# Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics

Authors: Sunhong Lee; Hyun Chol Lee; Mi Ran Lee; Seungpil Jeong; Gwang-Il Kim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We present an algorithm forC1 Hermite interpolation
using Möbius transformations of planar polynomial Pythagoreanhodograph
(PH) cubics. In general, with PH cubics, we cannot
solve C1 Hermite interpolation problems, since their lack of parameters
makes the problems overdetermined. In this paper, we
show that, for each Möbius transformation, we can introduce an
extra parameter determined by the transformation, with which we
can reduce them to the problems determining PH cubics in the
complex plane ℂ. Möbius transformations preserve the PH property
of PH curves and are biholomorphic. Thus the interpolants
obtained by this algorithm are also PH and preserve the topology
of PH cubics. We present a condition to be met by a Hermite
dataset, in order for the corresponding interpolant to be simple or
to be a loop. We demonstrate the improved stability of these new
interpolants compared with PH quintics.


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## Body

## 1. Introduction

Farouki and Sakkalis [1] introduced Pythagorean-hodograph (PH) curves, which are a special class of polynomial curves with a polynomial speed function. These curves have many computationally attractive features: in particular, their arc lengths and offset curves can be determined exactly. Farouki [2] reviews the abundant results on these curves obtained by many researchers. Hermite interpolation with PH curves is one of the main topics in this research (Farouki and Neff [3], Albrecht and Farouki [4], Jüttler [5], Jüttler and Mäurer [6], Farouki et al. [7], Pelosi et al. [8], and Šír et al. [9]).In this paper, we solve theC1 Hermite interpolation problem using the Möbius transformations of polynomial PH cubics in the plane. The use of Möbius transformation has been demonstrated in recent publications [10, 11]. In [11], Bartoň et al. used a general Möbius transformation in ℝ3, which is defined as a composition of an arbitrary number of inversions with respect to spheres or planes. They showed that (μ∘x)(t) is a rational PH curve for any general Möbius transformation μ(x1,x2,x3), if x(t) is a polynomial PH space curve in R3. (The preservation of PH properties under transformation is first studied by Ueda [12].) They also presented an algorithm for G1 Hermite interpolation. In this work, we use the classical Möbius transformation, a bijective linear fractional transformation in the extended complex plane ℂ∞=ℂ∪{∞}, that is,(1.1)Φ(z)=az+bcz+d,
for some complex numbers a, b, c, and d for which ad-bc≠0 [13]. Using this transformation, we can solve the C1 Hermite interpolation problems with PH cubics. In general, with PH cubics, we cannot solve C1 Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. But we can show that, for a C1 Hermite interpolation problem, we are always able to obtain four interplants which are constructed by PH cubics. The Möbius transformation makes this possible, since it permits the introduction of a new extra parameter into the problem, which is to be reduced to a simple problem to determine PH cubics as follows: here we adapt the complex representation method [14] to solve the C1 Hermite interpolation problem. The original problem is, for a Hermite dataset (pi,pf,vi,vf), to find a polynomial PH curve r(t) and a Möbius transformation Φ(z), which satisfy(1.2)(Φ∘r)(0)=pi,(Φ∘r)(1)=pf,(Φ∘r)′(0)=vi,(Φ∘r)′(1)=vf.
Next, by an appropriate translation, rotation, and scaling of the dataset, we can arrange that pi=0 and pf=1 and take a Möbius transformation(1.3)Φ(z)=αz(α-1)z+1,
which fixes 0 and 1, for some nonzero complex number α. Then the inverse image of the C1 Hermite dataset under a Möbius transformation Φ makes (1.2) into(1.4)r(0)=0,∫01r′(t)dt=1,r′(0)=1αvi,r′(1)=αvf,
which are suitable forms for adapting the complex representation method (for details, see Section 4). Farouki and Neff [3] already solved the C1 Hermite interpolation problem with PH quintics. According to (1.2), this is exactly the case in which r(t) is a quintic and Φ(z) is the identity, that is, α=1. On the other hand, our work in this paper is the case just when Φ(z) is not the identity, that is, α≠1. At the end of this paper, we will compare our interpolants with PH quintic ones for the same C1 Hermite dataset.The interpolants obtained by our method are specific rational curves represented by complex rational functions. For planar rational curves, there already exists a general theory, which were introduced by Pottmann [15] and Fiorot and Gensane [16]: they studied rational plane curves with rational offets. These curves are represented in the dual form, in which curves are specified using line coordinates instead of point coordinates. Pottmann showed how to design rational PH curves segments by G1 and G2 Hermite interpolations [17, 18]. However, in our work, what we need is only a suitable PH cubic and a PH-preserving transformation which is algebraically simple as possible and which can generate an extra-parameter, and the latter is completely settled by the classical Möbius transformation. Moreover, the transformation is biholomorphic. Thus it preserves the topology of the preimage curve (PH cubic). Therefore, the interpolants obtained by our method should have no cusp, although cusps are a generic feature of rational PH curves. They are simple curves or else loops. Hence, to obtain these, even avoiding the easy shortcut, there is no need to follow up the lengthy path with a far starting point. We just use the classical Möbius transformation of PH cubics, that is all.The rest of this paper is organized as follows. In Sections2 and 3, we review some basic properties of Möbius transformations and planar PH cubics. In Section 4, we solve the C1 Hermite interpolation problem using the Möbius transformations of planar PH cubics. In Section 5, we present the condition on a Hermite dataset, which determine whether the corresponding Hermite interpolant has a loop, we also compare these new interpolants with PH quintics and show that the former have improved stability. We conclude this paper in Section 6.


