Download Circular Approximation by Trigonometric Bézier Curves

World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:1, 2015
Circular Approximation by Trigonometric Bézier
Curves
Maria Hussin, Malik Zawwar Hussain, Mubashrah Saddiqa

International Science Index, Computer and Information Engineering Vol:9, No:1, 2015 waset.org/Publication/10000225
Abstract—We present a trigonometric scheme to approximate a
circular arc with its two end points and two end tangents/unit
tangents. A rational cubic trigonometric Bézier curve is constructed
whose end control points are defined by the end points of the circular
arc. Weight functions and the remaining control points of the cubic
trigonometric Bézier curve are estimated by variational approach to
reproduce a circular arc. The radius error is calculated and found less
than the existing techniques.
Keywords—Control points, rational trigonometric Bézier curves,
radius error, shape measure, weight functions.
I.
INTRODUCTION
C
IRCLES are around us everywhere and inspired the
mankind even before the beginning of recorded history.
Circles are the main source of many wonderful inventions,
such as circular ripples, wheels, circular gears which run the
machines in urban factories and make our life much easier.
Circular arcs are basic tools in engineering design, web design
and mobile design. They are also used as user interface tool
and in many other projects due to their exceptional properties
such as constant curvature, constant distance from a fixed
point etc.
The trigonometric polynomial curves were first introduced
by Schoenberg [6]. The control point form of quadratic and
cubic trigonometric polynomial curves were presented by Han
in [3] and [4]. Wang, Chen and Zhou [7] defined
trigonometric B-splines, known as algebraic-trigonometric Bsplines (NUAT). These contributions only narrowed down the
properties of trigonometric polynomial curves and did not
focus on its applications. The authors in [7] used trigonometric
polynomial curves for the shape preservation of data.
In this research paper, the circular approximation problem is
considered. This problem is defined as: Given two end points
and end tangents/unit tangents of a circular arc, construct a
curve which interpolates the end points of circular arc and
close to it between these endpoints.
Integral Bézier curves were in account for the
approximation of circular arc by a number of researchers. Lee
,
and
[5], Goldapp [2] and Fang [1] presented the
approximations of circular arc by quadratic, cubic and quintic
Bézier curves respectively. The radius error between the
circular arc and the approximating Bézier was minimized to
obtain the best approximation.
In this research paper, rational cubic trigonometric Bézier
curves are developed for approximating circular arc,
particularly when its
1. Two end points and end tangents vectors are given.
2. Two end points and end unit tangents vectors are given.
In this section, rational quadratic and cubic trigonometric
Bézier curves are introduced over the interval 0, .
The parametric form of newly developed rational quadratic
,
0, , is defined by the
trigonometric Bézier curve
relation given in (1):
∑
,
∑
(1)
where
and
are the quadratic trigonometric basis
are the
functions and weight functions respectively. , ,
control points of rational quadratic trigonometric Bézier curve.
The quadratic trigonometric basis functions
of (1), are
defined as
1
2 1
,
.
,
It can be easily verified that
∑
1 and
0 for
0,1,2,
(2)
where
,
∑
and
0
,
.
(3)
Thus from (2) and (3), the rational quadratic trigonometric
Bézier curve (1), satisfies the convex-hull property and
endpoints interpolation property.
The parametric form of the developed rational cubic
trigonometric Bézier curve is given by the following relation.
Maria Hussain is an Assistant Professor in Department of Mathematics,
Lahore College for Women University, Lahore, Pakistan (phone: +92-03314930071; e-mail: [email protected]).
Malik Zawwar Hussain is a Professor in Department of Mathematics,
University
of
the
Punjab,
Lahore,
Pakistan
(e-mail:
[email protected]).
Mubashrah Saddiqa is a Lecturer in Department of Mathematics, Lahore
College for Women University, Lahore, Pakistan.
International Scholarly and Scientific Research & Innovation 9(1) 2015
RATIONAL TRIGONOMETRIC BÉZIER CURVE
II.
∑
∑
30
,0
scholar.waset.org/1999.7/10000225
,
(4)
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:1, 2015
and
are the control points, weight functions
where ,
and basis functions respectively. These cubic trigonometric
, are defined as
basis functions
3
1
,
0,1,2,3.
