Interpolating rational bézier spline curves with local shape control

International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013

DOI : 10.5121/ijcga.2013.3401 1

INTERPOLATINGRATIONAL BÉZIER SPLINECURVES WITH LOCAL SHAPE CONTROL

Alexander P. Pobegailo1

1Department of Applied Mathematics and Computer Science, Belarusian StateUniversity, Minsk, Belarus

ABSTRACT

The paper presents a technique for construction of Cn interpolating rational Bézier spline curves by meansof blending rational quadric Bézier curves. A class of polynomials which satisfy special boundaryconditions is used for blending. Properties of the polynomials are considered. The constructed splinecurves have local shape control that make them useful in such geometric applications as real-timetrajectory generation and fast curve sketching.

KEYWORDSBlending Curves, Rational Bézier curves, Interpolation, Splines, Interpolating Rational Splines

1. INTRODUCTION

Interpolating spline curves play important role in different geometric applications. This paperpresents an approach to construction of interpolating rational Bézier spline curves with localcontrol which have Cn continuity. This property makes the spline curves suitable for using indifferent real-time geometric applications concerned with trajectory generation. A shape ofthe constructed spline curve can be modified by means of weights which are assigned to knotpoints of the spline curve. This feature enables using of the presented spline curves for fastsketching.

Segments of the presented spline curves are constructed by means of blending rationalquadric Bézier curves represented in homogeneous coordinates. The blending is performed bymeans of special polynomials which are considered in the paper. The polynomials ensure anecessary parametric continuity of the designed spline curves. The presented approach can beconsidered as generalization of the approach to construction of interpolating spline curves inlinear spaces considered by the author [1].

Firstly construction of spline curves by linear blending of parabolic arcs was proposed byOverhauser [2] and considered by Rogers and Adams [3]. Using linear blending of conics forconstruction of spline curves was considered by Chuan Sun [4]. Polynomial blending whichensures Gn continuity is considered in other articles of Hartmann [5] and Meek, Walton [6].Some other works concerned with interpolation with rational spline curves can be mentioned.Tai, Barsky and Loe presented an interpolation method that is based on blending a nonuniformrational B-spline curve with a singularly reparameterized linear spline [7]. Interpolating rationalspline curves of cubic degree with shape control are considered in works [8-10]. Rationalcubic spline curves with G2 continuity is considered in the work [11]. Weighted rational cubicspline interpolation and its application are considered in the articles [12-15].


2

2. BLENDING POLYNOMIALS

The purpose of this section is to define polynomials which will be used for blending of parametriccurves. For this purpose consider the following knot sequences

)1,,1,1,0,,0,0(

nn

and define the following polynomials:

∑−

=−=

12

,12 )()(n

niinn ubuw , ]1,0[∈u ,

where )(, ub mn are Bernstein polynomials

mmnmn uu

mnm

nub −−

−= )1(

)!(!

!)(, .

It follows from this definition that the polynomials wn(u) meet the following boundary conditions:

0)0( =nw , 1)1( =nw , (2.1)

0)1()0( )()( == mn

mn ww , }1,,2,1{ −∈∀ nm . (2.2)

The polynomials wn(u) have the following properties:

1)1()( =−+ uwuw nn , (2.3)

1)2/1()2/1( =−++ vwvw nn (2.4)

which follow from the property

1)(0

, =∑=

n

mmn ub

of Bernstein polynomials. It follows from Equation (2.4) that the polynomials wn(u) aresymmetric with respect to the point (1/2,1/2).Besides it can be proven that

0)(lim2/1

0

=∫∞→duuwn

n

and that the polynomial wn(u) is a minimum of the functional

∫=1

0

2)( |)(|)( duuffJ nn , Nn ∈ .

Proofs of the properties can be found in the paper [1]. Figure 1 shows graphs of the polynomialswn(u).

The following polynomials of lower degrees:uuw =)(1 ,

322 )1(3)( uuuuw +−= ,

54323 )1(5)1(10)( uuuuuuw +−+−=


3

are often used in geometric applications.

Figure 1. Graphs of the polynomials )(uwn

The polynomials wn(u) were introduced by the author [16, 17]. The representation of thesepolynomials wn(u) by means of Bernstein polynomials was proposed by Wiltsche [18].

