Appl. Math. J. Chinese Univ.
2011, 26(2): 198-212
Rational cubic/quartic Said-Ball conics
HU Qian-qian
1,2
WANG Guo-jin1
Abstract. In CAGD, the Said-Ball representation for a polynomial curve has two advantages
over the Bézier representation, since the degrees of Said-Ball basis are distributed in a step type.
One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomial
curve runs twice as fast as the de Casteljau algorithm of Bézier curve. Another is that the
operations of degree elevation and reduction for a polynomial curve in Said-Ball form are simpler
and faster than in Bézier form. However, Said-Ball curve can not exactly represent conics which
are usually used in aircraft and machine element design. To further extend the utilization
of Said-Ball curve, this paper deduces the representation theory of rational cubic and quartic
Said-Ball conics, according to the necessary and sufficient conditions for conic representation in
rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball
basis. The results include the judging method for whether a rational quartic Said-Ball curve is a
conic section and design method for presenting a given conic section in rational quartic Said-Ball
form. Many experimental curves are given for confirming that our approaches are correct and
effective.
§1
Introduction
In 1974, Ball [1-3] introduced a kind of rational cubic curves to be the basis of CONSURF
surface lofting program, such that the traditional manual lofting technics have thoroughly been
reformed. In 1989, Said [12] extended the cubic basis proposed by Ball into arbitrary odd
degree basis to get a type of generalized Ball curves. Goodman and Said [8,9] proved that the
generalized Ball basis is NTP (normalized totally positive) and hence it possesses the same kind
of shape preserving properties as the Bernstein basis. Also its representation for a polynomial
curve is much better suited to degree raising and lowering than the Bézier representation. Later
Hu et al [11] suggested an extension of the generalized Ball basis to arbitrary even degree, and
Received: 2009-11-11.
MR Subject Classification: 16E05, 16E40, 16S37, 16W50.
Keywords: Rational Said-Ball curve, Rational Bézier curve, conics.
Digital Object Identifier(DOI): 10.1007/s11766-011-2417-z.
Supported by the National Natural Science Foundations of China (61070065, 60933007) and the Zhejiang
Provincial Natural Science Foundation of China (Y6090211).
Corresponding author: wanggj@zju.edu.cn(G. J. Wang).
HU Qian-qian, WANG Guo-jin.
Rational cubic/quartic Said-Ball conics
199
uniformly called it Said-Ball basis. They also presented a recursive algorithm for evaluating a
Said-Ball curve which is more efficient than the de Casteljau algorithm for evaluating a Bézier
curve. In 2006 Delgado and Peña proved in [5] the NTP property of the family of Said-Ball
basis. In a word, the research for about twenty years indicated that Said-Ball curve not only
possesses the same ”shape preserving” properties as Bézier curve, but also is better than Bézier
curve in evaluation of polynomial, degree elevation and reduction.
To extend the utilization of Said-Ball curve, Tien et al [15] defined rational Said-Ball curve.
Of course we can affirm that it is more suitable for evaluation of a conic section, degree elevation
or reduction of a rational curve than the Bézier representation. However, up to now, research
on rational conics focused on Bézier representation only. As we all know, conics have gained
widespread applications in CAD/CAM systems. They are an important design tool in the
aircraft industry and all machine manufacture; they are also used in areas such as font design
[7]. There are an amount of publications on the subject of the conditions and algorithms for
rational low degree Bézier representation of conics [6,10,13,16]. People have well known that
for Bézier curves with all positive weights, the largest center angle of a circular arc in cubic
Bézier form is not more than 4π/3 [4,16]; for quartic Bézier form, it is less than 2π [14]; and
the degree to form a full circle must be at least 5 [4]. There is an analogous situation for
