;…;s_k;…;s_m;…;s_mþk} respectively, and control points P_i ði¼0;1;…;nÞ and R_j ðj¼0;1;…;mÞ respect- ively, find an orderkB-spline curveFðuÞwith control points F_i ði¼0;1;…;nþm2kþ2Þ and knot vector U¼ {t₀;t₁;…;t_k;…;t_n;t_nþ1 ¼s⁰_k21;s⁰_k;…;s⁰_mþk}; where s⁰_i¼ fðs_iÞfor some linear functionf, such that a suitable distance functiondðF;FÞ betweenFðuÞand

FðuÞ ¼ PðuÞ; t_k21#u#t_nþ1 Rðf²¹ðuÞÞ; s⁰_k21 #u#s⁰_mþ1 (

;

is minimized.

The basic idea of our method is to first find the conditions for precise merging of two B-spline curves, and perturb the control points of the curves by constrained optimization subject to satisfying the precise merging conditions. We then reparametrize the resulting curves using a new knot adjustmentalgorithm we present. Such knot adjustment is needed for producing a merged B-spline curve efficiently without superfluous knots.

The remainder of the paper is organized as follows.

Section 2 presents the definition of the B-spline curves, the de Boor algorithm, and our knot adjustment algorithm.

Section 3 describes the method for approximate merging of a pair of B-spline curves; the more general problem of

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* Corresponding author. Tel.: þ86-10-627-82052; fax: þ86-10-627- 82025.

E-mail address:shimin@tsinghua.edu.cn (S.-M. Hu).

approximate merging with point constraints is also dis- cussed. Conclusions are given in Section 4.

2. B-spline curves, evaluation and knot adjustment A B-spline curve of order kwith control points P_i ði¼ 0;1;…;nÞcan be defined as

PðtÞ ¼Xⁿ

i¼0

P_iN_i;kðtÞ; t_k21#t#t_nþ1; ð1Þ where N_i;kðtÞare the B-spline basis functions of order k defined over the knot vectorT¼{t₀;t₁;…;t_k;…t_n;…;t_nþk}; which is defined by the following recursive de Boor-Cox formula[15]

N_i;1ðtÞ ¼ 1; t_i#t,t_iþ1 0; Otherwise (

;

N_i;kðtÞ ¼ t2t_i

t_iþk212t_iN_i;k21ðtÞ þ t_iþk2t

t_iþk2t_iþ1N_iþ1;k21ðtÞ: ð2Þ The point on the curvePðtÞat parametertðt[½t_j;t_jþ1Þ) can be evaluated using the following de Boor algorithm

andPðtÞ ¼P^k21_j2kþ1ðtÞ:

To compute the derivatives of a B-spline curve, by letting P⁰_i ¼P_i;and writing

P^ð0ÞðtÞ ¼PðtÞ ¼Xⁿ

i¼0

P⁰_iN_i;kðtÞ; we get (see p. 97 inRef. [15]) P^ðlÞðtÞ ¼X^n2l

i¼0

P^l_iN_iþ1;k2lðtÞ; ð4Þ where

P^l_i¼

P_i; l¼0

k2l

t_iþk2t_iþlðP^l21_iþ12P^l21_i Þ; l.0 8>

<

>: : ð5Þ

Suppose two kth order B-spline curvesPðtÞ andRðsÞwith knot vectors T¼{t₀;t₁;…;t_k;…;t_n;…;t_nþk} and S¼ {s₀;s₁;…;s_k;…;s_m;…;s_mþk} are to be merged. In our merging algorithm, after perturbing the control points to satisfy precise merging conditions, we adjust the knot vectors of the curves. Specifically, we adjust the last k knots of the curve PðtÞso that they match k knots of the curveRðsÞ;that is, the knot vector ofPðtÞmust be adjusted from T to T⁰¼{t₀;t₁;…;t_k;…;t_n;t_nþ1;s_k;s_kþ1;…;s_2k22}: Although this knot adjustment can be achieved by first clamping T, and then unclamping it to T⁰ [15], for efficiency, we propose a new general algorithm to directly

adjust the knot vector T to T₂¼{t₀;t₁;…;t_k;…;t_n;t_nþ1; t⁰_nþ2;…;t⁰_nþk};wheret_nþ1 #t⁰_nþ2#· · ·#t⁰_nþk21 #t⁰_nþk:

Suppose the new curve defined on the new knot vectorT2

is denoted byQðtÞwith control pointsQ₀;Q₁;…;Q_n:Since the shape of the curve remains unchanged, we havePðtÞ ¼ QðtÞ: We claim that Q_i¼P_i; 0#i#n2kþ2; and the otherQ_i;n2kþ3#i#ncan be computed recursively by Algorithm 1. The correctness proof is given in Appendix 1.

