Journal of Information & Computational Science 6: 6 (2009) 2423–2432
Available at http://www.joics.com
Geometric Parameters Based Cubic PH Curve
Interpolation and Approximation ★
Min Cheng a,b , Guojin Wang c,∗
a Department
b College
of mathematics, Zhejiang University, Hangzhou, Zhejiang, P. R. China
of Science, Zhejiang University of Technology, Hangzhou, Zhejiang, P. R. China
c Institute
of Computer Graphics and Image Processing, Zhejiang University
Hangzhou, Zhejiang, P. R. China
Abstract
The Pythagorean-hodograph curves offer unique computational advantages in computer aided design
and manufacturing. In this paper, geometric parameters based cubic PH curve interpolation and
approximation algorithms are presented. The essential geometric parameters include the ratio 𝜌 of
the adjacent control polygon legs, the control polygon angle 𝜃, the first control polygon leg 𝐿, the
included angle 𝛿 between the first control polygon edge and the edge joining the origin O and the
first control point, as well as the coefficient Dir of curve rotation. Furthermore the problem of PH curve
approximation with endpoints interpolation to a cubic non-PH curve is studied, including the algorithms
based on different geometric parameters input {𝛿, 𝜃} , {𝜌, 𝜃} and {𝜌, 𝛿} respectively.
Keywords: Geometric Parameter; Non-PH Curves; PH Curve Approximation
1
Introduction
In the year 1990, Farouki and Sakkalis [1] introduced a class of special planar polynomial curves
called Pythagorean hodograph (PH) curves r (𝑡) = (𝑥(𝑡), 𝑦(𝑡)), whose hodograph (derivative)
components satisfy the Pythagorean conditions: 𝑥′ (𝑡) = 𝑤(𝑡)[𝑢2 (𝑡) − 𝑣 2 (𝑡)], 𝑦”(𝑡) = 2𝑤(𝑡)𝑢(𝑡)𝑣(𝑡),
where 𝑢(𝑡), 𝑣(𝑡), 𝑤(𝑡) are polynomials satisfying 𝐺𝐶𝐷(𝑢(𝑡), 𝑣(𝑡)) = 1 and 𝑚𝑎𝑥(𝑑𝑒𝑔(𝑢(𝑡), 𝑑𝑒𝑔(𝑣(𝑡)))
≥ 1. Thus the PH curve has polynomial parametric speed 𝜎(𝑡) satisfying 𝜎 2 (𝑡) = (𝑥′ (𝑡))2 +(𝑦 ′ (𝑡))2 .
More researches on PH curves have been done later by Farouki et al. [2]∼[6]. L¨
𝑢 derives
offset-rational condition [7] and Pottmann achieves the rational surface with rational offsets [8].
Meek and Walton create the explicit formulae, which give the unique PH cubic that solves a 𝐺𝐶1
Hermite interpolation problem [9]. For single quintic PH curve segments, end-point interpolation
★
Project supported by the National Basic Research Program of China (No. 2004CB719400), National Natural
Science Foundation of China (No. 60673031 and 60333010), the Scientific Research Program of the Education Office
of Zhejiang Province (No. 20070309) and the Natural Science Foundation of Zhejiang Province (No.Y107311).
∗
Corresponding author.
Email address: wanggj@zju.edu.cn (Guojin Wang).
c 2009 Binary Information Press
1548–7741/ Copyright ⃝
December 2009
2424 M. Cheng et al. /Journal of Information & Computational Science 6: 6 (2009) 2423–2432
of Hermite data has been discussed by Farouki etc. [10] and Moon et al. [11]. J¨
𝑢ttler extends this
problem to the case of degree seven [12]. For PH splines interpolating a sequence of points, planar
cases as well as spatial cases are researched by Farouki et al. [13], [14]. The inner characteristics
of a curve determine the intrinsic shape of the curve [1], and the intrinsic shape of a curve
intuitively represents the demand of the customer’s target design. Thus geometric parameters
based PH curve construction can appropriately match the practical requirements of customers.