elliptic segments, and it is important to study rational quartic Bézier representation of conics.
A special representation for conics in this form which has the same weight for all control points
but the middle one was presented by Fang [6]. Based on the fact that all rational Bézier conics
except for degree two are all degenerate [13], Hu and Wang achieved in [10] the necessary and
sufficient conditions as well as the class conditions for rational quartic Bézier representation
of conics. Considering the two advantages of rational Said-Ball curve and the importance of
rational cubic/quartic conics. In this paper , we study the necessary and sufficient conditions
and corresponding class conditions for rational cubic and quartic Said-Ball representation of
conics. The idea is based on the necessary and sufficient conditions for rational low degree
Bézier representation of conics [10, 17], and the transformation formula from Bernstein basis
to Said-Ball basis. According to the relationship between the areas of some triangles which
are composed of the control points of rational low degree Bézier curve and rational Said-Ball
curve, this paper derives all corresponding positions of the control points and all expressions of
the relations between the weights, of rational quartic Said-Ball conics in five cases. Also two
algorithms are provided to judge whether a rational quartic Said-Ball curve is a conic section,
or design rational quartic Said-Ball representations for a given conic section.
This paper is arranged as follows. In Section 2, we give some preliminary knowledge. The
necessary and sufficient conditions for rational cubic Said-Ball representation of conics are given
in Section 3. Based on the necessary and sufficient conditions for representing conics in rational
quartic Bézier form are described in Section 4, we derive the necessary and sufficient conditions
for rational quartic Said-Ball case in Section 5. And then class condition for rational quartic
Said-Ball conics and two algorithms for designing and judging rational quartic Said-Ball conics
are introduced in Section 6 and Section 7 respectively. Finally several numerical examples are
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presented to validate the effectiveness of the method in Section 8 and Section 9 conclusion are
presented.
§2
Definitions of rational Bézier and rational Said-Ball curves
According to [7] and [15], rational cubic Bézier and rational cubic Said-Ball curves are
defined by
P3
3
i=0 Bi (t)σi Vi
,
(1)
v(t) = P
3
Bi3 (t)σi
i=0
P3
3
i=0 Si (t)τi Ui
u(t) = P
,
(2)
3
3
i=0 Si (t)τi
respectively. Here σi , τi (i = 0, 1, 2, 3) are the associated weights, and Vi , Ui (i = 0, 1, 2, 3) are
the associated control points. For degree three, the Bernstein basis functions are (1 − t)3 , 3(1 −
t)2 t, 3(1 − t)t2 , t3 , and the Said-Ball basis functions are (1 − t)2 , 2(1 − t)2 t, 2(1 − t)t2 , t2 .
Also for degree four, rational Bézier and Said-Ball curves are defined by
P4
4
i=0 Bi (t)ωi Ri
r(t) = P
,
(3)
4
4
i=0 Bi (t)ωi
P4
4
i=0 Si (t)wi Pi
p(t) = P
,
(4)
4
4
i=0 Si (t)wi
respectively. Here ωi , wi (i = 0, ..., 4) are the associated weights, and Ri , Pi (i = 0, ..., 4) are the
associated control points. The Bernstein basis functions are (1 − t)4 , 4(1 − t)3 t, 6(1 − t)2 t2 , 4(1 −
t)t3 , t4 , and the Said-Ball basis functions are (1 − t)3 , 3(1 − t)3 t, 6(1 − t)2 t2 , 3(1 − t)t3 , t3 . To
preserve the convex hull property of these curves, all the weights are set as positive.
§3
Rational cubic Said-Ball representation of conics
By the transformation formula from Bernstein basis to Said-Ball basis [15], the rational
cubic Said-Ball curve (2) is changed to the rational cubic Bézier curve (1). The corresponding
weights satisfy
σ0 =τ0 , 3σ1 =τ0 + 2τ1 , 3σ2 = 2τ2 + τ3 ,σ3 =τ3 ,
(5)
and the endpoints of both curves are identical respectively, the control points V1 , V2 of curve
(1) are the internal points of the line segments U0 U1 and U3 U2 with the internal ratios 2τ1 : τ0 ,
2τ2 : τ3 respectively. (See Figure 1)