International Science Index, Computer and Information Engineering Vol:9, No:1, 2015 waset.org/Publication/10000225
1,
0 for
0,1,2,3,0
,
1
,
.
0,1,2, and
,
2
,
,
cos .
, where
(8)
Thus the rational cubic trigonometric Bézier curve is
rewritten as
.

,
,
,
The weight functions
positive real numbers.
,
(9)
where control points ,
0,1,2,3 and weight functions ,
are defined in (8).
With reference to Fig. 1, the following notations are
introduced
where
/
,
(5)
The cubic trigonometric Bézier curve (4) satisfies the
following properties:
and
,
(i) End point interpolation property: 0
(ii) Convex-hull property:
∑
rational quadratic trigonometric Bézier curve (7) and rational
cubic trigonometric Bézier curve (4) is given as follows
. (10)
0,1,2,3, are
It can be seen from Fig. 1 that cos
, or
.
Also, putting the values of
and
from (7) in (10) to get
. Since
cos , so
can be written as a
function of
as
.
(11)
By vector algebra, from Fig. 1, we have
Fig. 1 The control polygon of rational quadratic and rational cubic
Bézier curve
.
(12)
From (10) and (11), we get
III.
REPRESENTATION OF A CIRCLE BY TRIGONOMETRIC
BÉZIER CURVES
cos
In this section, the rational quadratic and rational cubic
representations of a circular arc are established. If the control
form an isosceles triangle with base
and
points , ,
0,1,2 taken as
the weight functions ,
1,
cos ,
(7)
The control point
is the intersection of the lines given by
the end points ,
along with unit tangents , at
and
respectively as shown in Fig. 1. Now by degree elevation
of rational quadratic trigonometric Bézier curve (7), we get a
rational cubic trigonometric representation of the same
and weights
circular arc with control points , , and
1, , , 1. The relation between control points and weights of
International Scholarly and Scientific Research & Innovation 9(1) 2015
(13)
.
IV. RATIONAL CUBIC TRIGONOMETRIC APPROXIMATION OF
CIRCLE WHEN TWO END POINTS AND END TANGENTS ARE
GIVEN
(6)
then the rational quadratic trigonometric Bézier curve (1)
represents circle. Here is the base angle of the triangle 
(See Fig. 1). Thus the rational quadratic trigonometric
Bézier curve (1) is rewritten as
.
Similarly,
.
In this section, the problem of approximation of a circle by
rational cubic trigonometric Bézier curve (9) is discussed. If
two end points and end tangents of a circular arc are given,
then the end control points
and of the rational cubic
trigonometric Bézier curve (9) are equal to the end points of
the given circular arc. For solving this problem, we calculate
,
and the corresponding weights
the control points
, . From Fig. 1, we have
,
where
and
are the end tangents, given as
.
31
,
scholar.waset.org/1999.7/10000225
(14)
and
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:1, 2015
0
The derivatiive vectors
0
and
3
and
3
thhese derivativee vectors are rewritten
r
as
0
3
. Usingg (14),
3
and
.
This gives
and
,
(15)
w
where
International Science Index, Computer and Information Engineering Vol:9, No:1, 2015 waset.org/Publication/10000225
1
2 cos
1
annd
2 cos
.
(16)
is the anggle between 0 and
and is thhe angle
beetween
and
. Now, when
n the input data is
cooming from a circle, then the rational cubic trigonoometric
Bézier curve, with
w control points
p
and weeights definedd in (8),
w reproduce the
will
t circle.
V. RATIONALL TRIGONOMETTRIC CUBIC APPROXIMATIO
V
ON OF
CIRCLE WHEN
N TWO END POINTS
O
AND END
N UNIT TANG
GENTS
AR
RE GIVEN
In this sectioon, the probleem of approxiimation of a circle
c
is
diiscussed whenn its two endd points and end
e unit tangeents,
annd
are giveen. Again thee end points of
o the rationaal cubic
triigonometric Bézier
B
curve are identical to the end pooints of
thhe given circuular arc. Thus we have to fiind the Bézierr points
and and the
t weights
and . Thee unit tangentss and
make the angles and with the base
respeectively.
U
Using
(13), we can find the B
Bézier points and i.e.
,
.
