# C 1 Hermite Interpolation with PH Curves Using the Enneper Surface

Authors: Hyun Chol Lee; Jae Hoon Kong; Gwangil Kim
Journal: Abstract and Applied Analysis
 (2018)

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## Abstract

We show that the geometric and PH-preserving properties of the Enneper surface allow us to find PH interpolants for all regularC1 Hermite data-sets. Each such data-set is satisfied by two scaled Enneper surfaces, and we can obtain four interpolants on each surface. Examples of these interpolants were found to be better, in terms of bending energy and arc-length, than those obtained using a previous PH-preserving mapping.


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## Body

## 1. Introduction

Pythagorean-hodograph (PH) curves were first introduced by Farouki and Sakkalis [1] as polynomial curves in R2 with polynomial speed functions, which have polynomial arc-lengths, rational curvature functions, and rational offsets, all of which derive from their polynomial speed functions. These properties make PH curves good candidates for CAGD and CAD/CAM applications such as interpolation of discrete data and control of motion along curved paths [2–4]. Also, these PH curves have subsequently been extended, with several applications, to rational curves with rational speed functions in Rn [3, 5, 6].PH curves have been the subject of a great deal of study, both their formal representation [7–9] and their practical applications [7–11]. PH curves have been generalized [11] to participate in medial axis transforms [3, 12], becoming MPH curves in the Minkowski space Rm,1 [8, 13, 14], and this has motivated a lot of further research. There has also been a lot of work on the use of PH curves for interpolating planar [7, 8, 10, 15, 16] and spatial data-sets [17–21], in particular to meet G1 Hermite [20, 22, 23] and C2 Hermite conditions [7]. In particular, C1 Hermite interpolation problems have been solved by several techniques [13, 24–27] including PH-preserving mappings [24], which have recently been extended [13] to MPH-preserving mappings.In this paper, we show that we can use Enneper surfaces to solveC1 Hermite interpolation problems with PH curves, by exploiting two properties of the Enneper surface: the geometric property that it contains two straight lines and the PH-preserving nature of its parametrization. Since Farouki and Neff’s original work on C1 Hermite interpolation with PH curves, there have been many developments: in particular, it has been shown [24] that C1 Hermite interpolation problems with PH curves in R3 can be reduced to problems in R2 and generic interpolants can then be obtained to satisfy a given C1 Hermite data-set. This is achieved by a special cubic PH-preserving mapping which satisfies the data-set. However, significant drawbacks remain with this method: one is that the algebraic manipulations required are long and complicated; and the other is that this method is restricted to a special class of C1 Hermite data-sets. We will address both of these issues: using the Enneper surface, we can solve C1 Hermite interpolation problems more efficiently for all regular C1 Hermite data-sets; and we will show that the interpolants obtained by this method may be expected to have better shapes than those obtained by the special mapping, in terms of both bending energy and arc-length.The rest of this paper is organized as follows: In Section2, we define the Pythagorean-hodograph curve and the PH-preserving mapping and give examples. In Section 3, we show that the parametrization of the Enneper surface in standard form is PH-preserving and that, by rescaling the Enneper surface, we can find two cubic surfaces that satisfy any regular C1 Hermite data-set. We also prove that we can obtain eight interpolants on the two cubic surfaces that satisfy a regular C1 Hermite data-set. In Section 4, we compare our method with the use of PH-preserving cubic mappings [24], from two different perspectives: the amount of algebraic computation required and the geometric characterizations of the resulting curves. By empirical comparison of interpolants for the same data-set, we show that our method is more efficient and stable than the use of mappings. In Section 5, we summarize the results of this work and propose some themes for further study.