Theorem 3.1. A rational cubic Said-Ball curve (2) is a conic section if and only if the following three conditions hold simultaneously:
(a)The control polygon U0 U1 U2 U3 is self-intersecting, or non-convex but not self-intersecting;
T2
T2
(c) τ1τ2τ3 = T1 (2τ2 T32+τ3 T2 ) ,
(b) τ0τ1τ2 = T2 (2τ1 T01+τ0 T1 ) ;
where Ti (i = 0, 1, 2, 3) are the directed areas of ∆U1 U2 U3 , ∆U0 U2 U3 , ∆U0 U1 U3 and ∆U0 U1 U2
respectively (See Figure 1 ). Set U∗ as the point of intersection of lines U0 U1 and U3 U2 . Also
HU Qian-qian, WANG Guo-jin.
Rational cubic/quartic Said-Ball conics
201
set Ŷ,Ỹ satisfy U1 U∗ = Ŷ U0 U1 , U2 U∗ = Ỹ U3 U2 . Let
τ1 τ2
ih
i.
η∗2 = h
τ0 + (2τ1 + τ0 )Ŷ τ3 + (2τ2 + τ3 )Ỹ
Then the class conditions are that curve (2) is an elliptic segment, a parabolic segment, or a
hyperbolic segment respectively when η∗2 < 1, η∗2 = 1 or η∗2 > 1.
U1
U2
T0
T3
T2
V1
U1
U2
T1
V2
V2
V1
S0
S3
S2
V0(U0)
V3(U3)
V0(U0)
S1
V3(U3)
Figure 1: The directed areas Si , Ti (i = 0, 1, 2, 3).
Proof. According to [17], the necessary and sufficient conditions for conics in rational cubic
Bézier form are as follows:
S12
σ22
S22
σ12
=
,
=
,
(6)
σ0 σ2
3S0 S2
σ1 σ 3
3S1 S3
and V0 V1 V2 V3 is a convex polygon without any three vertices collinear. Here Si (i = 0, 1, 2, 3)
are the directed areas of the triangles ∆V1 V2 V3 , ∆V0 V2 V3 , ∆V0 V1 V3 and ∆V0 V1 V2 , respectively (See Figure 1). The key to the problem is to replace Si by Ti (i = 0, 1, 2, 3). Clearly
we have
S2 /T2 = 2τ1 /(2τ1 + τ0 ), S1 /T1 = S0 /S∆V1 U2 U3 = 2τ2 /(2τ2 + τ3 ).
(7)
(S∆V1 U2 U3 − T0 )/(T1 − T0 ) = τ0 /(2τ1 + τ0 ).
Substituting the second term of (7) into the above equation, and eliminating S∆V1 U2 U3 , S0 is
represented by Ti (i = 0, 1, 2, 3) as
2τ2 (2τ1 T0 + τ0 T1 )
.
S0 =
(2τ1 + τ0 )(2τ2 + τ3 )
S3 can be handled in a similar way, hence,
2τ1 (2τ2 T3 + τ3 T2 )
S3 =
.
(2τ1 + τ0 )(2τ2 + τ3 )
Substituting (5), (7) and the two above formulae into (6), and eliminating σi , Si (i =
0, 1, 2, 3), conditions (b) and (c) are proven. In addition, for rational cubic Bézier conics, the
condition ’V0 V1 V2 V3 is a convex polygon without any three vertices collinear’ is equivalent
to Si (i = 0, 1, 2, 3) having the same strict sign. So according to (7), we know that T1 ·T2 > 0.
Multiplying (b) by τ3 and (c) by τ0 gives equivalent equations
τ1 τ3
τ3 T12
T1 (2τ2 T3 + τ3 T2 )
,
=
=
τ0 τ 2
T2 (2τ1 T0 + τ0 T1 )
τ0 T22
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Vol. 26, No. 2
and the second equation is equivalent to
2τ1 τ2 T0 T3 + τ0 τ2 T1 T3 + τ1 τ3 T0 T2 = 0·
So we have T0 ·T3 < 0; or T0 ·T3 > 0, T0 ·T1 < 0. Without loss of generality, we assume T1 >
0, T2 > 0. Then it is either T0 < 0, T1 > 0, T2 > 0, T3 < 0 or T0 ·T3 < 0, T1 > 0, T2 > 0.
In other words, the control polygon U0 U1 U2 U3 is self-intersecting, or non-convex but not
self-intersecting. Then (a) is proved.
Next we discuss the class conditions for rational cubic Said-Ball conics. On one hand, the
class conditions for rational cubic Bézier conics are that curve (1) is an elliptic segment, a
parabolic segment, or a hyperbolic segment respectively when η∗2 < 1, η∗2 = 1 or η∗2 > 1, where
1
η∗2 =
.
(8)
V1 U∗ V2 U∗
4 · V0 V1 · V3 V2
On the other hand, note that
V1 U∗ = V1 U1 + U1 U∗ ,
and V1 is the internal points of the line segment U0 U1 with the internal ratio 2τ1 : τ0 , we have
τ0
2τ1
V1 U∗ =
U0 U1 + U1 U∗ , V0 V1 =
U0 U1 .
2τ1 + τ0
2τ1 + τ0
Dividing V1 U∗ by V0 V, it hold
V1 U∗
τ0 + (2τ1 + τ0 )Ŷ
=
.
V0 V1
2τ1
By the same reasoning, we have
V2 U∗
τ3 + (2τ2 + τ3 )Ỹ
=
.
V3 V2
2τ2
Inserting the above two formulae into (8) yields the class conditions. Then the theorem is
proven.
Remark 3.1. Rational cubic Said-Ball curve (2) can represent only minor arcs of ellipses, so
all rational cubic Said-Ball curves are obtained by degree elevation of rational quadratic form.
It is different from rational cubic Bézier case.
§4
Rational quartic Bézier representation of conics
According to [10], the necessary and sufficient conditions for rational quartic Bézier representation of conics are presented in the five cases as follows.
Case I, III: When the control points R1 , R2 , R3 are on the same side of the line R0 R4