3. BLENDING CONIC ARCS

Consider two conic arcs pi(u), i∈{1, 2}, which are represented by means of rational Béziercurves and have common boundary points that is

22

1,02

222

1,1,002

)1(2)1(

)1(2)1()(

wuuwuwu

wuuwuwuu

i

iii +−+−

+−+−=

pppp , ]1,0[∈u . (3.1)

The problem is to construct a parametric curve p(u) which has the following boundary points:

01 )0()0( ppp == , 22 )1()1( ppp == (3.2)and satisfies the following boundary conditions:

)0()0( )(1

)( mm pp = , )1()1( )(2

)( mm pp = , },,2,1{ nm ∈∀ , (3.3)

where n∈N. The parametric curve p(u) which satisfies Equations (3.2) and (3.3) is called aparametric curve blending the parametric curves p1(u) and p2(u).

In order to solve the problem represent the parametric curves pi(u), i∈{1, 2}, usinghomogeneous coordinates as follows:

22

1,02 )1(2)1()( xxxx uuuuu ii +−+−= , ]1,0[∈u ,

where the points x0, xi,1 and x2 have the corresponding weight coordinates w0, wi,1 and w2. Thendefine the parametric curve x(u) as follows:

)()()())(1()( 21 uuwuuwu nn xxx +−= , ]1,0[∈u , (3.4)It follows from this definition that the corresponding parametric curve p(u) which is obtainedfrom the parametric curve x(u) by transition to Cartesian coordinates has the followingrational representation:

)()()())(1(

)()()())(1()(

21

21

uruwuruw

uuwuuwu

nn

nn

+−+−

=rr

p , ]1,0[∈u , (3.5)

where


4

222

1,1,002 )1(2)1()( pppr wuuwuwuu iii +−+−= , }2,1{∈i ,

22

1,02 )1(2)1()( wuuwuwuur ii +−+−= , }2,1{∈i .

It follows form the definition of the polynomials wn(u) that the parametric curve p(u) satisfiesconditions (3.2) because

011

1

21

21 )0()0(

)0(

)0()0()0())0(1(

)0()0()0())0(1()0( pp

rrrp ===

+−+−

=rrwrw

ww

nn

nn

and

222

2

21

21 )1()1(

)1(

)1()1()0())1(1(

)1()1()1())1(1()1( pp

rrrp ===

+−+−

=rrwrw

ww

nn

nn .

Derivatives of the parametric curve p(u) depend on derivatives of the numerator anddenominator of Equation (3.5). In order to simplify further considerations introduce thefollowing denotations:

)()()())(1()( 21 uuwuuwu nn rrr +−= , ]1,0[∈u ,

)()()())(1()( 21 uruwuruwur nn +−= , ]1,0[∈u .

Now determine derivatives of the numerator r(u) and denominator r(u). It is obtained usingLeibnitz's formula that

∑=

−− +−−

=m

i

imin

imin

m uuwuuwimi

mu

0

)(2

)()(1

)()( )())(()())(1(()!(!

!)( rrr ,

∑=

−− +−−

=m

i

imin

imin

m uruwuruwimi

mur

0

)(2

)()(1

)()( )())(()())(1(()!(!

!)(

for any m∈N. Substitution of Equations (2.2) into these equations yields that the derivativeshave the following values at the boundaries of the interval [0,º1]:

)0()0()0()0())0(1()0( )(1

)(2

)(1

)( mmn

mn

m ww rrrr =+−= , (3.6)

)1()1()1()0())1(1()1( )(2

)(2

)(1

)( mmn

mn

m ww rrrr =+−= (3.7)

and analogously

)0()0()0()0())0(1()0( )(1

)(2

)(1

)( mmn

mn

m rrwrwr =+−= , (3.8)

)1()1()1()0())1(1()1( )(2

)(2

)(1

)( mmn

mn

m rrwrwr =+−= (3.9)

for any }1,,2,1{ −∈∀ nm . Now show the derivatives of the order n also have necessaryvalues at the boundaries of the domain [0,º1]. It can be seen that

)0()0()0()0()0()0())0(1()0()0()0( )(1

)(22

)()(11

)()( nnn

nn

nn

nn

n wwww rrrrrr =++−+−=

)1()1()1()1()1()1())1(1()1()1()1( )(2

)(22

)()(11

)()( nnn

nn

nn

nn

n wwww rrrrrr =++−+−=

and analogously


5

)0()0( )(1

)( nn rr = , )1()1( )(2

)( nn rr = .