Taking the cubic PH curves which are most commonly used in design as an example, a system
of geometric parameters based cubic PH curve interpolation and approximation algorithms are
proposed in this paper. Here several basic geometric parameters are taken into use, including the
ratio 𝜌 of the adjacent control polygon legs, the control polygon angle 𝜃, the first control polygon
leg 𝐿, the included angle 𝛿 between the first control polygon edge and the edge joining the origin
O and the first control point, as well as the coefficient Dir of curve rotation.
The PH curve construction algorithms in this paper include end-points interpolation and approximation. First for the problem of end points interpolation of a cubic PH curve, the condition
equations represented by geometric parameters in terms of Bernstein-Bézier forms are presented.
Next the problem of PH curve approximation with endpoints interpolation to a cubic non-PH
curve is studied, including the algorithms based on three different geometric parameters {𝛿, 𝜃}
, {𝜌, 𝜃} and {𝜌, 𝛿} respectively. The corresponding error bounds are also obtained. The idea
can also be extended to the PH curve interpolation and approximation problem of higher degree
curves.
2
Basic Properties of Cubic PH Curves
With 𝜆 = 𝑑𝑒𝑔(𝑤(𝑡)) = 0, 𝜇 = 𝑚𝑎𝑥{𝑑𝑒𝑔(𝑢(𝑡)), 𝑑𝑒𝑔(𝑣(𝑡))} = 1, cubic PH curves can be constructed
as follows [1].
Lemma 1 Choosing 𝑤(𝑡) = 1, 𝑢(𝑡) = 𝑢0 𝐵01 (𝑡) + 𝑢1 𝐵11 (𝑡), 𝑣(𝑡) = 𝑣0 𝐵01 (𝑡) + 𝑣1 𝐵11 (𝑡), the control
3
∑
points in Bézier form of the cubic PH curve P(𝑡) =
𝐵𝑖3 (𝑡)P 𝑖 are:
{
𝑖=0
P 0 = (𝑥0 , 𝑦0 ), arbitrarily chosen; P 1 = P 0 + (𝑢20 − 𝑣02 , 2𝑢0 𝑣0 )/3;
P 2 = P 1 + (𝑢0 𝑢1 − 𝑣0 𝑣1 , 𝑢0 𝑣1 − 𝑢1 𝑣0 ); P 3 = P 2 + (𝑢21 − 𝑣12 , 2𝑢1 𝑣1 )/3;
(1)
Cubic PH curves also have the following three geometric properties.
Lemma 2 A PH curve of degree 𝑛 has 𝑛 + 3 freedoms. Particularly a cubic PH curve enjoys 6
freedoms.
Lemma 3 Planar cubic PH curves have no real inflection points.
Lemma 4 For a planar cubic Bézier curve P(𝑡) with control points {P 𝑖 }3𝑖=0 , let 𝐿𝑗 )(𝑗 = 1, 2, 3)
be the lengths of the control polygon legs, and let 𝜃1 , 𝜃2 be the control polygon angles at the interior
vertices P 1 , P 2 . Then the conditions
√
(2)
𝐿2 = 𝐿1 𝐿2 , 𝜃1 = 𝜃2
are sufficient and necessary to ensure that P(𝑡) has a Pythagorean hodograph.
M. Cheng et al. /Journal of Information & Computational Science 6: 6 (2009) 2423–2432 2425
Proofs for Lemma 1 ∼ 4 can be found in the reference [1]. According to Lemma 1 and Lemma 2,
it is obvious that a cubic PH curve can be uniquely determined by selecting {𝑥0 , 𝑦0 , 𝑢0 , 𝑣0 , 𝑢1 , 𝑣1 }.
The traditional construction techniques are usually based on the above mentioned parameters
which are closely linked with the choice of coordinate system and thus lack intrinsic geometric
property. Here new geometric parameters based construction methods for cubic PH curves are
discussed in the following.