(Figure 2(1), 2(2)) or R1 , R3 and R2 are on the both sides of the line R0 R4 respectively
(Figure 2(3)), the control points and weights satisfy
ω12
3 B2 B3
ω22
4 B3 D0
4 B1 D2
ω32
3 B1 B2
=
,
=
=
,
=
;
ω0 ω2
8 B1 D0 ω1 ω3
9 B2 D1
9 B2 D3 ω2 ω4
8 B3 D2
ω32
ω12
3 B3 D1
3 B32 D3
ω22
2
3 B1 D3
3 B12 D1
¶,
=
=
,
= µq
=
=
;
2
2
ω0 ω2
2 D0
2 B1 D0 D2 ω1 ω3
ω2 ω4
2 D2
2 B3 D0 D2
B22 D1 D3
2D1 D3
−
9
B 1 B 3 D0 D2
D0 D2
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Rational cubic/quartic Said-Ball conics
Case II: When R2 is the internal point of the line segment R0 R4 , and R1 coincides with
R3 (Figure 2(4)), the control points and weights satisfy
Ãr
!
r
r
D3
D1
D3
1
ω1
ω2 =
+
=
;
,
6
D1
D3
ω3
D1
Case IV: When R0 coincides with R1 and R2 is the internal point of the line segment R1 R3
(Figure 2(5)), the control points and weights satisfy
µ
¶
2
B2
ω3
2B2
ω2
=
1+
,
=
;
2
3
ω1
3
D1
ω1
D1
Case V: When R3 coincides with R4 and R2 is the internal point of the line segment R1 R3
(Figure 2(6)), the control points and weights satisfy
µ
¶
ω2
2
B2
ω1
2B2
=
1+
,
=
.
2
3
ω3
3
D3
ω3
D3
Here Bi are the directed areas of ∆R0 Ri R4 (i = 1, 2, 3), and Dj (j = 0, ..., 3) are the directed
areas of ∆R1 R3 R4 , ∆R2 R3 R4 , ∆R0 R1 R3 , and ∆R0 R1 R2 respectively (See Figure 2).
R
1
R3
R2
R3
R
R1
2
R4
R
0
R2
R
2
R0
R4
R1
(1)
R (R )
1
R3
R2
R2
R0
(2)
R3
3
R2
(3)
R4
R
1
R2
R2
R0
R2
(4)
R4
R4
R0(R1)
(5)
R3(R4)
R0
(6)
Figure 2: The control polygon of a rational quartic Bézier conic section in six different forms.
§5
Conics in rational quartic Said-Ball form by basis transformation
By the transformation formula from Bernstein basis to Said-Ball basis [15], the rational
quartic Said-Ball curve shown as (4) is changed to the rational quartic Bézier curve (3), and
the corresponding control points and weights satisfy
ω0 =w0 , 4ω1 =w0 + 3w1 , ω2 =w2 , 4ω3 = 3w3 + w4 , ω4 =w4 ;
P3 +w4 P4
0 +3w1 P1
Ri =Pi (i= 0, 2, 4), R1 = w0 P
, R3 = 3w33w
.
w0 +3w1
3 +w4
(9)
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Appl. Math. J. Chinese Univ.
Vol. 26, No. 2
The necessary and sufficient conditions for rational quartic Bézier representation of conics
are divided into two parts: control points and weights. We discuss the two parts as follows.
5.1
Conditions about weights
According to Section 4, the necessary and sufficient conditions are only related with the
weights and the areas Bi , Dj (i = 1, 2, 3, j = 0, ..., 3). Since (9) provides the relationship
between the weights of the curves (3) and (4), then the key to the problem is to replace
Bi , Dj (i = 1, 2, 3, j = 0, ..., 3) by Ai , Cj (i = 1, 2, 3, j = 0, ..., 3). Here Ai are the directed areas
of ∆P0 Pi P4 (i = 1, 2, 3), and Cj (j = 0, ..., 3) are the directed areas of ∆P1 P3 P4 , ∆P2 P3 P4 ,
∆P0 P1 P3 , and ∆P0 P1 P2 respectively (See Figure 3).
R2(P2)
R2(P2)
R2(P2)
P3
P1
P
1
0
A
R
1
R1
B1
R0(P0)
3
P
C
3
P
C2
A3
R3
R
1
D
0
D2
C1
C3
1
A2(B2)
R D
1
3
P
3
D1
R3
B
3
R4(P4)
R0(P0)
R4(P4)
R (P )
0
0
R (P )
4
4
Figure 3: The directed areas Ai , Bi (i = 1, 2, 3) and Ci , Di (i = 0, 1, 2, 3).
By (9) we know that the control points R1 , R3 of the curve (3) are the internal points of
the line segments P1 P0 and P3 P4 with internal ratios w0 : 3w1 , w4 : 3w3 respectively, and
R2 = P2 . Therefore clearly we have
3w1
3w3
3w3
3w1
B1 =
A1 ,B2 =A2 ,B3 =
A3 ,D1 =
C1 ,D3 =
C3 .
(10)
w0 + 3w1
3w3 + w4
3w3 + w4
w0 + 3w1
Obviously the key to the problem is to find the relationship between D0 , D2 and Ai , Cj (i =
1, 2, 3, j = 0, ..., 3). According to Section 4, the necessary and sufficient conditions for rational
quartic Bézier conics are related with D0 , D2 only when the control polygon is shown as in
Figure 2(1)-(3). In either case, according to (9), it satisfies
D0 :S∆R1 P3 P4 = 3w3 : (3w3 + w4 ).
(11)
In the case of Figure 2(1)-(3), according to the relation of the control points of the curves
(3) and (4) shown as in Figure 3 or the left of Figure 4, there is
(A3 − S∆R1 P3 P4 ) : (A3 − C0 ) = 3w1 : (w0 + 3w1 ),
that is
w0 A3 + 3w1 C0
.
w0 + 3w1
Substituting the above formula into (11), D0 is represented by A3 and C0 as
3w3 (w0 A3 +3w1 C0 )
D0 =
.