Algorithm 1. Computing the new control points after knot adjustment fromTtoT₂

1. ComputeP^l_n2kþ2;l¼1;2;…;k21 by Eq. (5).

2. LetQ^l_n2kþ2¼P^l_n2kþ2;forl¼0;1;…;k21:

3. Compute Q⁰_i for n2kþ3#i#n by (see top row of Fig. 1)

Q^l_i¼ t⁰_iþk212t_iþl

k2l21 Q^lþ1_i21þQ^l_i21;

i¼n2kþ3; n2kþ4;…;n; l¼0;1;…;n2i; ð6Þ

which can be derived from Eq. (5)

4. LetQ_i¼P_ifori#n2kþ2;andQ_i¼Q⁰_i forn2kþ 3#i#n:

Fig. 2 shows the results of adjusting the knot vectors of two cubic B-spline curves. The original control polygons are shown in solid line, while the new control polygons are shown in dotted line. In Fig. 2(a), the knot vector is adjusted from {0;0;0;0;0:5;1;1;1;1} to {0;0;0;0;0:5;1;1:3;1:3;1:3}: In Fig. 2(b), is adjusted to {20:6;20:4;20:2;0;0:5;1;1:5;1:6;1:7}:

3. Merging by constrained optimization

Suppose PðtÞ and RðsÞ are two B-spline curves to be approximately merged. Let RðsÞ be a B-spline curve with control points R_i ði¼0;1;…;mÞ and knot vector

Fig. 1. Recursive computation ofQi¼Q⁰i;i¼n2kþ2;…;n:

P^r_iðtÞ ¼

P_i; r¼0;i¼0;1;…;n t_iþk2t

t_iþk2t_iþrP^r21_i þ t2t_iþr

t_iþk2t_iþrP^r21_iþ1; r¼1;2;…;k21;i¼j2kþ1;…;j2r 8>

<

>: ; ð3Þ

S¼{s₀;s₁;…;s_k;…;s_m;…;s_mþk}:Without loss of general- ity, we suppose that the order ofRðsÞis alsok, andt_nþ1¼ s_k21: To obtain a good parametrization for the merged curve, we perform a linear transformation f on the knot vectorSsuch that

arc length ofPðtÞover½t_n;t_nþ1

arc length ofRðsÞover½s_k2s_k21 ¼ t_nþ12t_n s_k2s_k21:

3.1. Conditions for precise merging

We first derive the conditions for the curvesPðtÞandRðsÞ to be precisely merged into one B-spline curve. We let the curves share common derivatives, that is, to precisely merge atPðt_nþ1Þ ¼Rðs_k21Þ;we need

P^ðlÞðt_nþ1Þ ¼R^ðlÞðs_k21Þ; l¼0;1;…;k22; i.e.

X

n2l

i¼n2kþ1

P^l_iN_iþ1^T _;k2lðt_nþ1Þ ¼^k212lX

i¼0

R^l_iN_iþ1^S _;k2lðs_k21Þ;

l¼0;1;…;k22;

ð7Þ

whereN_i^T_;kðtÞN_i^S_;kðsÞare B-spline basis functions defined on the knot vectorsTandS, respectively. According to Eq. (6), P^l_iandR^l_iin Eq. (7) can be rewritten asPn

j¼n2kþ1a^ðlÞ_ijP_jand Pk21

j¼0 b^ðlÞ_ijRj; respectively, where a^ðlÞ_ij and b^ðlÞ_ij can be computed by Algorithms 2 and 3, respectively.