## 2. Möbius Transformations

AMöbius transformationΦ(z) is a bijective linear fractional transformation in the extended complex plane ℂ∞=ℂ∪{∞}, that is,(2.1)Φ(z)=az+bcz+d,
for some complex numbers a, b, c, and d for which ad-bc≠0 [13]. Then Φ(z) is a one-to-one correspondence on the extended complex plane ℂ∞ with its inverse(2.2)Φ-1(z)=dz-b-cz+a.
The derivative of Φ(z) is(2.3)Φ′(z)=ad-bc(cz+d)2.
For any Möbius transformations Φ(z) and Ψ(z), (Ψ∘Φ)(z) is also a Möbius transformation. Thus the set ℳ of all Möbius transformations forms a group under composition.A rational plane curver(t)=x(t)+-1y(t) is called a Pythagorean-hodograph (PH) curve [1] if there exists a rational function σ(t) such that(2.4)x′(t)2+y′(t)2=σ(t)2.Lemma 2.1.
LetΦ(z) be a Möbius transformation and  r(t) be a polynomial PH curve. Then  s(t)=(Φ∘r)(t) is a rational PH curve.Proof.
Sinces′(t)=Φ′(r(t))r′(t), we have
(2.5)|s′(t)|=|ad-bc||cr(t)+d|2|r′(t)|=|ad-bc|Re(cr(t)+d)2+Im⁡(cr(t)+d)2|r′(t)|.
This completes the proof.Lemma2.1 means that Möbius transformations preserve the PH property, which is a special case of the result of Bartoň et al. [11].For a polynomial PH curver(t) of degree n, a Möbius transformation of r(t)(2.6)(Φ∘r)(t)=ar(t)+bcr(t)+d
is a rational curve, also of degree n, with coefficients in the complex plane ℂ. However, if we associate the complex plane ℂ with ℝ2, and express (Φ∘r)(t) as a rational curve with real coefficients in ℝ2 then the result is generally a rational PH curve of degree 2n or n. If we perform a further Möbius transformation Ψ(z), the rational curve (Ψ∘(Φ∘r))(t)=((Ψ∘Φ)∘r)(t) retains a degree of 2n or n, since (Ψ∘Φ)(z) is a Möbius transformation.Lemma 2.2.
Let(Φ∘r)(t) be a Möbius transformation  Φ(z) of a polynomial curve  r(t), such that  (Φ∘r)(0)=0 and  (Φ∘r)(1)=1. Then there exist a polynomial curve  s(t) and a Möbius transformation  Ψ(z), such that  s(0)=0,  s(1)=1, and  (Φ∘r)(t)=(Ψ∘s)(t).Proof.
We can find a Möbius transformationΦ1(z)=az+b for some complex constants a and b such that a≠0 and s(t)=(Φ1∘r)(t) is a polynomial curve with s(0)=0 and s(1)=1. Consequently, we can obtain the Möbius transformation Ψ(z)=(Φ∘Φ1-1)(z) such that (Φ∘r)(t)=(Ψ∘s)(t).A Möbius transformationΨ(z) of this sort also fixes 0 and 1.Lemma 2.3.
LetΦ(z) be a Möbius transformation with  Φ(0)=0 and  Φ(1)=1. Then there exists a nonzero complex constant  α such that
(2.7)Φ(z)=αz(α-1)z+1.
If τ=|α| and η=arg(α), then Φ(z)=(Φτ∘Φη)(z)=(Φη∘Φτ)(z), where
(2.8)Φτ(z)=τz(τ-1)z+1,Φη(z)=eiηz(eiη-1)z+1.Proof.
LetΨ(z)=(az+b)/(cz+d) be a Möbius transformation with Ψ(0)=0 and Ψ(∞)=∞. Then from Ψ(0)=0 we get b=0, and from Ψ(∞)=∞ we get c=0. Thus Ψ(z)=αz, where α=a/d. Let τ=|α| and η=arg(α). Then we obtain Ψ(z)=(Ψτ∘Ψη)(z)=(Ψη∘Ψτ)(z), where Ψτ(z)=τz and Ψη(z)=eiηz.
Now letΦ(z) be a Möbius transformation with Φ(0)=0 and Φ(1)=1. Let S(z)=z/(z-1), so that S(0)=0 and S(∞)=1. Then, since (S-1∘Φ∘S)(0)=0 and (S-1∘Φ∘S)(∞)=∞, we get (S-1∘Φ∘S)(z)=Ψ(z), where Ψ(z)=αz for some nonzero α. Thus we obtain
(2.9)Φ(z)=(S∘Ψ∘S-1)(z)=αz(α-1)z+1.
Moreover, in the same way, we can obtain Φτ(z)=(S∘Ψτ∘S-1)(z) and Φη(z)=(S∘Ψη∘S-1)(z), so that Φ(z)=(Φτ∘Φη)(z)=(Φη∘Φτ)(z).SinceΦ′(z)=α/((α-1)z+1)2, we have Φ′(0)=α and Φ′(1)=1/α.