## 2. Preliminary

LetRn be the n-dimensional Euclidean space, for n∈N, and let P[t] be the set of polynomial functions with real coefficients. We express a polynomial curve in Rn as a mapping r:R→Rn from the space of real numbers R to Rn, such that the component functions of r, which are x1(t),x2(t),…,xn(t), are members of P[t].Definition 1.
A polynomial curver(t)=(x1(t),x2(t),…,xn(t)) is said to be aPythagorean-hodograph (PH) curve if its velocity vector or hodograph r′(t)=(x1′(t),…,xn′(t)) satisfies the Pythagorean condition (1)r′tn2=x1′t2+x2′t2+⋯+xn′t2=σt2, where ∗n denotes the Euclidean norm of Rn and σ(t)∈P[t].Definition 2.
A polynomial mappingϕ:Rn→Rm is said to bePH-preserving if, for every PH curve r(t) in Rn, ϕ(r(t)) is a PH curve in Rm.Example 3.
LetΨ be an affine transformation given by (2)Ψp=λRp+k,for∀p∈R3, where R is an orthogonal matrix in R3, λ∈R∖{0} is a scaling factor, and k is a constant vector in R3. Then, for a PH curve r(t)=(u(t),v(t)) in R2, the mapping Ψ~:R2→R3 defined by (3)Ψ~rt=Ψpt,wherept=ut,vt,0 is PH-preserving, since (4)Ψ~rt′32=Ψ~rt′,Ψ~rt′=λRp′t,λRp′t=λ2p′t,p′t=λ2p′t32=λ2u′t2+v′t2, where , denotes the usual inner product in R3.In addition, letP(u,v)=(x(u,v),y(u,v),z(u,v)) be a polynomial mapping given by (5)xu,v=u5-10u3v2+5uv4+u3-3uv2,yu,v=v5-10v3u2+5vu4+v3-3vu2,zu,v=215uvu2-v2. Then, for a r(t)=(u(t),v(t)) in R2, since (6)ddtxut,vt=5ut4u′t-30ut2vt2u′t-20ut3vtv′t+5vt4u′t+20utvt3v′t+3ut2u′t-3vt2u′t-6utvtv′t,(7)ddtyut,vt=5vt4v′t-30vt2ut2v′t-20vt3utu′t+5ut4v′t+20vtut3u′t+3vt2v′t-3ut2v′t-6utvtu′t,(8)ddtzut,vt=615ut2vtu′t+215ut3v′t-215vt3u′t-615utvt2v′t, we obtain (9)Prt′32=ddtxut,vt2+ddtyut,vt2+ddtzut,vt2=ut2+vt225ut2+5vt2+32u′t2+v′t2.This means that P(u,v) is PH-preserving. Figure 1(a) shows the image of P in R3, which is the PH-preserving surface parameterized by P. The blue curve on the surface is the image of the curve γ(t)=(t-1/3t3,t2) shown in Figure 1(b), which is a segment of the Tchirnhausen cubic. This is a PH curve in R2, and thus the curve on the surface is a PH curve in R3.Figure 1
(a) A PH-preserving surface parameterized byP, derived in Example 3, when 0≤t≤1; the blue curve on the surface is the image of γ(t), shown in (b).
(a) (b)