Thus it is proved that

)0()0( )(1

)( mm rr = , )1()1( )(2

)( mm rr = ,

)0()0( )(1

)( mm rr = , )1()1( )(2

)( mm rr = , },,2,1{ nm ∈∀ .

Using obtained results compute higher order derivatives of the parametric curve p(u). Itfollows form Equation (3.5) using introduced denotations for the numerator and denominatorthat

)()()( uuur rp = , ]1,0[∈u .

Differentiation of the last equation using Leibnitz's formula yields that

)()()()!(!

! )(

0

)()( uuurimi

m mm

i

imi rp =−∑

=

− , Nm∈ .

Substitution of Equations (3.6), (3.8) and (3.7), (3.9) into the last equation yields thefollowing values of derivatives at the boundaries of the interval [0,º1]:

)0()0()0()!(!

! )(1

0

)()(1

mm

i

imirimi

mrp =

−∑=

− , (3.10)

)1()1()1()!(!

! )(2

0

)()(2

mm

i

imirimi

mrp =

−∑=

− . (3.11)

On the other hand it follows from Equations (3.1) that

)()()( uuur iii rp = , }2,1{∈i ,

and therefore derivatives of the parametric curves pi(u), i∈{1, 2}, satisfy the followingequations:

)0()0()0()!(!

! )(1

0

)(1

)(1

mm

i

imirimi

mrp =

−∑=

− , (3.12)

)1()1()1()!(!

! )(2

0

)(2

)(2

mm

i

imirimi

mrp =

−∑=

− . (3.13)

Show that Equations (3.10) and (3.12) are equivalent. Consider the first derivatives of theparametric curves p(u) and p1(u). It follows from Equations (3.10) and (3.12) that the firstderivatives satisfy the following two equations:

)0()0()0()0()0( )1(1

)1(1

)1(1 rpp =+ rr , )0()0()0()0()0( )1(

11)1(

1)1(

11 rpp =+ rr .It follows from these two equations taking into account Equations (3.2) that

)0()0( )1(1

)1( pp = .

Now assume that the following equation:


6

)0()0( )1(1

)1( −− = mm pp

is also fulfilled. Then consider the m-th order derivatives of the parametric curves p(u) andp1(u). It follows from Equations (3.10) and (3.12) that the m-th order derivatives satisfy thefollowing equation:

∑∑=

−

=

−

−=

−

m

i

imim

i

imi rimi

mr

imi

m

0

)(1

)(1

0

)()(1 )0()0(

)!(!

!)0()0(

)!(!

!pp

which is equivalent to the equation

=−−

−+ ∑−

=

−−1

0

)1()(1

)(1 )0()0(

)!1(!

)!1()0()0(

m

i

imim rimi

mr pp

∑−

=

−−

−−−

+=1

0

)1(1

)(1

)(11 )0()0(

)!1(!

)!1()0()0(

m

i

imim rimi

mr pp .

It follows from the last equation taking into account the assumption that

)0()0( )(1

)( mm pp = .

Therefore the last equation is fulfilled for all m∈N by the principle of mathematicalinduction. Analogously it can be proven using Equations (3.11) and (3.13) that

)1()1( )(2

)( mm pp = .

for all m∈N. Thus Equations (3.3) are also fulfilled.

Figure 2 shows some curves constructed by means of blending two conic arcs with thepolynomials wn(u).