## 3. Planar Pythagorean-Hodograph Cubics

A planar polynomial curver(t)=x(t)+-1y(t) is a PH curve [19] if and only if there exist polynomials h(t), u(t), and v(t), which satisfy(3.1)x′(t)=h(t)[u(t)2-v(t)2],y′(t)=h(t)[2u(t)v(t)].
Note that, if gcd(u(t),v(t))=1, then gcd(u(t)2-v(t)2,2u(t)v(t))=1. In this paper, we will assume that h(t) is monic, meaning that its leading coefficient is 1.A polynomial curver(t) is a PH curve [14] if and only if there exists a polynomial h(t) and a polynomial curve w(t) such that(3.2)r′(t)=h(t)w(t)2.
Suppose that the PH cubic r(t) is a line. Then the hodograph r′(t) can be expressed as h(t)(x0+-1y0), where x0+-1y0 is a nonzero point and h(t) is the quadratic monic polynomial(3.3)h(t)=h0(1-t)2+h12(1-t)t+h2t2,
and h0, h1, and h2 are real constants such that h0+h2≠2h1.Letr(t) be a PH cubic for which r′(t)=w(t)2. Since w(t) is linear, we can write w(t) in Bernstein form:(3.4)w(t)=w0(1-t)+w1t,
where w0 and w1 are distinct complex constants. The hodograph r′(t) can then be expressed as(3.5)r′(t)=w02(1-t)2+w0w12(1-t)t+w12t2.
If we represent the PH cubic r(t) in the Bernstein form(3.6)r(t)=p0(1-t)3+p13(1-t)2t+p23(1-t)t2+p3t3,
then we obtain(3.7)p1=p0+13w02,p2=p1+13w0w1,p3=p2+13w12,
where p0 can be chosen arbitrarily.


## 4. First-Order Hermite Interpolation

We will now solve theC1 Hermite interpolation problem using Möbius transformations of PH cubics.Letpi and pf be the initial and final points to be interpolated, where pi≠pf. Let vi=rie-1θi and vf=rfe-1θf, respectively, be the initial vector at pi and the final vector at pf, where ri>0 and rf>0. For this Hermite dateset (pi,pf,vi,vf), we want to find planar PH cubics r(t) and Möbius transformations Φ(z) which satisfy (1.2), which are equivalent to(4.1)(Φ∘r)(0)=pi,∫01(Φ∘r)′(t)dt=pf-pi,(Φ∘r)′(0)=vi,(Φ∘r)′(1)=vf.
By an appropriate translation, rotation, and scaling of the data-set, we can arrange that pi=0 and pf=1. Then, from Lemmas 2.2 and 2.3, we seek some nonzero constants α and PH cubics r(t), which satisfy (1.4).