## 3. Construction ofC1 Hermite PH Interpolants on the Scaled Enneper Surface

Definition 4.
LetΣ∈R3 be a surface with the parametrization Φ:R2→R3 given by(10)Φu,v=u33-uv2+u,v33-vu2+v,2uv.Σ is called the Enneper surface in standard form.It is known [28] that PH curves in the domain of the Enneper surface can be mapped to PH curves on the surface: thus the parametrization of the Enneper surface is PH-preserving. We revisit this result briefly.Theorem 5.
Φ is PH-preserving.Proof.
Assume thatr(t)=(u(t),v(t)) is a PH curve in R2. Then, since(11)Φrt′32=ut2u′t-u′tvt2-2utvtv′t+u′t2+vt2v′t-v′tut2-2vtutu′t+v′t2+2u′tvt+2utv′t2=ut2+vt2+12u′t2+v′t2, the curve Φ(r(t)) in R3 is a PH curve. This completes the proof.Theorem 6.
Σ contains two straight lines ly={(x,y,z)∈R3∣x=0,z=0} and lx={(x,y,z)∈R3∣y=0,z=0} on the xy-plane {(x,y,z)∈R3∣z=0}.Proof.
Ifz(u,v)=2uv=0, or equivalently u=0 or v=0, we immediately obtain Φ(0,v)=(0,v3/3+v,0) or Φ(u,0)=(u3/3+u,0,0). This completes the proof.Remark 7.
The Enneper surface is the nontrivial polynomial minimal surface of the lowest possible degree; equivalently it is area-minimizing, and the parametrizationΦ is conformal (i.e., angle-preserving). Now we consider the surfaces obtained by rescaling the Enneper surface. Let Σλ be a polynomial surface of degree 3 given by the parametrization Φλ=λΦ where Φ is the standard parametrization of the Enneper surface and λ is a nonzero real number. Then Σλ is also a polynomial minimal surface of degree 3, and Φλ is PH-preserving, as shown in Example 3. From now on, we will call the surface Σλ the scaled Enneper surface (or s-Enneper surface) associated with a scaling factor λ. Note that all s-Enneper surfaces contain lx and ly, which we will use in Theorem 9.Definition 8.
AC1 Hermite data-set HC1={p0,p1,v0,v1} consists of two end-points p0 and p1, and two velocities v0 and v1 at those end-points, where p0, p1, v0 and v1∈R3. HC1 is said to beregular if p1-p0, v0 and v1 are linearly independent.Also note that ifp0 and p1 lie on a surface Σ, and v0 and v1 are tangents to that surface, then we can say that Σ satisfies HC1. If ϕ is a mapping which generates the surface Σ, then we can also say that ϕ satisfies HC1.Theorem 9.
For a regularC1 Hermite data-set HC1={(0,0,0),(1,0,0),v0=(v01,v02,0),v1=(v11,v12,v13)}, we can find two s-Enneper surfaces satisfying HC1.Proof.
LetΣλ be the s-Enneper surface obtained by the parametrization Φλ=λΦ where Φ is the parametrization of the Enneper surface in standard form. First, since Φλ(0,0)=λΦ(0,0)=(0,0,0) for all λ∈R∖{0}, it is obvious that p0=(0,0,0) lies on Σλ. In addition, as stated in Remark 7, by Theorem 5, we can find a point in lx given by u03/3+u0,0,0 and a suitable scaling factor λ, such that p1=(1,0,0)∈Σλ: since lx∈Σλ, we can assume the existence of some u0 and λ such that (12)Φλu0,0=λΦu0,0=λu033+u0,0,0. Hence, if (13)λ=3u03+3u0for  some  non-zerou0∈R, we have Φλ(u0,0)=p1=(1,0,0)∈Σλ. That is, p0=(0,0,0) and p1=(1,0,0) lie on the s-Enneper surface parameterized by Φλ. Next we note that, since Φuλ(0,0)=λ(1,0,0) and Φvλ(0,0)=λ(0,1,0), the unit vector n0 normal to Σλ at (0,0,0) is (0,0,1). This means that n0,v0=0. Moreover, since(14)Φuλu0,0=λu02+1,0,0,Φvλu0,0=λ0,-u02+1,2u0,we can obtain the unit vector n1 normal to Σλ at (1,0,0) as follows:(15)n1=0,-2u01+u02,1-u021+u02.Here, since (16)n1,v1=-2u01+u02v12+1-u021+u02v13, we can solve(17)v13u02+2v12u0-v13=0when v13≠0 and we obtain(18)u0=-v12v13±1+-v12v132.Then, since the given data-set is regular, it is obvious that v13≠0. Therefore v0 and v1 are tangent to the s-Enneper surface Σλ when(19)u0=-v12v13±1+-v12v132,λ=3u03+3u0. Consequently, Σλ satisfies HC1. This completes the proof.Remark 10.