Algorithm 2. Computing a^ðlÞ_ij; n2kþ1#i#n2l; n2kþ1#j#n;0#l#k22:

1. Forn2kþ1#j#n;letp₀;p₁;…;p_n be scalars, and

p_i¼d_ij¼ 0; i–j 1; i¼j (

:

2. Let

p^l_i¼

p_i; forl¼0

k2l

t_iþk2t_iþlðp^l21_iþ12p^l21_i Þ forl.0 8>

<

>: :

3. Leta^ðlÞ_ij ¼p^l_i;n2kþ1#i#n2l;0#l#k22: Algorithm 3. Computing b^ðlÞ_ij; 0#i#k212l; 0# j#k21;0#l#k22:

1. For 0#j#k21;letp₀;p₁;…;p_nbe scalars and p_i¼d_ij¼ 0; i–j

1; i¼j (

:

2. Let

p^l_i¼

p_i; forl¼0

k2l

s_iþk2s_iþlðp^l21_iþ12p^l21_i Þ; forl.0 8>

<

>: :

3. Letb^ðlÞ_ij ¼p^l_i;0#i#k212l;0#l#k22: Then the precise merging conditions in Eq. (7) can be rewritten as

Xⁿ

j¼n2kþ1

X

n2l

i¼n2kþ1

a^ðlÞ_ijN_iþ1^T _;k2lðt_nþ1Þ

! P_j

2^k21X

j¼0

X

k212l

i¼0

b^ðlÞ_ijN_iþ1^S _;k2lðS_k21Þ

! R_j¼0;

l¼0;1;…;k22:

ð8Þ

3.2. Merging by constrained optimization

To merge two arbitrary curves PðtÞ and RðsÞ; we first perturb their control points so that the curves can be precisely merged. We achieve this by minimizing the total perturbation of the control points subject to the precise merging constraints Eq. (8). Let the perturbation of the control points of PðtÞ and RðsÞ be denoted by ei¼ {e^x_i;e^y_i;e^z_i} i¼n2kþ2;n2kþ3;…;n and di¼{d^x_i;d^y_i;d^z_i};

Fig. 2. Two examples of knot adjustment of B-spline curves.

i¼0;1;…;k22 respectively.¹ That is, the modified curves are

PðtÞ ¼^ Xⁿ

i¼0

P^_iN_i^T_;kðtÞ ¼Xⁿ

i¼0

ðP_iþeiÞN_i^T_;kðtÞ; t_k21#t#t_nþ1;

and RðsÞ ¼^ X^m

i¼0

R^_iN_i^S_;kðsÞ ¼X^m

i¼0

ðR_iþd_iÞN_i^S_;kðsÞ; s_k21#s#s_mþ1;

and the precise-merging conditions from Eq. (8) become Xⁿ

j¼n2kþ1

X

n2l

i¼n2kþ1

a^ðlÞ_ijN_iþ1^T _;k2lðt_nþ1Þ

!

ðP_jþe_jÞ

2^k21X

j¼0

X

k212l

i¼0

b^ðlÞ_ijN_iþ1^S _;k2lðs_k21Þ

!

ðR_jþd_jÞ ¼0;

l¼0;1;…;k22:

ð9Þ

More generally, we consider the approximate merging problem in which the merged curve is constrained to pass through some target points on the original curves. This can be achieved by adding point constraint conditions as follows

Xⁿ

i¼n2kþ1

P_iN_i^T_;kðt_jÞ ¼ Xⁿ

i¼n2kþ1

ðP_iþeiÞN_i^T_;kðt_jÞ; j¼0;1;…;g:

X

k21

i¼0

R_iN_i^S_;kðs_jÞ ¼^k21X

i¼0

ðR_iþdiÞN_i^S_;kðs_jÞ; j¼0;1;…;h:

where t_j; j¼0;1;…;g and s_j; j¼0;1;…;h are parameters of the target points onPðtÞandRðsÞ;respectively.²So we have

Xⁿ

i¼n2kþ1

eiN_i^T_;kðt_jÞ ¼0; j¼0;1;…;g: ð10Þ

X

k21

i¼0

diN^S_i;kðs_jÞ ¼0; j¼0;1;…;h: ð11Þ

We determine e_i d_i by setting the optimization objective Oðei;djÞ as

Min Oðe_i;d_jÞ ¼ Xⁿ

i¼n2kþ2

e_iþ^k22X

j¼0

d_j; ð12Þ

and define the Lagrange function as L¼ Xⁿ

i¼n2kþ2

e_iþX^k22

j¼0

d_jþ^k22X

l¼0

l_l ^X

n j¼n2kþ1

X

n2l i¼n2kþ1

a^ðlÞ_ijN^T_iþ1;k2lðt_nþ1Þ 0 !