(w0 + 3w1 )(3w3 + w4 )
S∆R1 P3 P4 =
(12)
HU Qian-qian, WANG Guo-jin.
Rational cubic/quartic Said-Ball conics
P1
P3
R3
P3
205
P1
R1
R1(R3)
R0(P0)
R4(P4)
R0(P0)
R2(P2)
R4(P4)
R2(P2)
Figure 4: The control polygons of curves (1) and (2) under the case of Figure 2(3)(L) and
Figure 2(4)(R).
In a similar way, D2 is represented by A1 and C2 as
3w1 (3w3 C2 +w4 A1 )
D2 =
.
(w0 + 3w1 )(3w3 + w4 )
5.2
(13)
Conditions about control points
Figure 2 illustrates the distribution types of the control points of the rational quartic Bézier
conics. Since the position relation of control points of cases I and III are discussed in the above
subsection, next we only need to discuss cases II, IV and V.
For case II, to meet the point R1 coinciding with R3 , i.e., the lines P1 P0 and P3 P4 intersect
at R1 (See the right of Figure 4), if and only if the expressions
A1
w0 + 3w1
A3
3w3 + w4
=
,
=
,
B1
3w1
B1
3w3
hold. Then eliminating B1 , we have
A1
w3 (w0 + 3w1 )
.
(14)
=
A3
w1 (3w3 + w4 )
For case IV, to meet the point R0 coinciding with R1 , and R2 being the internal point of
the line segment R1 R3 , according to (9), if and only if P0 coincides with P1 , and P2 is inside
the triangle ∆P0 P3 P4 . And case V can be treated analogously.
Finally substituting (10), (12), and (13) into the necessary and sufficient conditions for
rational quartic Bézier representation of conics in Section 4 and adding (14) yields
Theorem 5.1. Suppose a rational quartic Said-Ball curve is expressed as (4), and Q is the
point of intersection of the two end tangent lines of the curve. Then the necessary and sufficient
conditions for the curve (4) being a conic section are one of the following conditions of the
control points and weights satisfying
I) (1a) Pi (i = 0, 1, ..., 4) are coplanar;
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Appl. Math. J. Chinese Univ.
Vol. 26, No. 2
(1b) The positions of the three points P1 , P2 , P3 are determined by one of the following three
conditions:
(1b1 ) The points P1 , P3 are the internal points of division of the directed line segments
P0 Q, P4 Q or on their extension lines respectively, P2 is inside the triangle ∆P0 QP4 ;
(1b2 ) The points P1 , P3 are on the extension lines of the directed line segments QP0 , QP4
respectively, P2 is in the domain determined by the extension lines of the directed line segments
QP0 , QP4 and the line segment P0 P4 ;
(1b3 ) The points P1 , P3 are on the extension lines of the directed line segments P0 Q, P4 Q
respectively, P2 is in the domain determined by the extension lines of the directed line segments
QP0 , QP4 and the line segment P0 P4 ;
(1c) ww0 w1 2 =
2A2 A3
A1 (w0 A3 +3w1 C0 ) ;
2A1 A2
A3 (3w3 C2 +w4 A1 ) .
(1d)12w22 =
w3 A3 (w0 A3 +3w1 C0 )
A2 C 1
=
w1 A1 (3w3 C2 +w4 A1 )
;
A2 C 3
(1e) ww2 w3 4 =
II) (2a)-(2b) Being the same as (1a)-(1b);
0 w 2 A3
(2c) w024w
A3 +3w1 C0 =
w0 A3 +3w1 C0
C1
3w3 C2 +w4 A1
C3
=
w1 A1 (3w3 C2 +w4 A1 )
;
w 3 A3 C 3
w3 A3 (w0 A3 +3w1 C0 )
;
w 1 A1 C 1
2 w 4 A1
(2d) 3w24w
=
=
3 C2 +w4 A1
q
A22 C1 C3
1
(2e) 24w2 = w1 w3 A1 A3 (w0 A3 +3w1 C0 )(3w3 C2 +w4 A1 ) −
2
III) (3a) Being the same as (1a);
6C1 C3
(w0 A3 +3w1 C0 )(3w3 C2 +w4 A1 ) .
(3b) P2 is the internal point of division of the line segment P0 P4 , and the line segments
P0 P1 and P3 P³q
4 cut cross each other;q
´
3w3 +w4 w1 C3
w0 +3w1 w3 C1
1
(3c)w2 = 6
w0 +3w1 · w3 · C1 +
3w3 +w4 · w1 · C3 ;
3
w3 (w0 +3w1 )
3 (w0 +3w1 )
3
=C
(3d) w
C1 ; (3e) w1 (3w3 +w4 ) =
w1 (3w3 +w4 )3
IV) (4a) Being the same as (1a);
A1
A3 .
(4b) P0 coincides
with P1 , ´
and P2 is inside the triangle ∆P0 P3 P4 ;
³
3w3
24w2
24w3
A2
2
(4c) 3w3 +w4 (w +3w )2 − 1 = A
C1 ; (4d) (w0 +3w1 )3 = C1 .
0
1
V) (5a) Being the same as (1a);
(5b) P3 coincides
with P4 ,´and P2 is inside the triangle ∆P0 P1 P4 ;
³
3w1
24w2
24w1
A2
2
(5c) w0 +3w1 (3w +w )2 − 1 = A
C3 ; (5d) (3w +w )3 = C3 .
3
§6
4
3
4
Class conditions of rational quartic Said-Ball conics
The class conditions for rational quartic Bézier conics are as follows [10]: the curve (4) is an
elliptic segment, a parabolic segment, or a hyperbolic segment respectively when η 2 < 1, η 2 = 1
or η 2 > 1 respectively , where