3
Geometric Parameters Based Cubic PH Curve Design
Let P(𝑡) be a planar cubic Bézier curve with control points {P 𝑖 }3𝑖=0 (see Fig.1). Let O be the
origin of coordinates and 𝐿 be the length of the first control polygon leg (i.e.𝐿 = 𝐿1 ). Define
𝜌 as the ratio of the adjacent control polygon legs. It is not difficult to see from Lemma 4 that
𝐿3
𝐿2
=
. Let 𝜃 be the control polygon angle at the interior vertices P 1 , P 2 (i.e. the angle
𝜌=
𝐿1
𝐿2
between vectors P 1 P 0 and P 1 P 2 , as well as the angle between vectors P 2 P 1 and P 2 P 3 ). Let 𝛿
be the included angle between the first control polygon edge and the edge joining the origin O
and the first control point (i.e. the angle between vectors P 0 O) and P 0 P 1 )). Similarly define 𝜁
be the included angle between the last control polygon edge and the edge joining the origin O
and the last control point (i.e. the angle between vectors P 3 O) and P 3 P 2 )). If the first control
point P 0 coincides with the origin O, select vector OP 0 ) as the positive axis of 𝑥.
Fig. 1: Geometric parameters of cubic PH curves
According to Lemma 3, cubic PH curves are always convex, which leads to the conclusion that a
cubic PH curve preserves the direction of rotation. Define Dir as the coefficient of curve rotation.
If it rotates clockwise, let Dir be +1, while if it rotates counterclockwise, set Dir be -1.
From the geometric properties of cubic PH curves, it is obvious to see that apart from the
initial starting point (𝑥0 , 𝑦0 ), the shape of the curve and the rotation direction can be determined
by the coefficient Dir and some geometric parameters including the control polygon legs and the
control polygon angle. Precisely speaking, the shape and the rotation direction of a curve can
be uniquely determined by Dir and {𝛿, 𝜃, 𝐿, 𝜌}. Consequently we choose {𝛿, 𝜃, 𝐿, 𝜌} as the basic
geometric parameters for cubic PH curves in this paper and a cubic PH curve can be designed as
following Theorem 1.
2426 M. Cheng et al. /Journal of Information & Computational Science 6: 6 (2009) 2423–2432
Theroem 1 Given the initial starting point P 0 = (𝑥0 , 𝑦0 ) , geometric parameters {𝛿, 𝜃, 𝐿, 𝜌} and
3
∑
the coefficient Dir of curve rotation, a cubic PH curve P(𝑡) =
𝐵𝑖3 (𝑡)P 𝑖 can be constructed as:
𝑖=0
⎧
(
𝐴1

P
0

P 1 = 𝐿0 ⋅
⎨ P 0 = (𝑥0 , 𝑦0 );
(
)
( −𝐵1
𝐴2 𝐵2
𝐴3

0
0

⋅
; P3 = P
⋅
⎩ P2 = P
𝐿0
𝐿0
−𝐵2 𝐴2
−𝐵3
where
4
)
𝐵1
;
𝐴1 )
𝐵3
;
𝐴3
⎧
𝐵1 = 𝐿 sin 𝛿 ⋅ 𝐷𝑖𝑟;
⎨ 𝐴1 = 𝐿0 − 𝐿 cos 𝛿;
𝐴2 = 𝐴1 + 𝜌𝐿 cos(𝛿 + 𝜃); 𝐵2 = 𝐵1 − 𝜌𝐿 sin(𝛿 + 𝜃) ⋅ 𝐷𝑖𝑟;
⎩
𝐴3 = 𝐴2 − 𝜌2 𝐿 cos(𝛿 + 2𝜃); 𝐵3 = 𝐵2 + 𝜌2 𝐿 sin(𝛿 + 2𝜃) ⋅ 𝐷𝑖𝑟.
(3)
(4)
Geometric Parameters Based Cubic PH Curve Approximation with Constraints of End-points
In practical applications, construction of a cubic PH curve with end-points interpolation to a
given cubic curve is often needed. There are two cases in this problem. One is that the given
curve itself satisfies the PH conditions already and the other case is non-PH. In this section the
condition equations for interpolated curve in Bézier form are presented, which is of the first case.
Under the second case, algorithm of PH curve approximation to a non-PH curve with end-points
interpolation is provided.