@

ðP_jþe_jÞ2^k21X

j¼0

X

k212l i¼0

b^ðlÞ_ijN_iþ1^S _;k2lðs_k21Þ

! ðR_jþd_jÞ

1 A

þX^g

j¼0

lk21þj

Xⁿ

i¼n2kþ1

e_iN_i;k^Tðt_jÞ

! þX^h

j¼0

lkþgþj

X

k21 i¼0

d_iN_i;k^Sðs_jÞ

!

; ð13Þ where l_i¼bl^x_i;l^y_i;l^z_ic are the Lagrange multipliers. By setting

›L=›e^x_i; ›L=›e^y_i;›L=›e^z_i; ›L=›d^x_j;›L=›d^y_j; ›L=›d^z_j; ›L=›l^x_i; ›L=›l^y_i;

›L=›l^z_ito zero, and writing the derived equations in vector form, we obtain a system of linear equations, which includes Eqs. (9) – (11), and the following two equations

2e_i¼2X^k22

l¼0

X

n2l j¼n2kþ1

l_la^ðlÞ_ijN_iþ1;k2l^T ðt_nþ1ÞþX^g

j¼0

lk21þjN_i;k^Tðt_jÞ;

n2kþ2#i#n;

ð14Þ

2d_j¼^k22X

l¼0

X

k212l i¼0

l_lb^ðlÞ_ijN^S_iþ1;k2lðs_k21ÞþX^h

j¼0

lkþgþjN_i;k^Sðs_jÞ;

0#j#k22:

ð15Þ

By solving the linear system, the constrained optimization solution can be obtained.

After obtaining two modified curves that can be precisely merged, we let

T_new¼{t₀;t₁;…;t_k;…;t_n;t_nþ1¼s_k21;s_k;s_kþ1;…;s_m;…;s_mþk} be the knot vector of the merged curve to be constructed.

Using Algorithm 1, we adjust the knot vectorTof curvePðtÞ^ toT⁰¼{t₀;t₁;…;t_k;…;t_n;t_nþ1;s_k;s_kþ1;…;s_2k22} and the knot vector S of curve RðsÞ^ to S⁰¼{t_n2kþ2;…;t_n;t_nþ1¼ s_k21;s_k;s_kþ1;…;s_mþk};respectively. For simplicity, we still denote the control points after knot adjustment as P^_i; i¼ 0;1;…;n; ^R_i; i¼0;1;…;m: Since PðtÞ^ andRðsÞ^ satisfy the precise merging conditions, i.e. they have common derivatives at parameter t_nþ1¼sk21; we conclude that the control points of the merged curve are

{P^₀;P^₁;…;P^_n2kþ1;P^_n2kþ2¼R^₀;P^_n2kþ3¼R^₁;…;P^_n

¼R^k22;R^k21;…;R^_m}:

3.3. Error estimation and examples

B-spline curves of order 4 are among the most commonly used parametric curves in shape design. Here, we give an error analysis for merging two B-spline curves of order 4;

error analysis for order 2 and 3 curves can be similarly derived. For higher-order curves, it seems hard to obtain explicit estimation.

1 From precise merging conditions, only k21 control points are involved in each curve, see Eq. (7). Thus, for convenience in subsequent derivation, we just set ei¼0 ði¼0;1;…;n2kþ1Þ and di¼0 ði¼ k21;k22;…;mÞ:

2 Note that the total number of point constraints for two curves should be less thank21;otherwise the solution of the equation system does not exist.

Theorem 1.SupposePðtÞandRðsÞare two fourth-order B- spline curves, and t_nþ1¼· · ·¼t_nþk; s₀¼· · ·¼s_k21; we have the following estimation

kP2Fk#3 4

u₁D1

u₁þD₁^kd1kþ 1 3

ðu₁þD1Þ²þu₁D1þu²₁ ðu₁þD1Þ²

ðu₁D₁^kd₂kþlu₁2D₁^lkd₁kÞ; ð16Þ

kR2Fk#3 4

u₁D1

u₁þD₁^kd1kþ 1 3

ðu₁þD1Þ²þu₁D1þu²₂ ðu₁þD1Þ²

ðu₁D₁^kd₂kþlu₁2D₁^lkd₁kÞ; ð17Þ where u₁¼t_nþ12t_n; D1¼s_k2s_k21; d₁¼P^ð1Þðt_nþ1Þ2R^ð1Þ ðt_k21Þ; d₂¼P^ð2Þðt_nþ1Þ2R^ð2Þðs_k21Þ; kP2Fk¼max_t[½t

k21;tnþ1

kPðtÞ2FðtÞk; and kR2Fk¼maxs[½s_k21;snþ1kRðsÞ2Fðt_nþ1þsÞk: A proof of Theorem 1 is given in Appendix B.