±
±

B2 B3 (4E2 · S∆R0 QR3 ) = B1 B2 (4E0 · S∆R1 QR4 ), for (I)




for (II)

 4ω1 ω3±,
2
η =
(15)
B1 B3 (S∆R1 QR4 · S∆R0 QR3 ),
for (III)



ω3 /ω1 ,
for (IV)




ω1 /ω3 ,
for (V)
HU Qian-qian, WANG Guo-jin.
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Rational cubic/quartic Said-Ball conics
Here Ei (i = 0, 1, 2, 3) are the directed areas of ∆P0 QP2 , ∆P1 QP4 , ∆P2 QP4 and ∆P0 QP3
respectively (See Figure 5).
Q
Q
E0
E2
P1
P2(R2)
P0(R0)
Q
E1
E3
R1
P4(R4)
P0(R0)
P3
R3
P4(R4)
P0(R0)
P4(R4)
Figure 5: The directed areas of Ei (i = 0, 1, 2, 3).
In order to obtain the class conditions for conics in rational quartic Said-Ball form, we need
to replace the areas of ∆R1 QR4 , ∆R0 QR3 , and Bi (i = 1, 2, 3) in (15) by Ei (i = 1, 3) and
Ai (i = 1, 2, 3), and the weights ωi (i = 1, 3) by wi (i = 0, 1, ..., 4).
According to Figure 5, it follows that
S∆R1 QR4 + B1 = A1 + E1 , S∆R0 QR3 + B3 = A3 + E3 .
Substituting the first and third terms in (8) into above two equations respectively yields
w0 A1
w4 A3
S∆R1 QR4 =
+ E1 , S∆R0 QR3 =
+ E3 .
w0 + 3w1
3w3 + w4
Substituting (9), (10) and the above formulae into (15), we directly deduce the following
Theorem 6.1. Suppose the rational quartic Said-Ball curve (4) is a conic section. Let

3w3 A2 A3
3w1 A1 A2

(I)

4E2 [3w3 E3 +w4 (A3 +E3 )] = 4E0 [w0 (A1 +E1 )+3w1 E1 ]


9w
w
A
A

1 3 1 3

,
(II)
 [w0 (A1 +E1 )+3w1 E1 ][3w3 E3 +w4 (A3 +E3 )]
2
η =
(w0 + 3w1 )(3w3 + w4 )/4,
(III) ,




(3w3 + w4 )/(w0 + 3w1 ),
(IV)



(w0 + 3w1 )/(3w3 + w4 ),
(V)
(16)
where Ai (i = 1, 2, 3), Ei (i = 1, 3) are defined as in Figure 3 and Figure 5 respectively. Then
when η 2 < 1, η 2 = 1 or η 2 > 1, the curve (4) is an elliptic segment, a parabolic segment, or a
hyperbolic segment respectively.
§7
Judgment and design of conics in rational quartic Said-Ball form
Algorithm 7.1. (Judging whether a rational quartic Said- Ball curve is a conic section)
Given a rational quartic Said-Ball curve (4) with control points Pi and weights wi (i =
0, 1, ..., 4).
Step1: If Pi (i = 0, 1, ..., 4) satisfy (1a), jump to Step 2, else return ”No”.
Step2: If there are at least two coincident points, then judge whether the control points and
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Appl. Math. J. Chinese Univ.
Vol. 26, No. 2
weights satisfy (4c)-(4d) or (5c)-(5d).
if (4c)-(4d) hold, return ”Yes, case (IV)”,
elseif (5c)-(5d) hold, return ”Yes, case (V)”, else return ”No”.
If Pi (i = 0, 1, ..., 4) are different to each other:
if P2 is located on the line P0 P4 , judge whether conditions (3c)-(3e) hold.
if (3c)-(3e) hold, return ”Yes, case (III)”, else return ”No”.
if P2 is not located on the line P0 P4 , judge whether conditions (1c)-(1e) or (2c)-(2e) hold.
if (1c)-(1e) hold, return ”Yes, case (I)”,
elseif (2c)-(2e) hold, return ”Yes, case (II)”,
else return ”No”.
Step3: If curve (4) is a conic section, calculate η 2 by Eq.(16) under five different cases. If
η 2 < 1, the curve is an elliptic segment; if η 2 = 1, the curve is a parabolic segment; if η 2 > 1,
the curve is a hyperbolic segment.
Algorithm 7.2. (Designing a given conic section in rational quartic Said-Ball form)
Given a conic section.
Step1: If the two end tangent lines are not parallel, then calculate its corresponding control
points Qi (i = 0, 1, 2) and weights 1, u1 , u2 in rational quadratic Said-Ball form(also in rational
quadratic Bézier form), and then jump to Step 2, else jump to Step 3.
Step2: Input the type of the quartic curve we need: (I)-(V). If the type is one of (II), (IV)
±√
or (V), change the weights 1, u1 , u2 to {1,u1 u2 , 1} by a linear parameter transformation
±
√
t = s [s + u2 (1 − s)], and denote the new middle weight as u1 .
If the type is (I),

−1)Q0 +2u1 Q1
1 −1
w = 2a1 +2u
, P1 = (2a12a
,


3
1 +2u1 −1
 1
2
Q0 +4a1 u1 u2 Q1 +u22 Q2
1+4a1 u1 u2 +u2
, P2 =
w2 =
,
6u2
1+4a1 u1 u2 +u22

−1
−1

1 +(2a1 u2 −1)Q2
 w3 = 2u1 u2 +2a1 u2 −1 , P3 = 2u1 u2 Q−1
.
3
2u1 u2 +2a1 u2 −1
a1 > max{0, 1/2 − u1 , 1/(2u2 ) − u1 u−2
2 }, and the position of the control
−2
choose −min{u1 − 1/2, u1 u2 − 1/(2u2 ), (1 + u22 )/(4u1 u2 )} < a1 < 0, and
If u1 > 0, choose
points is as (1b1 );
the position of the control points is as (1b3 ).
2
If u1 < 0,choose −min{u1 − 1/2, u1 u−2
2 − 1/(2u2 )} < a1 < −(1 + u2 )/(4u1 u2 ), and the position
of the control points is as (1b2 ).
If the type is (II), choose a1 , b1 such that the weights are positive.