### 4.1. Case ofr′(t)=h(t)=h0(1-t)2+h12(1-t)t+h2t2

In this case, (4.1) become(4.2)r(0)=0,h0+h1+h2=3,αh0=vi,h2α=vf.
From the second and third of these equations, we can see that Hermite interpolants r(t) exist if and only if θi+θf=mπ for some integers m. In this case, for α=τe-1θi or α=τe-1(θi+π), where τ is any positive number, we have(4.3)h0=viα,h2=αvf,h1=3-h0-h2.
Consequently, we can obtain the PH cubics(4.4)r(t)=h033(1-t)2t+h0+h133(1-t)t2+h0+h1+h23t3
and their Möbius transformations(4.5)(Φ∘r)(t)=αr(t)(α-1)r(t)+1.


### 4.2. Case ofr′(t)=w(t)2 Where w(t)=w0(1-t)+w1t

From (3.5) and (3.6), (4.1) become(4.6)r(0)=0,w02+w0w1+w12=3,αw02=vi,w12α=vf.
If we let(4.7)v=rirf3e-1(θi+θf)/2,
and also let a=w02/3, b=w12/3 and k=w0w1/3, then second and third equations in (4.6) imply that k=v or k=-v, and so we have(4.8)ab=k2,a+b=1-k.
Now let(4.9)am=1-k+(1+k)(1-3k)2,bm=1-k-(1+k)(1-3k)2,
where m=1 if k=v, and m=-1 if k=-v. Then we have a=am and b=bm, or a=bm and b=am. Consequently, we can obtain the four PH cubics(4.10)rm,1(t)=am3(1-t)2t+(am+k)3(1-t)t2+(am+k+bm)t3,rm,2(t)=bm3(1-t)2t+(bm+k)3(1-t)t2+(bm+k+am)t3,
where k=v or k=-v. Note that rm,1=rm,2 if and only if k is −1 or 1/3. From the PH cubics rm,j(t) we can obtain the Möbius transformations of the PH cubics Φm,j(rm,j(t))(m=1,-1,andj=1,2), where(4.11)Φm,j(z)=αm,jz(αm,j-1)z+1,αm,1=13viam,αm,2=13vibm.
If k is nonreal, then both rm,1(t) and rm,2(t) are nonlinear. But if k is a real number, then -1≤k≤1/3 if and only if both rm,1(t) and rm,2(t) are linear.We can summarize these results.Theorem 4.1.
Let(0,1,vi=rie-1θi,vf=rfe-1θf) be a  C1 Hermite data-set such that  ri>0 and  rf>0. (a)
Letv be the vector given by (4.7), and let k be v or -v. Then all C1 Hermite interpolants using Möbius transformations of planar PH cubics r(t), such that r′(t)=w(t)2 for some linear curve w(t), are Φm,j(rm,j(t))(m=1,-1,andj=1,2), from (4.10) and (4.11), where am and bm are given by (4.9).(b)
C1 Hermite interpolants using Möbius transformations of planar PH cubics r(t), such that r′(t)=h0(1-t)2+h12(1-t)t+h2t2 for some real number h0,h1, and h2 such that h0+h2≠h1, exist if and only if θi+θf=mπ for some integers m. In this case, the interpolants (Φ∘r)(t) are given by (4.5), where r(t) is given by (4.3) and (4.4), where α=τe-1θi or α=τe-1(θi+π) for any positive number τ.


## 4.1. Case ofr′(t)=h(t)=h0(1-t)2+h12(1-t)t+h2t2

In this case, (4.1) become(4.2)r(0)=0,h0+h1+h2=3,αh0=vi,h2α=vf.
From the second and third of these equations, we can see that Hermite interpolants r(t) exist if and only if θi+θf=mπ for some integers m. In this case, for α=τe-1θi or α=τe-1(θi+π), where τ is any positive number, we have(4.3)h0=viα,h2=αvf,h1=3-h0-h2.
Consequently, we can obtain the PH cubics(4.4)r(t)=h033(1-t)2t+h0+h133(1-t)t2+h0+h1+h23t3
and their Möbius transformations(4.5)(Φ∘r)(t)=αr(t)(α-1)r(t)+1.