We give some examples to illustrate our method.Figs. 3 and 4 show some examples of merging clamped and unclamped B-spline curves, respectively. The original curves are rendered in solid line, and the new merged curves are rendered in dotted line. In Fig. 3(a), the knot vectors of the original curves are both [0,0,0,0,0.5,1,1,1,1], and that of the merged curve is [0,0,0,0,0.5,1,1.48588,1.972, 1.972,1.972,1.972], and in Fig. 3(b), the knot vectors of the original curves are both [0,0,0,0,0.33,0.67,1,1,1,1], and

that of the merged curve is [0,0,0,0,0.33,0.67, 1,1.338,1.675, 2.012,2.012,2.012,2.012]. In Fig. 4(a), the original knot vectors are [0,0,0,0,0.5,1,1.2,1.4,1.6] and [0,0, 0,0,0.5,1,1,1,1], and the resulting one is [0,0,0,0,0.5,1,1.523, 2.045,2.045,2.045,2.045]. In Fig. 4(b), the original knot vectors are [0,0,0,0,0.33,0.67,1,1.2,1.4,1.6] and [20.3,20.2,20.1,0,0.33,0.67,1,1,1,1], and the resulting one for the merged curve is [0,0,0,0,0.33,0.67,1,1.342, 1.684,2.025, 2.025,2.025,2.025].

To increase the accuracy of merging, we can add geometric constraints, i.e. constraining the resulting curve to pass through some target points on the original curves.

Fig. 5shows the effect of merging with point constraints, the knot vectors of the original two curves are both [0,0,0,0,0.33,0.67,1,1,1,1], the constrained points are Pð0:75ÞandQð0:25Þ;which are marked as squares in figure.

4. Conclusion and future works

This paper presents an algorithm for approximate merging of two B-spline curves. We derive the precise merging conditions by letting the curves share common derivatives. The curves are modified by constrained optimization to satisfy the precise merging conditions. By applying the knot adjustment algorithm, the control points of the merged curve can be directly obtained from those of the curves satisfying precise merging conditions. The resulting merged curve has no superfluous knots.

We have used ‘Discrete coefficient norm’ inL₂sense in this paper, as in Ref. [14], but merging with the ‘squared difference integral norm’ is also possible. As a future work, merging of multiple B-spline curves and of surfaces can be further investigated.

Acknowledgements

This research was partially supported by the Natural Science Foundation of China (Project Number 60225016), NKBRSF (Project Number 1998030600) and the Hong Kong Research Grant Council (HKUST6215/99E).

Appendix A. Correctness proof of the knot adjustment algorithm

By the recursive definition ofP^l_i andQ^l_i in Eqs. (5) and (6), respectively, it is sufficient to prove that (I)Q_i¼P_ifor

Fig. 3. Approximate merging of two clamped B-spline curves.

Fig. 4. Approximate merging of two unclamped B-spline curves.

Fig. 5. Approximate merging with point constraints.

i#n2kþ1;(II)Q^l_n2kþ2¼P^l_n2kþ2forl¼0;1;…;k22: We prove them for two possible cases.

(a)n$2k22;i.e. at leastkcurve segments.

From the local property of B-spline curves, for t[

½t_s;t_sþ1Þ;only kbasis functions N_i;kðtÞ;i¼s2kþ1;…;s are non-zero, i.e.

PðtÞ ¼ X^s

i¼s2kþ1

P_iN_i;kðtÞ; t[½t_s;t_sþ1Þ: ðA1Þ LetN^_i;kðtÞdenote the B-spline basis functions defined on the knot vectorT₂:From the construction of the knot vectorsT andT₂;we haveN_i;kðtÞ ¼N^_i;kðtÞ;s2kþ1#i#s;for all s#n2k:

Consider the first B-spline curve segment in½t_k21;t_k:We have N_i;kðtÞ ¼N^_i;kðtÞ; 0#i#k21 (note n$2k22);

hence we getQ_i¼P_i;0#i#k21:By applying the same argument to the intervals t[½t_s;t_sþ1 for k22#s# n2kþ1:we haveQi¼Pifori#n2kþ1:Thus (I) is true.