Suppose the given curve Q(𝑡) is already represented in cubic Bézier form with control points
{Q 𝑖 }3𝑖=0 . Geometric parameters 𝐿1 , 𝐿2 , 𝐿3 , 𝛿, 𝜁, 𝜃1 , 𝜃2 with overline on the notation are defined
similarly with which are shown in Fig.1. They are defined respectively as 𝐿1 = ∥Q 0 Q 1 ∥, 𝐿2 =
∥Q 1 Q 2 ∥, 𝐿3 = ∥Q 2 Q 3 ∥, 𝛿 = ∠OQ 0 Q 1 , 𝜁 = ∠OQ 3 Q 2 , 𝜃1 = ∠Q 0 Q 1 Q 2 , 𝜃2 = ∠Q 1 Q 2 Q 3 .
4.1
Geometric Parameters Based PH Curve End-points Interpolation
The construction of end-points interpolation with cubic PH curves is given as follows. Obviously
the coefficient of curve rotation Dir is already uniquely determined by the given curve. For the
demand of end-points interpolation, we have
P 0 = Q 0 = (𝑥0 , 𝑦0 ), P 3 = Q 3 = (𝑥3 , 𝑦3 ),
(5)
It can be deduced from equations (3) ∼ (5) that here the geometric parameters {𝛿, 𝜃, 𝐿, 𝜌}
should satisfy the following condition equations:
{
𝐴3 −𝐿0
𝐿 = − cos 𝛿+𝜌 cos(𝛿+𝜃)−𝜌
2 cos(𝛿+2𝜃) ,
(6)
𝐵3 ⋅𝐷𝑖𝑟
𝐿 = 𝑠𝑖𝑛𝛿−𝜌 sin(𝛿+𝜃)+𝜌2 sin(𝛿+2𝜃) .
From Lemma 2 we learn that a cubic PH curve enjoys 6 freedoms. As end-points interpolation
is demanded, there are still 2 freedoms left. Thus there are infinitely many existing cubic PH
M. Cheng et al. /Journal of Information & Computational Science 6: 6 (2009) 2423–2432 2427
curve satisfying the demand of end-points interpolation. If the given curve is a non-PH curve, we
can choose two parameters from the set of geometric parameters {𝛿, 𝜃, 𝐿, 𝜌} to construct a PH approximating curve. Different selection of parameters involved will lead to different approximation
curve. Three algorithms of cubic PH curve approximation with end-points interpolation are given
in the following chapters. The algorithms are respectively based on the geometric parameters as
{𝛿, 𝜃},{𝜌, 𝜃} as well as {𝜌, 𝛿}.
4.2
Geometric Parameters {𝛿, 𝜃} Based PH Curve Approximation
The following quadratic equation of 𝜌 can be deduced from equations (6):
𝛼1 ⋅ 𝜌2 + 𝛽1 ⋅ 𝜌 + 𝛾1 = 0,
where
(7)
⎧
⎨ 𝛼1 = (𝐴3 − 𝐿0 ) sin(𝛿 + 2𝜃) + 𝐵3 cos(𝛿 + 2𝜃) ⋅ 𝐷𝑖𝑟,
𝛽1 = −(𝐴3 − 𝐿0 ) sin(𝛿 + 𝜃) − 𝐵3 cos(𝛿 + 𝜃) ⋅ 𝐷𝑖𝑟,
⎩
𝛾1 = (𝐴3 − 𝐿0 ) sin 𝛿 + 𝐵3 cos 𝛿 ⋅ 𝐷𝑖𝑟.
(8)
If 𝛽12 − 4𝛼1 𝛾1 ≥ 0, the real roots of the above quadratic equation exist as:
√
−𝛽1 ± 𝛽12 − 4𝛼1 𝛾1
𝜌=
.
2𝛼1
(9)
Now we give the geometric parameters {𝛿, 𝜃} based PH cubic approximation algorithm with
end-points interpolation.
Algorithm 1 Let Q(𝑡) be the given cubic Bézier curve with control points {Q 𝑖 }3𝑖=0 . Given
geometric parameters {𝛿, 𝜃}, if a positive root 𝜌 of equation (7) exists and simultaneously 𝐿 calculated by (6) is also positive, then the geometric parameters {𝛿, 𝜃} based PH cubic approximation
curve with end-points interpolation can be constructed according to Theorem 1.
Remark 1 By inverting the control points sequence, the geometric parameters {𝜁, 𝜃} based
PH cubic approximation curve with end-points interpolation can be obtained in a similar way as
described in Algorithm 1.