Figure 2. Blending conic arcs by means of the polynomials wn(u)

4. BÉZIER REPRESENTATION OF BLENDED CONIC ARCS

The purpose of this section is to obtain a rational Bézier representation of the blendingparametric curve p(u) described by Equation (3.5). In order to solve the problem consider ahomogeneous representation x(u) of the parametric curve p(u) which is described by Equation(3.6). Using Equation (2.3) the homogeneous representation can be transformed as follows:

=+−= )()()())(1()( 21 uuwuuwu nn xxx

=+−= )()()()1( 21 uuwuuw nn xx


7

=+= ∑∑−

=−

−

=−

12

2,12

1

01,12 )()()()(

n

nkkn

n

kkn uubuub xx

+++= ∑−

=−

1

022,21,11,200,2,12 ))()()()((

n

kkn ubububub xxx

=+++ ∑−

=−

12

22,21,21,200,2,12 ))()()()((n

nkkn ubububub xxx

++= ∑∑−

=−

−

=−

1

01,11,2,12

12

000,2,12 )()()()(

n

kkn

n

kkn ubububub xx

=++ ∑∑−

=−

−

=−

12

022,2,12

12

1,21,2,12 )()()()(n

kkn

n

nkkn ubububub xx

++= ∑∑=

+

−

=+

n

kkkn

n

kkkn cubcub

11,1,1,12

12

00,0,12 )()( xx

∑∑+

=+

+=+ ++

12

22,2,12

2

11,2,1,12 )()(

n

kkkn

n

nkkkn cubcub xx

Where

)12(2

)12)(2(,0 +

+−−=nn

knknc k , 120 −≤≤ nk ,

)12(

)12(,1 +

+−=nn

knkc k , nk 21 ≤≤ ,

)12(2

)1(,2 +

−=nn

kkc k , 122 +≤≤ nk .

It follows from the last equations that the parametric curve x(u) has the following Bézierrepresentation:

+++= ++ ))(()()( 1,11,101,01,1200,12 xxxx ccububu nn

++++ ∑=

+

n

kkkkkn cccub

22,21,1,10,0,12 ))(( xxx

++++ ∑−

+=+

12

12,21,2,10,0,12 ))((

n

nkkkkkn cccub xxx

212,1222,21,22,12,12 )())(( xxx ubccub nnnnnn +++ +++ .

Then transition to Cartesian coordinates yields that the blending parametric curve p(u) has thefollowing rational Bézier representation:

)(

)()(

ur

uu

rp = , ]1,0[∈u , (4.1)

where+++= ++ ))(()()( 1,11,11,1001,01,12000,12 pppr wcwcubwubu nn

++++ ∑=

+

n

kkkkkn wcwcwcub

222,21,11,1,100,0,12 ))(( ppp

++++ ∑−

+=+

12

122,2,21,21,2,100,0,12 ))((

n

nkkkkkn wcwcwcub ppp


8

2212,12222,21,21,22,12,12 )())(( ppp wubwcwcub nnnnnn +++ +++ (4.2)

and+++= ++ ))(()()( 1,11,101,01,1200,12 wcwcubwubur nn

++++ ∑=

+

n

kkkkkn wcwcwcub

22,21,1,10,0,12 ))((

++++ ∑−

+=+

12

12,2,21,2,10,0,12 ))((

n

nkkkkkn wcwcwcub

212,1222,21,22,12,12 )())(( wubwcwcub nnnnnn +++ +++ (4.3)

For example, the numerator and denominator of cubic and quintic blending parametric curveshave the following Bézier representations:

+++= )2(3

1)()()( 1,11,1001,3000,3 pppr wwubwubu

223,3221,21,22,3 )()2(3

1)( ppp wubwwub +++ ,

23,321,22,31,101,300,3 )()2(3

1)()2(

3

1)()()( wubwwubwwubwubur +++++=

and

+++= )23(5

1)()()( 1,11,1001,5000,5 pppr wwubwubu

+++++++ )36(10

1)()63(

10

1)( 221,21,2003,5221,11,1002,5 pppppp wwwubwwwub

225,5221,21,24,5 )()32(5

1)( ppp wubwwub +++ ,

+++= )23(5

1)()()( 1,101,500,5 wwubwubur

+++++++ )36(10

1)()63(

10

1)( 21,203,521,102,5 wwwubwwwub

25,521,24,5 )()32(5

1)( wubwwub +++

respectively.