4b0 (b1 +a1 u1 )−1
b1 −1)Q0 +4a1 b0 u1 Q1
, P1 = (4b04b

 w1 = 2 23
0 b1 −1+4a1 b0 u1
2b +2a +(4a b +b )u
2b2 Q0 +(4a1 b1 +b0 )u1 Q1 +2a21 Q2
w2 = 1 1 3 1 1 0 1 , P2 = 1 2b
2 +(4a b +b )u +2a2
1 1
0
1
1
1


4b1 u1 Q1 +(4a1 −1)Q2
4(a1 +b1 u1 )−1
,
P
=
w3 =
3
3
4b1 u1 +4a1 −1
If u1 > 0 and b0 = 1, the position of the control points is as (2b1 ); if u1 > 0 and b0 = −1, the
position of the control points is as (2b3 ); if u1 < 0, then b0 = 1, and the position of the control
points is as (2b2 ).
If the type is (III), then choose u1 > 1/2max{1, u2 }, the weights are 1, (2u1 − 1)/3, (1 +
HU Qian-qian, WANG Guo-jin.
Rational cubic/quartic Said-Ball conics
209
u22 )/(6u2 ), (2u1 /u2 − 1)/3, 1, and the control points are
−Q0 + 2u1 Q1 Q0 + u22 Q2 2u1 Q1 − u2 Q2
Q0 ,
,
,
,Q2 .
2u1 − 1
1 + u22
2u1 − u2
If the type is (IV), then choose b1 > 1/4max{1, 1/u1 }, the weights are 1, (4b1 − 1)/3, (2b21 +
u1 )/3, (4b1 u1 − 1)/3, 1, and the control points are
2b2 Q0 + u1 Q1 4b1 u1 Q1 − Q2
Q0 ,Q0 , 1 2
,
,Q2 .
2b1 + u1
4b1 u1 − 1
If the type is (V), then choose a1 > 1/4max{1, 1/u1 }, the weights are 1, (4a1 u1 −1)/3, (2a21 +
u1 )/3, (4a1 − 1)/3, 1, and the control points are
−Q0 + 4a1 u1 Q1 u1 Q0 + 2a21 Q1
Q0 ,
,
,Q2 ,Q2 .
4a1 u1 − 1
u1 + 2a21
Step3: In this case, the curve is a semi-circle or semi -ellipse. Suppose the implicit function
of the curve is x2 /a2 + y 2 /b2 = 1, whose two end parametric angles are θ, π + θ respectively.
Input the type of the quartic curve we need: (I) or (II).
For case (I), choose u2 > 0, a1 > 1/2max{1, 1/u2 }, then the weights are 1, (2a1 − 1)/3, (1 +
u22 )/(6u2 ), (2a1 u2 − 1)/3, 1 and the control points are
´ 
´ 
 ³
 ³
"
#
3/2
1/2
a (1 − u22 ) cos θ − 4a1 u2 sin θ
a (2a1 − 1) cos θ − 2u2 sin θ
a cos θ
´ ,
´ , 1 2  ³
, 2a11−1  ³
1+u2
3/2
1/2
b sin θ
b (1 − u22 ) sin θ + 4a1 u2 cos θ
b (2a1 − 1) sin θ + 2u2 cos θ
´  "
 ³
#
−1/2
a (2a1 u2 − 1) cos θ + 2u2
sin θ
−1
´  , −a cos θ .
 ³
2a1 u2 −1
−1/2
−b sin θ
b (2a1 u2 − 1) sin θ − 2u2
cos θ
For case (II), choose a1 , b1 > 1/2, then the weights are 1, (4b1 − 1)/3, 2(a21 + b21 )/3, (4a1 −
1)/3, 1, and the control points are
"
#
" ¡
"
#
¢ #
a cos θ
a 2(b21 − a21 ) cos θ − (4a1 b1 + 1) sin θ
a ((4b1 − 1) cos θ − 4a1 sin θ)
1
1
¡
¢ ,
, 4b1 −1
, 2a2 +2b2
1
1
b sin θ
b ((4b1 − 1) sin θ + 4a1 cos θ)
b 2(b21 − a21 ) sin θ + (4a1 b1 + 1) cos θ
"
# "
#
a ((4a1 − 1) cos θ + 4b1 sin θ)
−a cos θ
−1
,
.
4a1 −1
b ((4a1 − 1) sin θ − 4b1 cos θ)
−b sin θ
§8
Numerical examples
Example 8.1. Given a rational quartic Said-Ball curve, judge whether it is a conic section by
Algorithm 7.1.
1) The control points: (−1, 0), (−4, −1.6), (−0.8333, −1.3333), (2, −1.6), (1, 0),
the weights: 1, 0.3333, 0.5, 0.3333, 1;
2) The control points: (−1, 0), (2.3333, 1.7778), (−0.6957, −0.0696), (0.3220, 1.0847), (1, 0),
the weights: 1, 0.3, 0.3833, 0.9833, 1.
For case 1), since P2 is not located on the line segment P0 P4 , we need to confirm whether the
control points and weights satisfy the conditions (1c)-(1e) or (2c)-(2e). By a simple calculation,
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Appl. Math. J. Chinese Univ.
P0
Vol. 26, No. 2
P4
P1
P3
P2
P1
P3
P0
P2
P4
Figure 6: A rational quartic Said-Ball representation for a conic section.
the directed areas are
A1 = 1.2857, A2 = 1.2706, A3 = 1, C0 = 0.1429, C1 = 0.0471, C2 = 0.4286, C3 = 0.1361.
Obviously, (1c)-(1e) hold, i.e., this curve is a conic section. Further it is easy to obtain the
coordinates of the point of intersection of the end-sides of the control polygon are (0.5, 1.5),