Remark 2 If ∠Q 0 Q 1 Q 2 = ∠Q 1 Q 2 Q 3 for the given curve, we can set 𝜃 = ∠Q 0 Q 1 Q 2 =
∠Q 1 Q 2 Q 3 , then choosing 𝛿(0 ≤ 𝛿 ≤ 𝜋) increasingly from 0 to 𝜋 will get different approximating
PH curves. According to the following Theorem 3 in Section 5, the PH curve with minimum
approximation error can be obtained.
(3𝜋 − 𝛿 − 𝜁 − 𝜙)
), the
Remark 3 If we choose 𝛿 = 𝛿, 𝜁 = 𝜁, 𝜙 = ∠P 3 OP 0 , (i.e. 𝜃 =
2
obtained cubic PH approximating curve in this paper matches with the one with GC1 Hermite
interpolation presented by Meek and Walton [9].
4.3
Geometric Parameters {𝜌, 𝜃} Based PH Curve Approximation
The following equation of 𝛿 can be deduced from equations (6):
𝛼2 ⋅ sin 𝛿 + 𝛽2 ⋅ cos 𝛿 = 0.
(10)
2428 M. Cheng et al. /Journal of Information & Computational Science 6: 6 (2009) 2423–2432
where
{
𝛼2 = (𝐴3 − 𝐿0 )[𝜌2 cos(2𝜃) − 𝜌 cos 𝜃 + 1] + 𝐵3 [−𝜌2 sin(2𝜃) + 𝜌 sin 𝜃] ⋅ 𝐷𝑖𝑟,
𝛽2 = (𝐴3 − 𝐿0 )[𝜌2 sin(2𝜃) − 𝜌 sin 𝜃] + 𝐵3 [𝜌2 cos(2𝜃) − 𝜌 cos 𝜃 + 1] ⋅ 𝐷𝑖𝑟.
(11)
𝛽
ˆ ≤ 𝛿ˆ ≤ 2𝜋) be the angle satisfying (cos 𝛿,
ˆ sin 𝛿)
ˆ = ( √ 𝛼2
√ 2
Denote 𝛿(0
,
). It can be
𝛼22 + 𝛽22
𝛼22 + 𝛽22
ˆ = 0. Thus we get 𝛿 = 𝜋 − 𝛿ˆ 𝑜𝑟 𝛿 = 2𝜋 − 𝛿.
ˆ
deduced from (10) that sin(𝛿 + 𝛿)
Now we give the geometric parameters {𝜌, 𝜃} based PH cubic approximation algorithm with
end-points interpolation.
Algorithm 2 Let Q(𝑡) be the given cubic Bézier curve with control points {Q 𝑖 }3𝑖=0 . When
ˆ 0 ≤ 𝛿ˆ ≤ 𝜋) or 𝛿 = 2𝜋 − 𝛿ˆ (if
geometric parameters {𝜌, 𝜃} are given, choose 𝛿 = 𝜋 − 𝛿(if
𝜋 ≤ 𝛿ˆ ≤ 2𝜋). If 𝐿 calculated by (6) is also positive, then the geometric parameters {𝜌, 𝜃} based
PH cubic approximation curve with end-points interpolation can be constructed according to
Theorem 1.
𝐿2
Remark 4 If the adjacent control polygon legs of the given curve are proportional, i.e.
=
𝐿1
𝐿3
𝐿2
𝐿3
, we can choose 𝜌 =
=
. Then increasingly select 𝜃(0 ≤ 𝜃 ≤ 𝜋) from 0 to 𝜋, the cubic
𝐿2
𝐿1
𝐿2
PH curve with minimum approximation error can also be obtained according to Theorem 3 (see
Section 5). Under this circumstance, if the control polygon angles of the given curve also satisfy
∠Q 0 Q 1 Q 2 = ∠Q 1 Q 2 Q 3 , the curve itself is already a PH curve.