5. CONSTRUCTION OF TWO SMOOTHLY JOINED CONIC ARCS

The purpose of this section is to introduce analytical expressions for construction smoothly joinedconic arcs. The expressions will be used for construction spline curves in the next section.Consider three distinct points p0, p1, and p2 with the corresponding weights w0, w1 and w2. Theproblem is to construct two conic arcs pi(u), i∈{1, 2}, which are represented by means ofrational Bézier curves

12

1,102

112

1,11,1002

1)1(2)1(

)1(2)1()(

wuuwuwu

wuuwuwuu

+−+−+−+−

=ppp

p , ]1,0[∈u , (5.1)

22

1,202

222

1,21,2112

2)1(2)1(

)1(2)1()(

wuuwuwu

wuuwuwuu

+−+−+−+−

=ppp

p , ]1,0[∈u (5.2)

and are smoothly joined at the common point p1 that is


9

)0()1( )(2

)(1

nn pp = , Nn ∈ . (5.3)

In order to find control points p1,1, and p2,1 which ensure the necessary smooth junction representthe conic arcs arcs p1(u) and p2(u) using homogeneous coordinates as follows

12

1,102

1 )1(2)1()( xxxx uuuuu +−+−= , ]1,0[∈u ,

22

1,212

2 )1(2)1()( xxxx uuuuu +−+−= , ]1,0[∈u .

The conic arcs arcs x1(u) and x2(u) are smoothly joined at the point x1 only provided that thefollowing two conditions:

)0()1( 21 xx ′=′ , )0()1( 21 xx ′′=′′ . (5.4)

are fulfilled. Resolution of these equations yields the following values of unknown controlpoints x1,1 and x2,2 of the quadric Bézier curves x1(u) and x2(u):

402

11,1

xxxx

−−= ,

402

11,2

xxxx

−+= .

The values of knot points p1,1, and p2,1 can be obtained from these equations by transition totransition to Cartesian coordinates as follows:

40022

111,1

pppp

www

−−= ,

40022

111,2

pppp

www

−+= . (5.5)

402

11,1

wwww

−−= ,

402

11,2

wwww

−+= . (5.6)

Show that in this case the conic arcs arcs p1(u) and p2(u) are also smoothly joined at the point p1.For this purpose represent the conic arcs p1(u) and p2(u) as follows:

)(

)()(

ur

uu

i

ii

rp = , }2,1{∈i (5.7)

where

112

1,11,1002

1 )1(2)1()( pppr wuuwuwuu +−+−= ,

222

1,21,2112

2 )1(2)1()( pppr wuuwuwuu +−+−=and

12

1,102

1 )1(2)1()( wuuwuwuur +−+−= ,

22

1,212

2 )1(2)1()( wuuwuwuur +−+−= .

It follows from Equations (5.4) that the first two derivatives of the numerators ri(u) anddenominators ri(u) satisfy the following conditions at the common point p1:

)0()1( 21 rr ′=′ , )0()1( 21 rr ′′=′′ (5.8)and

)0()1( 21 rr ′=′ , )0()1( 21 rr ′′=′′ . (5.9)

It is obvious that any of all other higher order derivatives of the numerators ri(u) anddenominators ri(u) is equal to zero. Now transform Equation (5.7) as follows:


10

)()()( uuur iii rp = , }2,1{∈i ,

and find the first derivatives of the equation parts. It is obtained that

)()()()()( uuuruur iiiii rpp ′=′+′ , }2,1{∈i .

It follows from these equations taking into account Equations (5.8) and (5.9) that

)0()0(

)0()0()0(

)1(

)1()1()1()1( 2

2

222

1

1111 p

prprp ′=

′−′=

′−′=′

r

r

r

r.

Analogously it can be proven that

)0()1( 21 pp ′′=′′ .

Now notice that all higher order derivatives of the conic arcs pi(u) at the common point p1

depends only values of these parametric curves at the common point and the first two derivativesof the numerators ri(u) and denominators ri(u) at the common point. Then taking into accountthe last two equations it can be stated that Equations (5.3) are also fulfilled. Thus it is obtainedthat the conic arcs p1(u) and p2(u) are smoothly joined at the common point p1.Since spline curves will be constructed by means of blending conic arcs it is reasonable toconsider shape modification of the conic arc by means of changing weights of its knot points.This modification behaves just opposite to the modification of a weight of the conic control point.That is if a weight of the conic control point increases then the conic pulls toward the controlpoint. Otherwise if a weight of the conic control point decreases then the conic pushes away fromthe control point. Figure 3 shows modification of a conic shape depending on a weight of itscontrol point. Values of the weight are depicted near the knot points.