## 4.2. Case ofr′(t)=w(t)2 Where w(t)=w0(1-t)+w1t

From (3.5) and (3.6), (4.1) become(4.6)r(0)=0,w02+w0w1+w12=3,αw02=vi,w12α=vf.
If we let(4.7)v=rirf3e-1(θi+θf)/2,
and also let a=w02/3, b=w12/3 and k=w0w1/3, then second and third equations in (4.6) imply that k=v or k=-v, and so we have(4.8)ab=k2,a+b=1-k.
Now let(4.9)am=1-k+(1+k)(1-3k)2,bm=1-k-(1+k)(1-3k)2,
where m=1 if k=v, and m=-1 if k=-v. Then we have a=am and b=bm, or a=bm and b=am. Consequently, we can obtain the four PH cubics(4.10)rm,1(t)=am3(1-t)2t+(am+k)3(1-t)t2+(am+k+bm)t3,rm,2(t)=bm3(1-t)2t+(bm+k)3(1-t)t2+(bm+k+am)t3,
where k=v or k=-v. Note that rm,1=rm,2 if and only if k is −1 or 1/3. From the PH cubics rm,j(t) we can obtain the Möbius transformations of the PH cubics Φm,j(rm,j(t))(m=1,-1,andj=1,2), where(4.11)Φm,j(z)=αm,jz(αm,j-1)z+1,αm,1=13viam,αm,2=13vibm.
If k is nonreal, then both rm,1(t) and rm,2(t) are nonlinear. But if k is a real number, then -1≤k≤1/3 if and only if both rm,1(t) and rm,2(t) are linear.We can summarize these results.Theorem 4.1.
Let(0,1,vi=rie-1θi,vf=rfe-1θf) be a  C1 Hermite data-set such that  ri>0 and  rf>0. (a)
Letv be the vector given by (4.7), and let k be v or -v. Then all C1 Hermite interpolants using Möbius transformations of planar PH cubics r(t), such that r′(t)=w(t)2 for some linear curve w(t), are Φm,j(rm,j(t))(m=1,-1,andj=1,2), from (4.10) and (4.11), where am and bm are given by (4.9).(b)
C1 Hermite interpolants using Möbius transformations of planar PH cubics r(t), such that r′(t)=h0(1-t)2+h12(1-t)t+h2t2 for some real number h0,h1, and h2 such that h0+h2≠h1, exist if and only if θi+θf=mπ for some integers m. In this case, the interpolants (Φ∘r)(t) are given by (4.5), where r(t) is given by (4.3) and (4.4), where α=τe-1θi or α=τe-1(θi+π) for any positive number τ.