Next we show that (II) is true. We first consider the first derivative of the B-spline curves. For the segment

½t_n2kþ2;t_n2kþ3; PðtÞ ¼QðtÞ yields P^ð1ÞðtÞ ¼Q^ð1ÞðtÞ: Hence, from Eq. (4), we have

X

n2kþ1

i¼n22kþ3

P¹_iN_i;k21ðtÞ ¼ ^n2kþ1X

i¼n22kþ3

Q¹_iN^_i;k21ðtÞ:

By applying the same argument as for segmentt[½t_k21;t_k above, it is obvious thatN_i;k21ðtÞ ¼N^_i;k21ðtÞ for n22kþ 3#i#n2kþ1:We therefore conclude that Q¹_n2kþ1¼

P¹_n2kþ1: Analogously, for subsequent segments,

½t_n2kþ3;t_n2kþ4;…;½t_n;t_nþ1; we have Q^l_n2kþ1¼P^l_n2kþ1; l¼0;1;…;k21:Since

Q^l_n2kþ2¼ t⁰_nþ12t_n2kþ2þl

k2l21 Q^lþ1_n2kþ1þQ^l_n2kþ1; l¼0;1;…;k22;

P^l_n2kþ2¼ t_nþ12t_n2kþ2þl

k2l21 P^lþ1_n2kþ1þP^l_n2kþ1; l¼0;1;…;k22;

and t⁰_nþ1 ¼t_nþ1; we conclude Q^l_n2kþ2¼P^l_n2kþ2; l¼ 0;1;…;k22:

(b)n,2k22;i.e. fewer thanksegments.

From the equations Q^l_iþ1¼ t⁰_iþk2t_iþlþ1

k2l21 Q^lþ1_i þQ^l_i; l¼0;1;…;k22; P^l_iþ1¼ t_iþk2t_iþlþ1

k2l21 P^lþ1_i þP^l_i; l¼0;1;…;k22;

and ðt⁰_iþk2t_iþlþ1Þ=ðk2l21Þ ¼ ðt_iþk2t_iþlþ1Þ=ðk2l21Þ for i#n2kþ1; we know that, to prove (I) Q_i¼P_i

for i#n2kþ1 and (II) Q^l_n2kþ25P^l_n2kþ2 for

l¼0;1;…;k22;, it is sufficient to show thatQ^l₀¼P^l₀ð0# l#k21Þ:

We first prove Q₀¼P₀: For the B-spline curves PðtÞ and QðtÞ; we insert multi-knots in the spans ½t_n;t_nþ1and

½t⁰_n;t⁰_nþ1; respectively, for 2k222n times, and obtain a new curve PðtÞ with control points P_i ð0#i#2k22Þ; and another new curve QðtÞ with control points Q_i ð0# i#2k22Þ: Since knot insertion does not change the shape of the curves, we have PðtÞ ¼ QðtÞ : Investigating PðtÞ and QðtÞ from the viewpoint of knot adjustment, we conclude that P₀ ¼Q₀ by part (a) of the correctness proof. From the knot insertion formula by Boehm (see p.

141 in Ref. [15]), we haveP₀¼P₀;Q₀¼Q₀;thusP₀¼ Q₀:By similar discussion for the curvesP^ðlÞðtÞandQ^ðlÞðtÞ; l¼1;2;…;k21;we haveQ^l₀¼P^l₀ ð1#l#k21Þ:This completes the proof.

Appendix B. Proof of Theorem 1

Proof.Let the perturbation of the control points P_n22; P_n21;P_nin curvePðtÞandR₀;R₁;R₂inRðsÞbee1;e2;eand e d1;d2;respectively. Due to the precise merging condition, we have

e2e₂

u₁ 2 D₁2e D1

¼d; ðB1Þ

e2e₂

u₁ 2 e₂2e₁ u₁þu₂

u₁ 2

d₂2d₁ D1þD2

2 e2D₂ D1

D1

¼d₂: ðB2Þ Let a₁¼1; a₂¼D1=ðu₁þD1Þ; a₃¼u₁=ðu₁þD1Þ a₄¼ D1=ðu₁þu₂Þ; a₅¼1þ ðD1=ðu₁þu₂ÞÞ; a₆¼1þ ðu₁=ðD1þ D2ÞÞ; a₇¼u₁=ðD1þD2Þ; we can rewrite formulas (B1) and (B2) in matrix form as follows