Note that, if a data-setHC1={p0,p1,v0,v1} isnot regular, the three vectors p1-p0, v0, and v1 must lie on the same plane. This means that the interpolation problem that must be solved to obtain PH curves satisfying the given data-set is equivalent to a planar C1 Hermite interpolation, which has been studied thoroughly [3, 10, 15, 16], in parallel, including the case of solving the interpolation problem for such a data-set with MPH curves [14, 29, 30].Example 11.
LetHC1={(0,0,0),(1,0,0),v0=(1,1,0),v1=(2,1,2)}. Then, using Theorem 9, we can obtain the two s-Enneper surfaces shown in Figure 2: Figure 2(a) is the s-Enneper surface when u0=-1+5/2 and λ=21+155/38 and Figure 2(b) is the s-Enneper surface when u0=-1-5/2 and λ=21-155/38.Figure 2
Two s-Enneper surfaces satisfyingHC1 in Example 11: (a) the s-Enneper surface Σ1λ that satisfies HC1 when u0=-1+5/2 and λ=21+155/38; and (b) the s-Enneper surface Σ2λ that satisfies HC1 when u0=-1-5/2 and λ=21-155/38.
(a) (b)Theorem 12.
For aC1 Hermite data-set HC1={(0,0,0),(1,0,0),v0=(v01,v02,0), v1=(v11,v12,v13)}, where v13≠0. We can obtain four PH interpolants satisfying HC1 on each of the two s-Enneper surfaces which satisfy HC1.Proof.
By Theorem9, for HC1={(0,0,0),(1,0,0),v0=(v01,v02,0), v1=(v11,v12,v13)} with v13≠0, we can obtain two s-Enneper surfaces satisfying HC1. Then, by using the parametrization Φλ of each s-Enneper surface, we can obtain a new C1 Hermite data-set H~C1={p~0=(0,0),p~1=(u0,0),v~0,v~1} on R2 from the given HC1, which satisfies Φλ(0,0)=(0,0,0), Φλ(u0,0)=(1,0,0), DΦλ(v~0)=v0, and DΦλ(v~1)=v1, where DΦλ denotes the derivative mapping of Φλ. Let u0=(-v12/v13)±1+-v12/v132 and λ=3/u03+3u0; then for each parametrization Φλ of the s-Enneper surfaces satisfying HC1, we have that Φλ(0,0)=(0,0,0) and Φλ(u0,0)=(1,0,0) as shown in the proof of Theorem 9. Next, assume that v~0=(v~01,v~02) and v~1=(v~11,v~12). Then, since (20)DΦλv~0=v~01DΦλ1,0+v~02DΦλ0,1=v~01Φuλ0,0+v~02Φvλ0,0,DΦλv~1=v~11DΦλ1,0+v~12DΦλ0,1=v~11Φuλu0,0+v~12Φvλu0,0, solving simultaneously DΦλ(v~0)=v0=(v01,v02,0) and DΦλ(v~1)=v1=(v11,v12,v13) with respect to v~01, v~02, v~12 and v~22, we obtain(21)v~01=v01λ,v~02=v02λ,(22)v~11=v11λu02+1,v~12=v132λu0.Finally, we solve the C1 Hermite interpolation problem for the reduced data-set H~C1 with planar PH quintics in R2. Note that if the given data-set HC1 is regular, then the reduced data-set H~C1 is also regular, since Φλ is conformal (i.e., angle-preserving). For the regular C1 Hermite data-set H~C1, we can always [10] find four interpolants: two of them are simple and the others are loops. Therefore, since Φλ is PH-preserving, if r is a PH interpolant satisfying H~C1, then Φλ(r) is also a PH interpolant satisfying the given data-set HC1, which lies on the s-Enneper surface with the parametrization Φλ. Consequently, we can obtain four PH interpolants satisfying HC1 on each of the two s-Enneper surfaces which satisfy the data-set. Moreover, since Φλ is conformal, the topological property of each interpolant on the s-Enneper surface, whether it is a simple curve or a loop, is the same as that of the preimage of itself obtained from the inverse of Φλ. This completes the proof.Example 13.