## 5. Best Interpolant

In this section we consider how to choose the best interpolant for a given Hermite data-set(pi=0,pf=1,vi,vf).We will begin by presenting a condition under which the Möbius transformation of a PH cubic(Φ∘r)(t) has a loop, where r′(t)=(w0(1-t)+w1t)2 for some distinct complex constants w0 and w1. Since Φ(z) represents a one-to-one correspondence on the extended complex plane, the condition that r(t) has a loop is both necessary and sufficient to establish that (Φ∘r)(t) has a loop. Under the conditions r(0)=0 and r(1)=1, the PH cubic r(t) is given by r(t)=A(t-B)3+C, where(5.1)A=(w0-w1)23,B=w0w0-w1,C=w033(w0-w1).
The condition that there exist constants t1 and t2, such that 0≤t1<t2≤1 and r(t1)=r(t2), is necessary and sufficient to establish that r(t) has a loop. From r(t1)=r(t2), we can obtain 3B2-3(t1+t2)B+(t12+t1t2+t22)=0, which implies(5.2)B=3(t1+t2)±-1(t2-t1)36.
This equation is equivalent to(5.3)t1+t2=2ReB,t2-t1=23|Im⁡B|,
and hence(5.4)t1=ReB-3|Im⁡B|,t2=ReB+3|Im⁡B|.
Consequently, r(t) has a loop if and only if B∈Ω1∪Ω2, (see Figure 1) where(5.5)Ω1={z∈C0≤Rez-3Im⁡z<Rez+3Im⁡z≤1},Ω2={z∈C∣0≤Rez+3Im⁡z<Rez-3Im⁡z≤1}.
On the other hand, the PH cubic r(t) can be represented by(5.6)r(t)=a3(1-t)2t+(a+k)3(1-t)t2+(a+k+b)t3,
where a=w02/3, k=w0w1/3, and b=w12/3. From k=w0w1/3 and B=w0/(w0-w1), we can obtain(5.7)B2-B=k1-3k=11/k-3,
and hence(5.8)k=13-13(3B2-3B+1).
Note that(5.9){13-13(3z2-3z+1)∈C∣0≤Rez-3Im⁡z<Rez+3Im⁡z≤1}={13-13(3z2-3z+1)∈C∣0≤Rez+3Im⁡z<Rez-3Im⁡z≤1}.
Therefore we conclude as following.Areas ofΩ1 and Ω2: B belongs to Ω1∪Ω2 if and only if r(t) has a loop.
(a)(b)Theorem 5.1.
Suppose that(Φ∘r)(t) is a Möbius transformation of a planar PH cubic, such that  r(0)=0,r(1)=1, and  r′(t)=(w0(1-t)+w1t)2 for some distinct complex constants  w0 and  w1 (see Figure 2). Then  (Φ∘r)(t)(0≤t≤1) is a simple curve if and only if  w0w1/3∉D, where
(5.10)D={13-13(3z2-3z+1)∈C∣0≤Rez-3Im⁡z<Rez+3Im⁡z≤1}.Figure 2
Area ofD: w0w1/3∉D if and only if (Φ∘r)(t) in Theorem 5.1 is a simple curve.For a given Hermite data-set(0,1,vi,vf), the term v in (4.7) belongs to D if and only if (Φ1,1∘r1,1)(t) and (Φ1,2∘r1,2)(t) have a loop; and -v belongs to D if and only if (Φ-1,1∘r-1,1)(t) and (Φ-1,2∘r-1,2)(t) have a loop. Note that D is a subset of the left half-plane, that is, D⊂{z∈ℤ:Rez<0}. Thus we can deduce that both (Φ1,1∘r1,1)(t) and (Φ1,2∘r1,2)(t), or both (Φ-1,1∘r-1,1)(t) and (Φ-1,1∘r-1,2)(t) are simple curves. From these simple curves we can choose a best interpolant, which is that with the least bending energy(5.11)E((Φ∘r)(t))=∫Φ∘rκ2ds=∫01κ(t)2|(Φ∘r)′(t)|dt,
where κ is the curvature of (Φ∘r)(t).Example 5.2.
Consider a Hermite data-set(0,1,2e--1π/4,2e--1π/8). Then the vectorvbecomes(5.12)v=rirf3e-1(θi+θf)/2=23e--13π/16.
Thus v∉D and -v∈D, which implies that (Φ1,1∘r1,1)(t) and (Φ1,2∘r1,2)(t) are simple but (Φ-1,1∘r-1,1)(t) and (Φ-1,2∘r-1,2)(t) each have a loop. See Figure 3.For the Hermite dataset(0,1,2e--1π/4,2e--1π/8), the graph on the left shows that -v∈D and v∉D; the central graph shows the PH cubics r(t) with their control polygons; the graph on the right shows the four interpolants.
(a)(b)(c)Example 5.3.
In the case of a Hermite data-set(0,1,e--13π/5,e--1π/5), the vectorvbecomes(5.13)v=rirf3e-1(θi+θf)/2=13e--12π/5.
Thus v∉D and -v∉D, which implies that (Φ1,1∘r1,1)(t), (Φ1,2∘r1,2)(t), (Φ-1,1∘r-1,1)(t), and (Φ-1,2∘r-1,2)(t) are all simple. See Figure 4.For the Hermite data-set(0,1,e--13π/5,e--1π/5), the graph on the left shows that -v∉D and v∉D; the central graph shows the PH cubics r(t) with their control polygons; the graph on the right shows the four interpolants.
(a)(b)(c)Example 5.4.
Consider a family ofC1Hermite data-sets(0,2,k(1+-1),1+2-1),wherek=1,5,10,20.We constructC1  Hermite interpolants that satisfy these data-sets using Möbius transformations of PH cubics, and also PH quintics, all shown in Figure 5. The Möbius transformations of the PH cubics always provide two S-shaped simple curves and two other curves; the latter are C-shaped simple curves whenk=1or5  and have a single loop in the other cases. As the parametric speed of the initial Hermite condition increases, the C-shaped interpolants change from simple curves to single loops, while the simple S-shaped interpolants retain their original shape characteristics. We also observe that, unlike the S-shaped interpolants produced by Möbius transformations of PH cubics, the S-shaped PH quintic interpolants may be simple (like the curve labeled 4 in Figure 5), or have one or two loops (some PH quintics labeled 2 in Figure 5 are S-shaped double loops).Figure 5