4.4
Geometric Parameters {𝜌, 𝛿} Based PH Curve Approximation
The following equation of 𝜃 can be deduced from equations (6):
ˆ − 𝜌 sin(𝜃 + 𝛿 + 𝜃)
ˆ + 𝑠𝑖𝑛(𝛿 + 𝜃)
ˆ = 0,
𝜌2 sin(2𝜃 + 𝛿 + 𝜃)
(12)
where 𝜃ˆ satisfies
ˆ sin 𝜃)
ˆ = (√
(cos 𝜃,
𝐴3 − 𝐿 0
,√
2
(𝐴3 − 𝐿0 )2 + 𝐵3
𝐵3 ⋅ 𝐷𝑖𝑟
(𝐴3 − 𝐿0 )2 + 𝐵32
).
(13)
Remark 5 The explicit solution of equation (12) does exist, while the expression is very
complicated. So here we will not give unnecessary details.
Now we give the geometric parameters {𝜌, 𝛿} based PH cubic approximation algorithm with
end-points interpolation.
Algorithm 3 Let Q(𝑡) be the given cubic Bézier curve with control points {Q 𝑖 }3𝑖=0 . When
geometric parameters {𝜌, 𝛿} are given, if the root 𝜃 of equation (12) exists in [0, 𝜋] and simultaneously 𝐿 calculated by (6) is also positive, then the geometric parameters {𝜌, 𝛿} based PH cubic
approximation curve with end-points interpolation can be constructed according to Theorem 1.
Remark 6 By inverting the control points sequence, the geometric parameters {𝜌, 𝜁} based
PH cubic approximation curve with end-points interpolation can be obtained in a similar way as
described in Algorithm 3.
M. Cheng et al. /Journal of Information & Computational Science 6: 6 (2009) 2423–2432 2429
Remark 7 If the adjacent control polygon legs of the given curve are proportional, i.e.
𝐿2
𝐿3
=
,
𝐿1
𝐿2
𝐿2
𝐿3
=
. Then increasingly select 𝛿(0 ≤ 𝛿 ≤ 𝜋) from 0 to 𝜋, the cubic PH
𝐿1
𝐿2
curve with minimum approximation error can also be obtained according to Theorem 3 in Section
5.
we can choose 𝜌 =
5
Error Bound of Cubic PH Curve Approximation
For geometric parameters based cubic PH curve approximation with end-points interpolation, the
approximation error bound can be calculated as following.
Theroem 2 Let Q(𝑡) be the given cubic Bézier curve with control points {Q 𝑖 }3𝑖=0 and P(𝑡) be
the cubic PH approximating curve with control points {P 𝑖 }3𝑖=0 . Then the approximation error
function 𝜖(𝑡) satisfies
3
∥𝜖(𝑡)∥ = ∥P(𝑡) − Q(𝑡)∥ ≤ 𝑚𝑎𝑥(∥P 1 − Q 1 ∥, ∥P 2 − Q 2 ∥)
4
(14)
From Theorem 2, it can be learned that the approximation error function lies on the distance
∥P 1 Q 1 ∥ and ∥P 2 Q 2 ∥. Therefore we can get the error function represented by the geometric
parameters as follows.
Theroem 3 Let Q(𝑡) be the given cubic Bézier curve with control points {Q 𝑖 }3𝑖=0 and P(𝑡) be
the cubic PH approximating curve with control points {P 𝑖 }3𝑖=0 . Then the approximation error
function 𝜖(𝑡) has the upper bound expression as
√
√
3
2
2
2
(15)
∥𝜖(𝑡)∥ ≤ 𝑚𝑎𝑥{ 𝐿1 + 𝐿1 − 2𝐿1 𝐿1 cos(𝛿 − 𝛿), 𝐿23 + 𝐿3 − 2𝐿3 𝐿3 cos(𝜁 − 𝜁)}.
4
where 𝐿1 , 𝛿, 𝐿3 , 𝜁 represent the geometric parameters of the given curve, as defined in section 4.