(18)
.
|
In this sectioon, a circular arc is estimaated by the scchemes
developed in Seections IV andd V.
(a)). Minimizatiion of
: In this case the
t values of
and
are obtaained by minim
mizing the inntegral given in
i (18)
for rational cubic trigonoometric Bézier curve (9). The
T end
1,0
1 and
1,0
0 . The
points of thhe circle are taaken as
optimized values of
1.6435an
nd
=1.47322, and
correspondiing to these values we have
h
1,1.6896 ,
1,2.0350
0 ,
0.3946 and
0..3276. The reesulting
graph is cloose to circle with radius error
e
betweenn 0 and
0.0836. Thee approximateed circle is shoown in Fig. 2.
(b)). Minimizatiion of
: In this case the values of
and
are obtaained by minim
mizing the inntegral given in
i (19)
1,0
for (9). Thee end points oof the circle arre taken as
and
1,0 . The optim
mized values of
1.57008 and
=1.5708, and correspoonding to theese values wee have
1,2 ,
1,2 ,
0.333 and
0.3333
3. The
r
error beetween
resulting grraph is close too circle with radius
0 and 9.98880 10 . The
T approximaated circle is shown
in Fig. 3.
(c)). Geometric Approach: When we used
u
the geoometric
t circle is rreproduced wiith
and
d =
approach, the
(See Fig. 4).
4 Corresponnding to thesse values, wee have
and
. In this casse, the
1,2 ,
1,2 ,
rational triggonometric cuurve (9) will reproduce
r
thee circle
(See. Fig. 5) which iss the same as
a obtained by
b the
minimizatioon of
. In Figs. 6 and
a
7 the currvature
derivative plots
p
of (9) arre shown for the
t above menntioned
cases.
(17)
The correspponding weigght functionns
and
are
caalculated by (16). The otherr weights are unity. The anngles
annd are approximated byy minimizing the shape measure
m
quuantities
and
defined in (18) andd (19)
reespectively.
′
VI. NUMER
RICAL EXAMPL
LES
are compuuted as
|
VII. CONCLUSION
In this research paper, we have discusseed the rational cubic
triggonometric teechniques to approximate the
t circle. We
W have
useed approachees, the variatioonal and the geometric onne. The
quuality of thee rational cuubic trigonom
metric probllem is
callculated usingg (20), which is found to be
b
17.6
6099. It
is 375.3 for pollynomial Herm
mite interpolaant it is 375.33. This
r
cubic trigonometricc schemes didd better
shoows that our rational
thaan integral and
a
rational polynomial
p
Hermite
H
interrpolant.
Raadius error is found
f
less thann many existing schemes [11].
(19)
Equation (188) shows thatt the change in
i curvature is
i more
siggnificant than
n its magnitudde. The disconntinuous curvature is
accceptable as loong as the sloppe ′
is conntinuous. The quality
off the develo
oped rationaal cubic trig
gonometric circular
c
appproximation scheme is measured by
, defined byy
,
.
F 2 A rationall cubic trigonom
Fig.
metric approxim
mation of circlee using
(20)
mitations of addmissible values for
Here, , , , are the lim
and . Heree
90 ,
90 and
90 ,
270 i.e.
alll acceptable
have a positive -ccomponent, and
a
all
addmissible haave a negativee -componennt.
International Scholarly and Scientific Research & Innovation 9(1) 2015
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scholar.waset.org/1999.7/10000225
World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering Vol:9, No:1, 2015
[7]
G. Wang, Q. Chenand M. Zhhou, “NUAT B-spline curves,”C
Computer
2 2004, pp. 193--205.
Aided Geomettric Design,vol. 21,
International Science Index, Computer and Information Engineering Vol:9, No:1, 2015 waset.org/Publication/10000225
F 3 A rationaal cubic trigonoometric approxim
Fig.
mation of circlee using
Fig. 4 Tan
ngents direction
n
Fig. 5 The graphical
g
represeentation of circcle using geomeetric
ap
pproach
F 6 Curvaturee derivative plo
Fig.
ot of cubic trigonometric Bézieer using
geomettric approach
Fig. 7 Cu
urvature derivative plot of cubiic trigonometricc
v
apprroach
Bézier using variational
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D
8, 1991, pp. 227-238.
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