On the other hand if a weight of the conic knot point increases then the conic pushes away fromthe control point. Otherwise if a weight of the conic knot point decreases then the conic pullstoward the control point. Figure 4 shows modification of a conic shape depending on a weight ofits knot point.

Now it is clear how weights of knot points influence on shape of two smoothly joined conics.Figure 5 shows modification of two smoothly joined conic shape by means of the weight which isprescribed to the knot point.

Figure 3. Modification of conic shape by means of a control point weight


11

Figure 4. Modification of conic shape by means of a knot point weight

Figure 5. Modification of two smoothly joined conic shape by means of weights

More detailed considerations concerned with modification of rational curve shape by means ofweights are presented in the works of Piegl [19], Sánchez-Reyes [20] and Juhász [21].

6. RATIONAL BÉZIER SPLINE CURVES WITH LOCAL SHAPE CONTROL

The purpose of this section is to present a technique for construction of a rational spline curvep(u)∈Cn, n∈N, which interpolates a sequence of knot points pi, i∈{0, 1, 2,..., k}, k∈N, with thecorresponding weights wi. In order to solve the problem construct a segment pi(u), 0 < i < k, of theparametric curve p(u) by means of blending two conic arcs pi,1(u) and pi,2(u) as was proposed inSection 3. Figure 6 explains construction a segment pi(u) of the spline curve p(u).

Figure 6. Construction of a spline curve segment


12

In order to ensure Cn smoothness of the spline curve p(u) it is necessary to ensure that that theconic arcs pi-1,1(u) and pi,2(u) are smoothly joined at the common point pi. This can be obtained byconstructing the conic arcs pi-1,1(u) and pi,2(u) as was proposed in Section 5. Taking into accountthese considerations define the conic arcs pi,1(u) and pi,2(u) as follows:

12

1,2

112

1,1,2

1, )1(2)1(

)1(2)1()(

+

++

+−+−+−+−

=iii

iiiiiii wuuwuwu

wuuwuwuu

pppp , ]1,0[∈u .

for i∈{0, 1,..., k-1} and

12

2,2

112

2,2,2

2, )1(2)1(

)1(2)1()(

+

++

+−+−+−+−

=iii

iiiiiii wuuwuwu

wuuwuwuu

pppp , ]1,0[∈u .

for i∈{1, 2,..., k} where the control points pi,1 and pi,2 with the corresponding weights wi,1 and wi,2

are defined using Equations

41111

1,−−++ −

−= iiiiiii

www

pppp ,

41111

2,−−++ −

+= iiiiiii

www

pppp .

411

1,−+ −

−= iiii

wwww ,

411

2,−+ −

+= iiii

wwww .

Then a segment pi(u) of the spline curve p(u) can be defined using Equation (3.5) as follows:

)()()())(1(

)()()())(1()(

2,1,

2,1,

uruwuruw

uuwuuwu

inin

inini +−

+−=

rrp , ]1,0[∈u ,

where

112

,,2

, )1(2)1()( +++−+−= iijijiiiji wuuwuwuu pppr , }2,1{∈j ,

12

,2

, )1(2)1()( ++−+−= ijiiji wuuwuwuur , }2,1{∈j .

It follows from Equations (3.3) and (5.3) that in this case segments of the parametric curve p(u)satisfy the following condition:

)0()1( )(1

)( ni

ni += pp , Nn ∈ .

Figure 7 shows how a shape of the spline curve depends on the weight prescribed to the knotpoint of the spline curve. Weights of all other knot points of the spline curve are equal to theunity.

Figure 7. Modification of a spline curve shape by means of weights


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7. CONCLUSIONS

The approach to construction of Cn continuous interpolating spline curves by means of blendingrational quadric Bézier curves is introduced. Spline curves are constructed locally whichimplies local shape control of them by means of weights which are assigned to the knotpoints of the constructed spline curve. Bézier representation of the considered spline curves isintroduced. The presented technique can be used for fast prototyping rational spline Bézier.Besides since the spline curves are constructed locally the presented technique can be alsoused in real-time geometric applications connected with computer graphics and animation.

REFERENCES

[1] Pobegailo, A. P., Polynomial blending of parametric curves in linear spaces. BSU, Minsk, 2013.