Comparison of pairs of PH interpolants, satisfying the sameC1 Hermite data-set (0,2,k(1+-1),1+2-1), when k=1,5,10,20: (a), (b), (c), and (d), respectively, show the Möbius transformations of the PH cubics MCi when k=1,5,10,20; and (a)′, (b)′, (c)′, and (d)′ show the correspoding PH quintics Qi.We observed the behavior of these interpolants as the parametric speed at the endpoints changes. As this speed increases, the arc-lengths of PH quintics increase rapidly, but the arc-length, of Möbius transformations of PH cubics are generally less affected. In particular, the simpleS-shaped interpolants, produced by Möbius transformations of PH cubics show little change in arc-length. Table 1 shows that these latter interpolants have both lower bending energies and shorter arc-lengths, than all the other interpolants we are considering. If we look at Table 1 and identify the most shapely interpolants with the lowest bending energies, we find that the best Möbius transformation of a PH cubic is always S-shaped and simple. However, the merit of the PH quintic interpolants depends on the parametric speeds at their end points. For example, in Figures 5(a)′ and 5(b)′, the curves labeled 4 are best, while the curves labeled 1 are best in Figures 5(c)′ and 5(d)′. Looking closely at the PH quintic interpolants, we see that the simple S-shaped curve with the best shape when k=1 becomes less and less acceptable as the parametric speeds at the end-points increase. But the interpolants labeled 1 in Figures 5(a)′, 5(b)′, 5(c)′, and 5(d)′ exhibit the opposite behavior: initially these curves are C-shaped loops with high bending energies when k=1; but as the parametric speed increases, they become C-shaped simple curves with lower bending energies. When k reaches 20, it has the best shape but the greatest arc-length. This suggests that the best-shaped interpolants, produced by Möbius transformations of PH cubics are more stable than the corresponding PH quintics, in the sense that the former largely achieve a lower arc-length and bending energy than the latter, except when the end-point speeds are significantly asymmetric, as we see when k=20 in this example.Table 1
Comparison of arc-length and bending energy for the interpolants of Figure5.
k=1MC1MC2MC3MC4Q1Q2Q3Q4arc-length3.032.193.102.292.342.162.342.16BE45.05.572.86.814931062735.3k=5MC1MC2MC3MC4Q1Q2Q3Q4arc-length2.932.284.502.313.052.403.052.40BE50.26.520.95.736.176247.310.0k=10MC1MC2MC3MC4Q1Q2Q3Q4arc-length2.892.315.472.364.423.024.423.02BE54.038.216.67.514.4345.919.336.9k=20MC1MC2MC3MC4Q1Q2Q3Q4arc-length2.852.346.132.407.915.397.915.39BE60.111.917.711.38.013610.797.9


## 6. Conclusions

Möbius transformations preserve Pythagorean-hodograph properties. For anyC1 Hermite data-set, we can generally obtain four C1 Hermite interpolants as Möbius transformations of PH cubics. We have proved that these interpolants are always simple curves or single loops, and that at least two of them must be simple. We have also presented the condition that an interpolant must meet if it is to be a simple curve.We compared the shape characteristics ofC1 Hermite interpolants, produced by Möbius transformations of PH cubics, together with their response to changes of parametric speed at their end points, with the same data for PH quintic interpolants satisfying an identical C1 Hermite dataset: we found that interpolants produced by Möbius transformations of PH cubics generally have lower bending energies and shorter arc-lengths than PH quintics.One avenue for further research is to look for ways of predicting how the geometry of Möbius transformation of PH cubics will be determined by a particularC1 Hermite data-set. Another avenue to explore would be the application of Möbius transformations to other interpolation problems involving PH (or MPH) curves, in both two and three dimensions. In particular, we might look to complete the geometric characterization of Möbius transformation of PH cubics in C1 Hermite interpolation.



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*Source: 560246-2012-04-23.xml*