LetHC1 be the same C1 Hermite data-set as in Example 11. Then, by Theorem 12, we can obtain eight interpolants on the two s-Enneper surfaces which satisfy HC1, as shown in Figure 3. They consist of four interpolants on the s-Enneper surface Σ1λ, with u0=-1+5/2 and λ=21+155/38, as shown in Figure 3(b), and four interpolants on the s-Enneper surface Σ2λ, with u0=-1-5/2 and λ=21-155/38, as shown in Figure 3(d). These eight interpolants are obtained from the images of the corresponding planar interpolants, shown in Figures 3(a) and 3(c), which satisfy the reduced data-sets H~Cj1 obtained by the parametrization of each s-Enneper surface Σjλ where j=1,2.Figure 3
Two s-Enneper surfaces and interpolants satisfyingHC1 and H~Cj1 (j=1,2) in Example 13: (a) four interpolants satisfying H~C11; (b) four interpolants satisfying HC1 on Σ1λ; (c) four interpolants satisfying H~C21; (d) four interpolants satisfying HC1 on Σ2λ.
(a) (b) (c) (d)Corollary 14.
For an arbitrary regularC1 Hermite data-set HC1={p0,p1,v0,v1} in R3, we can obtain eight interpolants on the surfaces that satisfy HC1, by scaling and exploiting the isometry of the two s-Enneper surfaces.Proof.
LetΨ be an affine mapping in R3, obtained by composing a translation, an orthogonal transformation, and a scaling, as shown in Example 3, so that Ψ(p0)=(0,0,0) and Ψ(p0)=(1,0,0). Then, using Ψ, we can obtain a new regular C1 Hermite data-set HC1∗ as follows: (23)HC1∗=Ψp0=0,0,0,Ψp0=1,0,0,σRv0,σRv1, where σ and R are, respectively, the scaling factor and the orthogonal matrix of Ψ. Here we can use Theorem 12 to obtain eight interpolants on two s-Enneper surfaces satisfying HC1∗. In addition, as shown in Example 3, since Ψ is PH-preserving, we can obtain eight interpolants on the cubic surfaces given by Ψ-1(Σjλ), where Σjλ is the s-Enneper surface satisfying HC1∗, for j=1,2.Example 15.
LetHC1={p0=(0,0,0), p1=(1,1,1), v0=(-1,1,1), v1=(-1,0,1)}. Then, as stated in Corollary 14, using the orthogonal matrix R given by (24)σR=131313-2326260-6666 with the scaling factor σ=1/3, we can reduce the given data-set to the C1 Hermite data-set HC1∗={(0,0,0),(1,0,0),σRv0=(1/3,22/3,0), σRv1=(0,2/2,6/6)}.By Theorems9 and 12, we can obtain two s-Enneper surfaces and eight interpolants satisfying HC1∗. So, using the inverse transformation Ψ-1=σ-1R-1 for the surfaces and interpolants, we finally obtain two cubic surfaces and interpolants satisfying the original data-set HC1, as shown in Figure 4: The cubic surface and the interpolants shown in Figure 4(b) are obtained by the inverse transformation Ψ-1(Σ1λ), where Σ1λ is the s-Enneper surface obtained when u0=2-3 and λ=48+273/26; and the cubic surface and the interpolants in Figure 4(d) are obtained by Ψ-1(Σ2λ), where Σ2λ is the s-Enneper surface obtained when u0=-2-3 and λ=-48+273/26.Figure 4
The s-Enneper surfaces and interpolants that satisfyHC1 in Example 15: (a) four planar interpolants when u0=2-3 and λ=48+273/26; (b) four interpolants satisfying HC1 on the cubic surface Ψ-1(Σ1λ); (c) four planar interpolants when u0=-2-3 and λ=-48+273/26; (d) four interpolants satisfying HC1 on the cubic surface Ψ-1(Σ2λ).
(a) (b) (c) (d)Remark 16.
Note that when the given data-setHC1 is not regular, that is, p1-p0, v0 and v1 are linearly dependent, these three vectors must lie on a plane in R3. That allows us to reduce this interpolation problem to a planar C1 Hermite interpolation problem using a suitable PH-preserving affine transformation. Thus we can obtain [10] four planar PH interpolants satisfying HC1.