and E0 = 0.1588, E1 = 0.2143, E2 = 0.0706, E3 = 0.5. By Theorem 6.1, it follows η 2 = 6, i.e.,
it is a hyperbolic segment. (Figure 6(1))
For case 2), since P2 is not located on the line segment P0 P4 , we need to confirm whether the
control points and weights satisfy the conditions (1c)-(1e) or (2c)-(2e). By a simple calculation,
the directed areas are
A1 = A3 = −2.4, A2 = −2, C0 = C2 = −7.2, C1 = C3 = −3.2.
Obviously, (2c)-(2e) hold, i.e., this curve is a conic section. Further it is easy to obtain the
coordinates of the point of intersection of the end-sides of the control polygon are (0.5, 1.2),
and E0 = E2 = 1.6, E1 = E3 = 3.6. By Theorem 6.1, it follows η 2 = 0.25, i.e., it is a hyperbolic
segment. (Figure 6(2))
Example 8.2. Given a parabolic segment in rational quadratic Said-Ball form with control
points (−1, 0), (−0.2, 0.8), (1, 0) and weights 1, 2, 4, represent it in rational quartic form.
By Algorithm 7.2, if the curve belongs to the condition (I) and set a1 = 1, then its weights
are 1, 1.6667, 2.0417, 2.6667, 1 and the control points are (−1, 0), (−0.36, 0.64), (0.1755, 0.5224),
(0.85, 0.1), (1, 0) (Figure 7(1)). If the curve belongs to the condition (IV) and set b1 =
1.5, then the weights are 1, 1.6667, 1.8333, 2.6667, 1 and the control points are (−1, 0), (−1, 0),
(−0.8545, 0.1455), (−0.44, 0.96), (1, 0)(Figure 7(2)).
Example 8.3. Given an elliptic segment satisfying x2 /9+y 2 /16 = 1, whose two end parameter
angles are π/3, 4π/3 respectively, represent its rational quartic Said-Ball form.
Obviously this curve is a semi-ellipse. By Algorithm 7.2, we choose the type of this curve
is (I), and set a1 = 2, u2 = 0.5. Then the weights are 1, 1, 0.4167, 0.3333, 1, and the control points are (1.5, 3.4641), (0.2753, 4.4069), (−4.9788, 6.6039), (−8.8485, 2.1928), (−1.5, 3.4641)
(Figure 8(1)). If the curve belongs to the condition (II), and set a1 = 2, b1 = 3, then the
HU Qian-qian, WANG Guo-jin.
211
Rational cubic/quartic Said-Ball conics
P3
P1
P2
P2
P3
P0
P4
P4
P0(P1)
Figure 7: A rational quartic Said-Ball representation for a conic section, (1): condition (I), (2):
condition (IV).
weights are 1, 3.6667, 8.6667, 2.3333, 1 and the control points are (1.5, 3.4641), (−0.3895, 4.9186),
(−1.9212, 3.2554), (−5.9538, −0.0355), (−1.5, 3.4641). (Figure 8(2))
P3
P2
P2
P4
P3
P4
P1
P1
P0
P0
Figure 8: A rational quartic Said-Ball representation for a semi ellipse, (1): condition (I), (2):
condition(II).
§9
Concluding remarks
We have presented the necessary and sufficient conditions for rational cubic and quartic SaidBall representation of conics in this paper. These conditions are divided into two categories:
Said-Ball weights and control points. Further more, two algorithms are provided to design and
judge rational quartic Said-Ball conics. One is to judge whether a rational quartic Said-Ball
curve is a conic section; another is to present positions of the control points and values of the
weights of the conic section in rational quartic Said-Ball form. These results have potential
valuable application benefits to computer graphics and geometric modeling.
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Appl. Math. J. Chinese Univ.
Vol. 26, No. 2
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1
Department of Mathematics, Zhejiang University, Hangzhou 310027.
College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018.
Email: qianqianhu@hotmail.com , wanggj@zju.edu.cn.
2