INTERPOLATING RATIONAL BÉZIER SPLINE CURVES WITH LOCAL SHAPE CONTROL

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

1. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
DOI : 10.5121/ijcga.2013.3401 1
INTERPOLATING RATIONALBÉZIERSPLINE
CURVESWITHLOCALSHAPECONTROL
Alexander P. Pobegailo
1
1
Department ofApplied Mathematics andComputerScience,Belarusian State
University,Minsk,Belarus
ABSTRACT
The paper presents a technique for constructionofC
n
interpolatingrational Bézierspline curves by means
of blendingrationalquadric Bézier curves. A class of polynomials which satisfy special boundary
conditions is used for blending. Properties of the polynomialsare considered.The constructed spline
curves have local shape control that make them useful in such geometric applications asreal-time
trajectory generation and fast curve sketching.
KEYWORDS
BlendingCurves,RationalBézier curves,Interpolation,Splines, Interpolating Rational Splines
1.INTRODUCTION
Interpolating spline curves play important role in different geometric applications. This paper
presents an approach to construction of interpolatingrationalBéziersplinecurves with local
controlwhich haveC
n
continuity. This propertymakes the spline curves suitable for using in
differentreal-timegeometricapplicationsconcerned withtrajectory generation.A shape of
the constructed spline curve can be modified by means of weights which are assigned to knot
points of the spline curve.This feature enables using of the presented spline curves forfast
sketching.
Segments of the presented spline curves are constructed by means of blendingrational
quadric Bézier curvesrepresented in homogeneous coordinates.The blending is performedby
means of specialpolynomialswhich are considered in the paper.The polynomials ensure a
necessary parametric continuity of the designedsplinecurves. The presented approachcan be
considered asgeneralization of theapproach to construction ofinterpolatingspline curves in
linear spaces considered by the author [1].
Firstly construction of spline curves by linear blending of parabolic arcs was proposed by
Overhauser[2]and considered byRogers and Adams[3].Using linear blending of conics for
construction of spline curves was considered byChuan Sun[4].Polynomial blending which
ensuresG
n
continuity is considered in other articlesofHartmann[5] andMeek,Walton[6].
Some other works concerned with interpolation with rational spline curves can bementioned.
Tai, Barsky and Loepresented an interpolation method that is based on blending a nonuniform
rational B-spline curve with a singularly reparameterized linear spline[7]. Interpolating rational
spline curves of cubic degree with shape control are considered in works [8-10]. Rational
cubic spline curves withG
2
continuity is considered in the work[11].Weighted rational cubic
spline interpolation and its applicationare considered in the articles [12-15].

2. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
2
2.BLENDINGPOLYNOMIALS
The purpose of this section is to define polynomials which will be used for blending of parametric
curves. For this purpose consider the following knot sequences
)1,,1,1,0,,0,0(




nn
and define the following polynomials:
å
-
=
-
=
12
,12
)()(
n
ni
inn
ubuw , ]1,0[Îu ,
where )(
,
ub
mn are Bernstein polynomials
mmn
mn
uu
mnm
n
ub
-
-
-
= )1(
)!(!
!
)(
,
.
It follows from this definition that the polynomialswn(u)meet the following boundary conditions:
0)0(=
nw , 1)1(=
nw , (2.1)
0)1()0(
)()(
==
m
n
m
n ww , }1,,2,1{ -Î" nm  . (2.2)
The polynomialswn(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
,
=å
=
n
m
mn
ub
of Bernstein polynomials.It follows from Equation (2.4) that the polynomialswn(u) are
symmetric with respect to the point(1/2,1/2).
Besides it can be proven that
0)(lim
2/1
0
=
ò
¥®
duuw
n
n
andthatthe polynomialwn(u)is a minimum of the functional
ò
=
1
0
2)(
|)(|)( duuffJ
n
n ,NnÎ.
Proofsof the properties can be found in the paper [1]. Figure 1 shows graphs of the polynomials
wn(u).
The following polynomialsof lower degrees:
uuw=)(
1 ,
32
2 )1(3)( uuuuw +-= ,
5432
3
)1(5)1(10)( uuuuuuw +-+-=

3. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
3
are often used in geometric applications.
Figure 1. Graphs of the polynomials)(uw
n
The polynomialswn(u)were introduced by the author [16,17]. The representation of these
polynomialswn(u)by means of Bernstein polynomials was proposed by Wiltsche[18].
3.BLENDINGCONICARCS
Consider twoconic arcspi(u),iÎ{1,2}, whichare representedby means of rational Bézier
curvesandhave common boundary points that is
2
2
1,0
2
22
2
1,1,00
2
)1(2)1(
)1(2)1(
)(
wuuwuwu
wuuwuwu
u
i
ii
i
+-+-
+-+-
=
ppp
p , ]1,0[Îu . (3.1)
The problem is to construct a parametric curvep(u)which hasthe 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 Î" , (3.3)
wherenÎN.The parametric curvep(u)which satisfies Equations (3.2) and (3.3)iscalled a
parametric curve blending the parametric curvesp1(u)andp2(u).
In order to solve the problem represent theparametric curvespi(u),iÎ{1,2},using
homogeneous coordinates as follows:
2
2
1,0
2
)1(2)1()( xxxx uuuuu
ii
+-+-= , ]1,0[Îu ,
wherethe pointsx0,xi,1andx2have thecorrespondingweight coordinatesw0,wi,1andw2.Then
define the parametric curvex(u)as follows:
)()()())(1()(
21 uuwuuwu
nn xxx +-= , ]1,0[Îu , (3.4)
It follows from this definition that the corresponding parametric curvep(u)which is obtained
from the parametric curvex(u)by transition to Cartesian coordinates has the following
rationalrepresentation:
)()()())(1(
)()()())(1(
)(
21
21
uruwuruw
uuwuuw
u
nn
nn
+-
+-
=
rr
p , ]1,0[Îu , (3.5)
where

4. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
4
22
2
1,1,00
2
)1(2)1()( pppr wuuwuwuu
iii +-+-= , }2,1{Îi ,
2
2
1,0
2
)1(2)1()( wuuwuwuur
ii
+-+-= , }2,1{Îi .
It follows form the definition of the polynomialswn(u)that the parametric curvep(u)satisfies
conditions (3.2) because
01
1
1
21
21
)0(
)0(
)0(
)0()0()0())0(1(
)0()0()0())0(1(
)0( pp
rrr
p ===
+-
+-
=
rrwrw
ww
nn
nn
and
22
2
2
21
21
)1(
)1(
)1(
)1()1()0())1(1(
)1()1()1())1(1(
)1( pp
rrr
p ===
+-
+-
=
rrwrw
ww
nn
nn
.
Derivatives of the parametric curvep(u)depend on derivatives ofthe numerator and
denominatorof Equation (3.5).In order to simplify further considerations introduce the
following denotations:
)()()())(1()(
21 uuwuuwu
nn rrr +-= , ]1,0[Îu ,
)()()())(1()(
21 uruwuruwur
nn +-= , ]1,0[Îu .
Nowdetermine derivativesof the numeratorr(u) and denominatorr(u).It is obtainedusing
Leibnitz's formulathat
å
=
--
+-
-
=
m
i
imi
n
imi
n
m
uuwuuw
imi
m
u
0
)(
2
)()(
1
)()(
)())(()())(1((
)!(!
!
)( rrr ,
å
=
--
+-
-
=
m
i
imi
n
imi
n
m
uruwuruw
imi
m
ur
0
)(
2
)()(
1
)()(
)())(()())(1((
)!(!
!
)(
for anymÎN.Substitution of Equations (2.2) into these equations yields that the derivatives
have the following values at theboundaries of the interval [0,º1]:
)0()0()0()0())0(1()0(
)(
1
)(
2
)(
1
)( mm
n
m
n
m
ww rrrr =+-= , (3.6)
)1()1()1()0())1(1()1(
)(
2
)(
2
)(
1
)( mm
n
m
n
m
ww rrrr =+-= (3.7)
and analogously
)0()0()0()0())0(1()0(
)(
1
)(
2
)(
1
)( mm
n
m
n
m
rrwrwr =+-= , (3.8)
)1()1()1()0())1(1()1(
)(
2
)(
2
)(
1
)( mm
n
m
n
m
rrwrwr =+-= (3.9)
for any }1,,2,1{ -Î" nm  .Nowshowthederivatives of the ordernalso have necessary
values 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
)()( nn
n
n
n
n
n
n
n
n
wwww rrrrrr =++-+-=
)1()1()1()1()1()1())1(1()1()1()1(
)(
2
)(
22
)()(
11
)()( nn
n
n
n
n
n
n
n
n
wwww rrrrrr =++-+-=
and analogously

5. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
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 Î" .
Using obtained results compute higher order derivatives of the parametric curvep(u). It
follows form Equation (3.5)using introduced denotations for thenumerator and denominator
that
)()()( uuur rp= , ]1,0[Îu .
Differentiation of the last equation usingLeibnitz's formulayields that
)()()(
)!(!
!
)(
0
)()(
uuur
imi
m
m
m
i
imi
rp =
-
å
=
-
,NmÎ.
Substitution of Equations (3.6),(3.8)and (3.7), (3.9)into thelastequation yields the
following valuesof derivatives atthe boundaries of the interval [0,º1]:
)0()0()0(
)!(!
! )(
1
0
)()(
1
m
m
i
imi
r
imi
m
rp =
-
å
=
-
, (3.10)
)1()1()1(
)!(!
! )(
2
0
)()(
2
m
m
i
imi
r
imi
m
rp =
-
å
=
-
. (3.11)
On the other handit follows from Equations (3.1) that
)()()( uuur
iii rp= , }2,1{Îi ,
and therefore derivatives of the parametric curvespi(u),iÎ{1,2},satisfy the following
equations:
)0()0()0(
)!(!
! )(
1
0
)(
1
)(
1
m
m
i
imi
r
imi
m
rp =
-
å
=
-
, (3.12)
)1()1()1(
)!(!
! )(
2
0
)(
2
)(
2
m
m
i
imi
r
imi
m
rp =
-
å
=
-
. (3.13)
Show thatEquations(3.10)and (3.12) areequivalent.Consider the first derivatives of the
parametric curvesp(u) andp1(u).It follows fromEquations (3.10) and (3.12) that thefirst
derivatives 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 accountEquations (3.2) that
)0()0(
)1(
1
)1(
pp = .
Nowassumethat the following equation:

6. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
6
)0()0(
)1(
1
)1( --
=
mm
pp
is also fulfilled. Then consider them-th order derivatives of the parametric curvesp(u) and
p1(u).It follows from Equations (3.10) and (3.12)that them-th orderderivatives satisfy the
following equation:
åå
=
-
=
-
-
=
-
m
i
imi
m
i
imi
r
imi
m
r
imi
m
0
)(
1
)(
1
0
)()(
1
)0()0(
)!(!
!
)0()0(
)!(!
!
pp
which is equivalent to the equation
=
--
-
+å
-
=
--
1
0
)1()(
1
)(
1 )0()0(
)!1(!
)!1(
)0()0(
m
i
imim
r
imi
m
r pp
å
-
=
--
--
-
+=
1
0
)1(
1
)(
1
)(
11 )0()0(
)!1(!
)!1(
)0()0(
m
i
imim
r
imi
m
r 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 allmÎNby the principle of mathematical
induction. Analogously it can be proven using Equations (3.11) and (3.13) that
)1()1(
)(
2
)( mm
pp = .
for allmÎN.Thus Equations (3.3) are also fulfilled.
Figure2shows some curves constructed by means of blending two conicarcs with the
polynomialswn(u).
Figure2. Blendingconicarcs by means of the polynomialswn(u)
4.BÉZIERREPRESENTATION OFBLENDEDCONICARCS
The purpose of this section isto obtainarationalBézierrepresentationof the blending
parametric curvep(u) described by Equation (3.5). In order to solve the problemconsider a
homogeneous representationx(u)of the parametric curvep(u)which is described by Equation
(3.6).UsingEquation (2.3) the homogeneous representation can be transformed as follows:
=+-= )()()())(1()(
21 uuwuuwu
nn xxx
=+-= )()()()1(
21 uuwuuw
nn xx

7. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
7
=+= åå
-
=
-
-
=
-
12
2,12
1
0
1,12
)()()()(
n
nk
kn
n
k
kn
uubuub xx
+++=å
-
=
-
1
0
22,21,11,200,2,12
))()()()((
n
k
kn
ubububub xxx
=+++å
-
=
-
12
22,21,21,200,2,12
))()()()((
n
nk
kn
ubububub xxx
++= åå
-
=
-
-
=
-
1
0
1,11,2,12
12
0
00,2,12
)()()()(
n
k
kn
n
k
kn
ubububub xx
=++ åå
-
=
-
-
=
-
12
0
22,2,12
12
1,21,2,12 )()()()(
n
k
kn
n
nk
kn ubububub xx
++= åå
=
+
-
=
+
n
k
kkn
n
k
kkn
cubcub
1
1,1,1,12
12
0
0,0,12
)()( xx
åå
+
=
+
+=
+ ++
12
2
2,2,12
2
1
1,2,1,12 )()(
n
k
kkn
n
nk
kkn cubcub xx
Where
)12(2
)12)(2(
,0
+
+--
=
nn
knkn
c
k
, 120 -££ nk ,
)12(
)12(
,1
+
+-
=
nn
knk
c
k
, nk21££ ,
)12(2
)1(
,2
+
-
=
nn
kk
c
k
, 122 +££ nk .
It follows from the last equations that the parametric curvex(u)has the following Bézier
representation:
+++=
++
))(()()(
1,11,101,01,1200,12
xxxx ccububu
nn
++++å
=
+
n
k
kkkkn cccub
2
2,21,1,10,0,12 ))(( xxx
++++å
-
+=
+
12
1
2,21,2,10,0,12
))((
n
nk
kkkkn
cccub xxx
212,1222,21,22,12,12 )())(( xxx ubccub
nnnnnn +++ +++ .
Then transition to Cartesian coordinates yields that the blending parametric curvep(u) has the
followingrational Bézier representation:
)(
)(
)(
ur
u
u
r
p= , ]1,0[Îu , (4.1)
where
+++=
++ ))(()()(
1,11,11,1001,01,12000,12 pppr wcwcubwubu
nn
++++å
=
+
n
k
kkkkn wcwcwcub
2
22,21,11,1,100,0,12 ))(( ppp
++++å
-
+=
+
12
1
22,2,21,21,2,100,0,12 ))((
n
nk
kkkkn wcwcwcub ppp

8. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
8
2212,12222,21,21,22,12,12 )())(( ppp wubwcwcub
nnnnnn +++ +++ (4.2)
and
+++=
++
))(()()(
1,11,101,01,1200,12
wcwcubwubur
nn
++++å
=
+
n
k
kkkkn
wcwcwcub
2
2,21,1,10,0,12
))((
++++å
-
+=
+
12
1
2,2,21,2,10,0,12
))((
n
nk
kkkkn
wcwcwcub
212,1222,21,22,12,12
)())(( wubwcwcub
nnnnnn +++
+++ (4.3)
For example,the numerator and denominatorofcubic andquinticblending parametric curves
have the followingBézier 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 OFTWOSMOOTHLYJOINEDCONICARCS
The purpose of this section is to introduceanalytical expressionsfor construction smoothly joined
conic arcs. Theexpressionswill be used for construction spline curves in the next section.
Consider three distinct pointsp0,p1, andp2withthe corresponding weightsw0,w1andw2.The
problem is to construct two conic arcspi(u),iÎ{1,2}, which are represented by means of
rational Bézier curves
1
2
1,10
2
11
2
1,11,100
2
1
)1(2)1(
)1(2)1(
)(
wuuwuwu
wuuwuwu
u
+-+-
+-+-
=
ppp
p , ]1,0[Îu , (5.1)
2
2
1,20
2
22
2
1,21,211
2
2
)1(2)1(
)1(2)1(
)(
wuuwuwu
wuuwuwu
u
+-+-
+-+-
=
ppp
p , ]1,0[Îu (5.2)
and are smoothly joined at the common pointp1that is

9. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
9
)0()1(
)(
2
)(
1
nn
pp = ,NnÎ. (5.3)
In order to find control pointsp1,1, andp2,1which ensure the necessary smooth junction represent
the conic arcs arcsp1(u) andp2(u) using homogeneous coordinates as follows
1
2
1,10
2
1 )1(2)1()( xxxx uuuuu +-+-= , ]1,0[Îu ,
2
2
1,21
2
2 )1(2)1()( xxxx uuuuu +-+-= , ]1,0[Îu .
The conic arcs arcsx1(u) andx2(u) are smoothly joined at the pointx1only provided thatthe
following two conditions:
)0()1(
21 xx ¢=¢ , )0()1(
21 xx ¢¢=¢¢ . (5.4)
are fulfilled.Resolution of these equations yields the following values of unknown control
pointsx1,1andx2,2of the quadric Bézier curvesx1(u) andx2(u):
4
02
11,1
xx
xx
-
-= ,
4
02
11,2
xx
xx
-
+= .
The values of knot pointsp1,1, andp2,1canbe obtained from these equations by transition to
transition to Cartesian coordinatesas follows:
4
0022
111,1
pp
pp
ww
w
-
-= ,
4
0022
111,2
pp
pp
ww
w
-
+= . (5.5)
4
02
11,1
ww
ww
-
-= ,
4
02
11,2
ww
ww
-
+= . (5.6)
Show that in this case the conic arcs arcsp1(u) andp2(u) are also smoothly joined at the pointp1.
For this purpose represent the conic arcsp1(u) andp2(u) as follows:
)(
)(
)(
ur
u
u
i
i
i
r
p= , }2,1{Îi (5.7)
where
11
2
1,11,100
2
1 )1(2)1()( pppr wuuwuwuu +-+-= ,
22
2
1,21,211
2
2
)1(2)1()( pppr wuuwuwuu +-+-=
and
1
2
1,10
2
1 )1(2)1()( wuuwuwuur +-+-= ,
2
2
1,21
2
2 )1(2)1()( wuuwuwuur +-+-= .
It follows from Equations (5.4) thatthefirst twoderivatives ofthenumeratorsri(u) and
denominatorsri(u) satisfy the following conditions at the common pointp1:
)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 thenumeratorsri(u) and
denominatorsri(u) is equal to zero.Now transform Equation (5.7) as follows:

10. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
10
)()()( uuur
iii
rp= , }2,1{Îi ,
and find the first derivatives of the equationparts. It is obtained that
)()()()()( uuuruur
iiiii
rpp ¢=¢+¢ , }2,1{Îi .
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
111
1 p
prpr
p ¢=
¢-¢
=
¢-¢
=¢
r
r
r
r
.
Analogously it can be proven that
)0()1(
21 pp ¢¢=¢¢ .
Now notice that all higher order derivatives of the conic arcspi(u) at the common pointp1
depends only values of theseparametric curves at the common point and the first two derivatives
ofthenumeratorsri(u) and denominatorsri(u) at the common point.Then taking into account
the last two equations it canbe stated that Equations (5.3) are also fulfilled.Thus it is obtained
that theconic arcsp1(u) andp2(u) are smoothly joined at the common pointp1.
Since spline curves will be constructed by means of blending conic arcs it is reasonable to
consider 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 control
point. Otherwise if a weight of the conic control point decreases then the conic pushes away from
the control point. Figure 3 shows modification of a conic shape depending on a weight of its
control 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 from
the control point. Otherwise if a weight of the conic knot point decreases then the conic pulls
toward the control point. Figure 4shows modification of a conic shape depending on a weight of
its 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 weightwhich is
prescribed to the knot point.
Figure 3. Modification of conic shape by means of a control point weight

11. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
11
Figure4.Modification of conic shape by means of a knot point weight
Figure5.Modification oftwo smoothly joinedconicshape by means of weights
More detailed considerationsconcerned with modification of rational curve shape by means of
weights are presented in the works of Piegl [19],Sánchez-Reyes [20] andJuhász [21].
6.RATIONALBÉZIERSPLINECURVES WITHLOCALSHAPECONTROL
The purpose of this section is to present a technique for construction of arationalspline curve
p(u)ÎC
n
,nÎN, which interpolates a sequence of knot pointspi,iÎ{0,1,2,...,k},kÎN, withthe
corresponding weightswi. In order to solve the problem construct a segmentpi(u), 0<i<k, of the
parametric curvep(u) by means of blending two conic arcspi,1(u) andpi,2(u)as wasproposedin
Section 3.Figure6explains construction a segmentpi(u) of the spline curvep(u).
Figure6. Construction ofaspline curvesegment

12. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
12
In order to ensureC
n
smoothness of the spline curvep(u) it is necessary to ensure that that the
conic arcspi-1,1(u) andpi,2(u) are smoothly joined at the common pointpi. This can be obtained by
constructing the conic arcspi-1,1(u) andpi,2(u) as was proposed in Section 5. Taking into account
these considerationsdefine the conic arcspi,1(u) andpi,2(u) as follows:
1
2
1,
2
11
2
1,1,
2
1,
)1(2)1(
)1(2)1(
)(
+
++
+-+-
+-+-
=
iii
iiiiii
i
wuuwuwu
wuuwuwu
u
ppp
p , ]1,0[Îu .
foriÎ{0,1,...,k-1} and
1
2
2,
2
11
2
2,2,
2
2,
)1(2)1(
)1(2)1(
)(
+
++
+-+-
+-+-
=
iii
iiiiii
i
wuuwuwu
wuuwuwu
u
ppp
p , ]1,0[Îu .
foriÎ{1,2,...,k}where the control pointspi,1andpi,2with the corresponding weightswi,1andwi,2
are defined using Equations
4
1111
1,
--++ -
-=
iiii
iii
ww
w
pp
pp ,
4
1111
2,
--++ -
+=
iiii
iii
ww
w
pp
pp .
4
11
1,
-+
-
-=
ii
ii
ww
ww ,
4
11
2,
-+
-
+=
ii
ii
ww
ww .
Thena segmentpi(u) of the spline curvep(u) can be defined using Equation (3.5) as follows:
)()()())(1(
)()()())(1(
)(
2,1,
2,1,
uruwuruw
uuwuuw
u
inin
inin
i
+-
+-
=
rr
p , ]1,0[Îu ,
where
11
2
,,
2
,
)1(2)1()(
++
+-+-=
iijijiiiji
wuuwuwuu pppr , }2,1{Îj ,
1
2
,
2
, )1(2)1()(
++-+-=
ijiiji wuuwuwuur , }2,1{Îj .
It follows from Equations (3.3) and (5.3) that in this case segments of the parametric curvep(u)
satisfy the following condition:
)0()1(
)(
1
)( n
i
n
i +=pp ,NnÎ.
Figure7showshow a shape of the spline curve depends ontheweight prescribed to the knot
point of the spline curve.Weights of all other knot points of the spline curve are equal to the
unity.
Figure7. Modification ofa spline curveshapeby means of weights

13. International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 2013
13
7.CONCLUSIONS
The approach to construction ofC
n
continuous interpolating splinecurves by means of blending
rational quadric Bézier curves is introduced. Spline curves are constructed locally which
implies local shape control of them by means of weights which are assigned to the knot
points of the constructed splinecurve. Bézier representation of the considered spline curves is
introduced. 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 also
used inreal-timegeometric 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-Hill
publisher.
[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)“ParametricG
n
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 relaxation
parameters”, in A. Cohen et al. (eds.),Curves and surfaceftting,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 rational
cubic 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 spline
interpolation 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 rational
cubic spline curves”,Computer-Aided Design, Vol.41,No.11,pp.825–829.
[15]Qi Duana, Liqiu Wangb, Twizellc, E.H.,(2005) “A new weightedrational cubic interpolation and its
approximation”,Applied Mathematics and Computation, Vol.168,No.2,pp.990–1003.
[16]Pobegailo, A.P., (1991)“Local interpolation with weight functions for variable-smoothness curve
design”,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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14
[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”,IEEEComputer
Graphics and Applications, Vol.17, No.1,pp.52-59.
[21]Juhász, I.,(1999) “Weight-based shape modification of NURBS curves”,Computer Aided Geometric
Design, Vol. 16, No. 5, pp. 377-383.
Authors
A. P. Pobegailo graduated from Belarusian State University in 1981.Thenhe
obtained Ph. Degree onsystem analysis and automatic control. Currentlyhe is an
assistant professor at Belarusian State University. He teachescourses on operating
systems and systems design.His research interests includecomputer graphics and
geometric design.