## 4. Comparison with Interpolants Obtained by Other PH-Preserving Mappings

In this section, we compare our method with the use of PH-preserving cubic mappings [24]. First note that, for a regular C1 Hermite data-set HC1 in R3, we can use mappings to obtain 16 interpolants satisfying HC1, which consist of four interpolants on each of four cubic surfaces satisfying HC1, as shown in Figure 5(a). Whereas, applying our method to the same data-set, we can obtain eight interpolants satisfying HC1, which consist of four interpolants on each of two cubic surfaces satisfying HC1, as shown in Figure 5(b). Now we will compare these two methods in terms of their requirements for algebraic computation and the geometries of the resulting curves.Figure 5
Comparison of interpolants satisfyingHC1={p0=(0,0,0),p1=(1,0,0),v0=(0,1,1),v1=(0,1,-1)} of Example 11 in [24]: (a) 16 interpolants obtained by using general PH-preserving cubic mappings; (b) 8 interpolants obtained using the s-Enneper surface. In each subfigure, the interpolant with the lowest bending energy is shown as the orange curve; the shortest interpolant is shown as the green curve; and the longest interpolant is shown as the curve denoted by dashed lines.
(a) (b)Our method requires less algebraic computation to determine the cubic surfaces satisfying the given data-set than the mapping method: when using the mapping method, to determine the PH-preserving surfaces satisfying the given data-set, lengthy computation processes are additionally required in fixing the free parameters of the surfaces, since they have many free parameters with complicated constraints unlike Enneper surfaces. Moreover, our method only requires the given data-set to meet the regularity condition, whereas mapping requires a data-set which meets several conditions.We will now examine the shapes of the interpolants produced by mapping and by our method. Looking at the interpolants in Figure5, it is clear that those in Figure 5(a) are longer and more complicated than those in Figure 5(b), which means that the surfaces satisfying the given data-set HC1 must be different.Next, we will compare the interpolants obtained using each method in terms of bending energy:(25)Eγ=∫γκ2+τ2ds, where κ and τ are the curvature and torsion of an interpolant γ. The bending energy E of a curve is an established measure of its fairness. We can consider an interpolant to have a better shape than another if it has lower bending energy with a similar arc-length. If we look at the bending energies and arc-lengths in Tables 1 and 2, we observe the following:(i)
While the curve with the lowest bending energy in Figure5(a) has the longest arc-length, which is about twice that of the shortest curve, the curve with the lowest bending energy in Figure 5(b) is not much longer than the shortest curve.(ii)
The diversity of arc-lengths among the curves in Figure5(a) is much larger than the diversity of those in Figure 5(b), where the lengths of the shortest and longest curves are in the ratio 1 : 2.36.(iii)
Finally, the lowest bending energy of any curve in Figure5(b) is 63% lower than that of the curve with the lowest bending energy in Figure 5(a).Table 1
Comparison of the bending energies and arc-length of the 3 important interpolants shown in Figure5(a).
Curve Bending energy Arc-length Dashed-orange 1 37.308 2.306 Dashed-orange 2 37.308 2.306 Green 153.924 1.228Table 2
Comparison of the bending energies and arc-lengths of the 3 important interpolants shown in Figure5(b).
Curve Bending energy Arc-length Orange 23.378 1.609 Green 599.572 1.251 Dashed-blue 71.977 1.805These results suggest that our method can produce better interpolants through a more convenient and shorter algebraic computation than mapping.


## 5. Concluding Remarks and Suggestions for Further Study

We have proved that the parametrization of the Enneper surface in standard form is PH-preserving and that it also preserves thex- and y-axes in the parametric plane. We went on to show how to produce interpolants which lie on two s-Enneper surfaces and satisfy a regular C1 Hermite data-set in R3. We also proved that eight interpolants can be obtained, four on each of the surfaces. We also compared our method with a previous method [24] based on PH-preserving cubic mappings and showed that we can obtain interpolants with lower bending energy without significant increase in arc-length.The work reported in this paper raises further questions: first, as shown in Section4, by using different parameterizations to solve the C1 Hermite interpolation problem satisfying a single data-set HC1, we obtain different cubic surfaces. Then the Enneper surface is a minimal surface: does that mean that all the cubic PH-preserving mappings are harmonic? Further, are the surfaces produced by the cubic PH-preserving mappings used in the mapping method also minimal? This brings us to the question of how to characterize all possible cubic PH-preserving mappings. Now, as shown in Example 3 of [24], PH-preserving mappings are not necessarily harmonic; and so PH-preserving surfaces need not to be minimal.Extending our perspective to include PH-preserving mappings raises another question which brings us nearer to core of the PH-preserving property: what is the key to the property of PH preservation: conformality, harmonicity, or both? finally, we might consider the following interesting questions: can we design new PH-preserving mappings by using specific PH curves inR2 such as PH-cuts of Laurent series [27]? We have been considering these last two questions and currently believe that the PH-preserving property of mappings is mainly dependent upon conformality. We have to be in a position to write on this topic shortly. In addition, PH and MPH curves are tightly connected, as stated in [11]. Hence, considering our recent work [13] on MPH-preserving mappings, we also naturally reach the following questions: what would be the equivalent of Enneper surfaces in the Minkowski setting? Can we again achieve C1 interpolation, but this time with MPH curves? These could also be the nice themes for further study.



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