a₁ 0 2a₂ 2a₃ 0 0 a₄ 2a₅ a₆ 2a₇ 0

@

1 A·

e e₁ e2

d1

d₂ 0 BB BB BB BB BB B@

1 CC CC CC CC CC CA

¼

u₁D1

u₁þD₁^d1 u₁D1d₂þðu₁2D1Þd₁ 0

BB

@

1 CC A;

ðB3Þ and

e e₁ e₂ d₁ d₂ 0 BB BB BB BB BB B@

1 CC CC CC CC CC CA

¼ 1 D·

a₁ða²₄þa²₅þa²₆þa²₇Þ a₁ða₃a₆2a₂a₅Þ

a₄ða₃a₆2a₂a₅Þ a₄ða²₁þa²₂þa²₃Þ

2a₂ða²₄þa²₅þa²₆þa²₇Þ2a₅ða₃a₆2a₂a₅Þ 2a₂ða₃a₆2a₂a₅Þ2a₅ða²₁þa²₂þa²₃Þ

2a₃ða²₄þa²₅þa²₆þa²₇Þ2a₆ða₃a₆2a₂a₅Þ 2a₃ða₃a₆2a₂a₅Þ2a₆ða²₁þa²₂þa²₃Þ

2a₇ða₃a₆2a₂a₅Þ 2a₇ða²₁þa²₂þa²₃Þ 0

BB BB BB BB BB BB

@

1 CC CC CC CC CC CC A

·

u₁D₁ u₁þD₁d₁ u₁D₁d₂þ ðu₁2D₁Þd₁ 0

BB

@

1 CC A; ðB4Þ

where

D¼ ða²₄þa²₅þa²₆þa²₇Þða²₁þa²₂þa²₃Þ2ða₃a₆2a₂a₅Þ²: By simple deduction, we have

a₁ða²₄þa²₅þa²₆þa²₇Þ

.maxðla₄ða₃a₆2a₂a₅Þl;l2a₂ða²₄þa²₅þa²₆þa²₇Þ2a₅ða₃a₆ 2a₂a₅Þl;l2a₃ða²₄þa²₅þa²₆þa²₇Þ2a₆ða₃a₆2a₂a₅Þl;l 2a₇ða₃a₆2a₂a₅ÞlÞ;

a₁ða²₄þa²₅þa²₆þa²₇Þ

D #3

4; a₄ða²₁þa²₂þa²₃Þ

D # 1

2þ ffiffi 2 p ;

a₇ða²₁þa²₂þa²₃Þ

D # 1

2þ ffiffi 2 p ;

a₂ða₃a₆2a₂a₅Þ þa₅ða²₁þa²₂þa²₃Þ

D #1þa₂a₃þa²₃

3 ;

a₃ða₃a₆2a₂a₅Þ þa₆ða²₁þa²₂þa²₃Þ

D #1þa₂a₃þa²₃

3 ;

a₁ða₃a₆2a₂a₅Þ D

#1 3

By substituting the above inequalities into Eqs. (B3) and (B4), we obtain Eqs. (16) and (17). This completes the proof. A

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Chiew-Lan Tai has been an Assistant Pro- fessor at the Computer Science Department of the Hong Kong University of Science and Technology since August 1997. She received her BSc and MSc in Mathematics from the University of Malaya, and her MSc in Computer and Information Sciences from the National University of Singapore. She earned her DSc degree in Information Science from the University of Tokyo under a fellowship awarded by the Japan Society for the Pro- motion of Science. Her research interests includes geometric modeling, computer graphics, and graphics recognition.

Shi-Min Hu is an associate professor in Department of Computer Science and Tech- nology at Tsinghua University. He received the BS in mathematics from Jilin university of China in 1990, the MS and the PhD in CAGD and Computer Graphics from Zhejiang uni- versity of China in 1993 and 1996, he finished his post-doctor research at Tsinghua Univer- sity in 1998. His current research interests are in computer-aided geometric design, computer graphics, and human-computer interaction.

Qi-Xing Huang is a graduate student in the Department of Computer Science and Technology, Tsinghua University, Beijing, People’s Republic of China, where he received a BS in 2001. His research interests are computer aided geometric design and computer graphics.