[2] Overhauser, A.W., (1968) “Analytic Definition of Curves and Surfaces by Parabolic Blending”, Tech.Rep. No. SL68-40, Ford Motor Company Scientific Laboratory.

[3] Rogers, D.F., Adams, J.A., (1989) Mathematical elements for computer graphics, McGraw-Hillpublisher.

[4] Chuan Sun, Huanxi Zhao, (2009) “Generating fair, C2 continuous splines by blending conics”,Computers & Graphics, Vol. 33, No. 2, pp. 173–180.

[5] Hartmann, E., (2001) “Parametric Gn blending of curves and surfaces”, The Visual Computer, Vol. 17,No. 1, pp. 1-13.

[6] Meek, D.S., Walton, D. J., (2009) “Blending two parametric curves”, Computer-Aided Design, Vol.41, No. 6, pp. 423-431.

[7] Tai, C.-L., Barsky, B.A. and Loe, K.F., (2000) “An interpolation method with weights and relaxationparameters”, in A. Cohen et al. (eds.), Curves and surface ftting, Saint-Malo, Vanderbilt Univ. Press,Nashville, pp. 393-402.

[8] Habib, Z., Sakai, M. and Sarfraz, M., (2004) “Interactive Shape Control with Rational Cubic Splines”,Computer-Aided Design & Applications, Vol. 1, No. 1-4, pp. 709-718.

[9] Habib, Z., Sarfraz, M. and Sakai, M., (2005) “Rational cubic spline interpolation with shape control”,Computers & Graphics, Vol. 29, No. 4, pp. 594-605.

[10] Sarfraz, M., (1994) “Generalized geometric interpolation for rational cubic splines”, Computers &Graphics, Vol. 18, No. 1, pp. 61–72.

[11] Karčiauskas, K., Peters, J., (2011) “Rational G2 splines”, Graphical Models, Vol. 73, No. 5, pp. 286–295.

[12] Fangxun Bao, Qinghua Sun, Qi Duan, (2009) “Point control of the interpolating curve with a rationalcubic spline”, Journal of Visual Communication and Image Representation, Vol. 20, No. 4, pp. 275–280.

[13] Qi Duan, Djidjeli, K., Price, W.G., Twizell, E.H., (2000) “Weighted rational cubic splineinterpolation and its application”, Journal of Computational and Applied Mathematics, Vol. 117, No.2, pp. 121–135.

[14] Qi Duan, Fangxun Bao, Shitian Du, Twizell, E.H., (2009) “Local control of interpolating rationalcubic spline curves”, Computer-Aided Design, Vol. 41, No. 11, pp. 825–829.

[15] Qi Duana, Liqiu Wangb, Twizellc, E.H., (2005) “A new weighted rational cubic interpolation and itsapproximation”, Applied Mathematics and Computation, Vol. 168, No. 2, pp. 990–1003.

[16] Pobegailo, A.P., (1991) “Local interpolation with weight functions for variable-smoothness curvedesign”, Computer-Aided Design, Vol. 23, No. 8, pp. 579-582.

[17] Pobegailo, A.P., (1992) “Geometric modeling of curves using weighted linear and circular segments”,The Visual Computer, Vol. 8, No. 4, pp. 241-245.


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[18] Wiltsche, A., (2005) “Blending curves”, Journal for Geometry and Graphics, Vol. 9, No. 1, pp. 67-75.

[19] Piegl, L., (1989) “Modifying the shape of rational B-splines. Part 1: curves”, Compuer.-Aided Design,Vol. 21, No. 8, pp. 509-518.

[20] Sánchez-Reyes, J., (1997) “A Simple Technique for NURBS Shape Modification”, IEEE ComputerGraphics and Applications, Vol. 17, No. 1, pp. 52-59.

[21] Juhász, I., (1999) “Weight-based shape modification of NURBS curves”, Computer Aided GeometricDesign, Vol. 16, No. 5, pp. 377-383.

Authors

A. P. Pobegailo graduated from Belarusian State University in 1981. Then heobtained Ph. Degree on system analysis and automatic control. Currently he is anassistant professor at Belarusian State University. He teaches courses on operatingsystems and systems design. His research interests include computer graphics andgeometric design.

Technology

Interpolating rational bézier spline curves with local shape control