6
Examples
Example 1 (Geometric parameters {𝛿, 𝜃} based PH curve approximation)
Let Q(𝑡) be a cubic Bézier curve with control points {{1.0, 1.0}, {0.0, 2.1}, {−1.41421, 1.9},
{−2.41421, 1.0}}, which is rendered in solid line (see Fig.2). The curve Q(𝑡) does not satisfy
the PH conditions. Applying Algorithm 1 of the geometric parameters {𝛿, 𝜃} based PH cubic
curve approximation with end-points interpolation, we get the cubic PH curve with minimum
3𝜋
𝜋
. The PH approximating curve and its control polygon
approximation error at 𝛿 = , 𝜃 =
2
4
7𝜋
are rendered in dash line in Fig.2 left figure. If we set 𝜃 =
, the corresponding cubic PH
10
3𝜋 𝜋 5𝜋 3𝜋
approximating curve with 𝛿 being chosen as
, , ,
are rendered in Fig.2 right figure.
8 2 8 4
2430 M. Cheng et al. /Journal of Information & Computational Science 6: 6 (2009) 2423–2432
Fig. 2: Cubic PH approximating curve with {𝛿 = 𝜋2 , 𝜃 =
3𝜋
4 }(left)
and with {𝛿, 𝜃 =
7𝜋
10 }(right)
Example 2 (Geometric parameters {𝜌, 𝜃} based PH curve approximation)
Let Q(𝑡) be a cubic Bézier curve with control points {{1.0, 1.0}, {0.0, 2.0}, {−1.3968, 2.22123},
{−2.03884, 0.961158}}, which is rendered in solid line (see Fig.3). The curve Q(𝑡) does not
satisfy the PH conditions. It can be seen that here the three control polygon legs are the same,
i.e. 𝜌 = 1. Set 𝜌 = 1, and apply Algorithm 2 of the geometric parameters {𝜌, 𝜃} based PH cubic
approximation with end-points interpolation, the corresponding cubic PH approximating curve
𝜋 13𝜋 7𝜋 5𝜋 2𝜋 17𝜋 3𝜋 19𝜋 5𝜋 7𝜋 11𝜋 23𝜋
with 𝜃 being chosen as ,
,
, ,
,
,
,
,
, ,
,
, 𝜋 are rendered in
2 24 12 8 3 24 4 24 6 8 12 24
17𝜋
dash line in Fig.3 left figure. The one with minimum approximation error achieves at 𝜃 =
(see
24
dash line in Fig.3 right figure).
Fig. 3: Cubic PH approximating curve with {𝜌 = 1, 𝜃}(left) and with {𝜌 = 1, 𝜃 =
17𝜋
24 }(right)
Example 3 (Geometric parameters {𝜌, 𝛿} based PH curve approximation)
Let Q(𝑡) be a cubic Bézier curve which is same as in Example 2, hence does not satisfy the PH
conditions but have the same control polygon legs, i.e. 𝜌 = 1. Set 𝜌 = 1, and apply Algorithm 3
of the geometric parameters {𝜌, 𝛿} based PH cubic approximation with end-points interpolation,
𝜋 7𝜋 𝜋 3𝜋 5𝜋 11𝜋
,
,
the corresponding cubic PH approximating curve with 𝛿 being chosen as , , , ,
4 24 3 8 12 24
M. Cheng et al. /Journal of Information & Computational Science 6: 6 (2009) 2423–2432 2431
𝜋 13𝜋 7𝜋 5𝜋 2𝜋 17𝜋
,
,
, ,
,
are rendered in dash line in Fig.4 left figure. The one with minimum
2 24 12 8 3 24
13𝜋
approximation error achieves at 𝛿 =
(see dash line in Fig.4 right figure).
24
Fig. 4: Cubic PH approximating curve with {𝜌 = 1, 𝛿}(left) and with {𝜌 = 1, 𝛿 =
7
13𝜋
24 }(right)
Conclusion
Construction of cubic PH curves based on geometric parameters is presented in this paper. And
the problems of cubic Bézier curve interpolation at two end-points and the PH curve approximation to non-PH Bézier curves with end-points interpolation are also researched. Three algorithms
for cubic PH curve approximation with end-points interpolation, which are based on different
geometric parameters are given. Geometric parameters based PH curve construction can well express the intrinsic shape of the curve and thus will appropriately match the practical requirements
of customers. This is surely an outset of the research on geometric parameters based PH curve
approximation in Bézier forms. How to extend this idea to geometric parameters based PH curve
approximation with higher degree (for example quintic or septic) still needs further investigation.
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