C 1 Hermite Interpolation with PH Curves Using the Enneper Surface

and Applied Analysis 3


Introduction
Pythagorean-hodograph (PH) curves were first introduced by Farouki and Sakkalis [1] as polynomial curves in R 2 with polynomial speed functions, which have polynomial arclengths, 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][3][4].Also, these PH curves have subsequently been extended, with several applications, to rational curves with rational speed functions in R  [3,5,6].
In this paper, we show that we can use Enneper surfaces to solve  1 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  1 Hermite interpolation with PH curves, there have been many developments: in particular, it has been shown [24] that  1 Hermite interpolation problems with PH curves in R 3 can be reduced to problems in R 2 and generic interpolants can then be obtained to satisfy a given  1 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  1 Hermite data-sets.We will address both of these issues: using the Enneper surface, we can solve  1 Hermite interpolation problems more efficiently for all regular  1 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 Section 2, we define the Pythagorean-hodograph curve and the PHpreserving 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  1 Hermite data-set.We also prove that we can obtain eight interpolants on the two cubic surfaces that satisfy a regular  1 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.

Preliminary
Let R  be the -dimensional Euclidean space, for  ∈ N, and let P[] be the set of polynomial functions with real coefficients.We express a polynomial curve in R  as a mapping r : R → R  from the space of real numbers R to R  , such that the component functions of r, which are  1 (),  2 (), . . .,   (), are members of P[].Definition 1.A polynomial curve r() = ( 1 (),  2 (), . . .,   ()) is said to be a Pythagorean-hodograph (PH) curve if its velocity vector or hodograph r  () = (  1 (), . . .,    ()) satisfies the Pythagorean condition where where R is an orthogonal matrix in R 3 ,  ∈ R \ {0} is a scaling factor, and k is a constant vector in R 3 .Then, for a PH curve r() = ((), V()) in R 2 , the mapping Ψ : R 2 → R 3 defined by is PH-preserving, since where ⟨, ⟩ denotes the usual inner product in R 3 .
In addition, let P(, V) = ((, V), (, V), (, V)) be a polynomial mapping given by Then, for a r() = ((), V()) in R 2 , since we obtain      P (r ())       which is a segment of the Tchirnhausen cubic.This is a PH curve in R 2 , and thus the curve on the surface is a PH curve in R 3 .

Construction of 𝐶 1 Hermite PH Interpolants on the Scaled Enneper Surface
Definition 4. Let Σ ∈ R 3 be a surface with the parametrization Φ : R 2 → R 3 given by 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 PHpreserving.We revisit this result briefly.
the curve Φ(r()) in R 3 is a PH curve.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 parametrization Φ is conformal (i.e., angle-preserving).Now we consider the surfaces obtained by rescaling the Enneper surface.Let Σ  be a polynomial surface of degree 3 given by the parametrization Φ  = Φ where Φ is the standard parametrization of the Enneper surface and  is a nonzero real number.Then Σ  is also a polynomial minimal surface of degree 3, and Φ  is PH-preserving, as shown in Example 3. From now on, we will call the surface Σ  the scaled Enneper surface (or s-Enneper surface) associated with a scaling factor .Note that all s-Enneper surfaces contain   and   , which we will use in Theorem 9.
Definition 8.A  1 Hermite data-set  1  = {p 0 , p 1 , k 0 , k 1 } consists of two end-points p 0 and p 1 , and two velocities k 0 and k 1 at those end-points, where p 0 , p 1 , k 0 and k 1 ∈ R 3 . 1   is said to be regular if p 1 −p 0 , k 0 and k 1 are linearly independent.Also note that if p 0 and p 1 lie on a surface Σ, and k 0 and k 1 are tangents to that surface, then we can say that Σ satisfies  1   .If  is a mapping which generates the surface Σ, then we can also say that  satisfies  1  .

we can obtain eight interpolants on the surfaces that satisfy 𝐻 1 𝐶 , by scaling and exploiting the isometry of the two s-Enneper surfaces.
Proof.Let Ψ be an affine mapping in R 3 , obtained by composing a translation, an orthogonal transformation, and a scaling, as shown in Example 3, so that Ψ(p 0 ) = (0, 0, 0) and Ψ(p 0 ) = (1, 0, 0).Then, using Ψ, we can obtain a new regular  1 Hermite data-set  1  * as follows: where  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  1  * .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 (Σ   ), where Σ   is the s-Enneper surface satisfying  1  * , for  = 1, 2.
Remark 16.Note that when the given data-set  1   is not regular, that is, p 1 − p 0 , k 0 and k 1 are linearly dependent, these three vectors must lie on a plane in R 3 .That allows us to reduce this interpolation problem to a planar  1 Hermite interpolation problem using a suitable PH-preserving affine transformation.Thus we can obtain [10] four planar PH interpolants satisfying  1   .

Comparison with Interpolants Obtained by Other PH-Preserving Mappings
In this section, we compare our method with the use of PHpreserving cubic mappings [24].First note that, for a regular  1 Hermite data-set  1  in R 3 , we can use mappings to obtain 16 interpolants satisfying  1   , which consist of four interpolants on each of four cubic surfaces satisfying  1   , as shown in Figure 5(a).Whereas, applying our method to the same data-set, we can obtain eight interpolants satisfying  1   , which consist of four interpolants on each of two cubic surfaces satisfying  1   , 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.
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 Figure 5, 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  1   must be different.
Next, we will compare the interpolants obtained using each method in terms of bending energy: where  and  are the curvature and torsion of an interpolant .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 arclengths in Tables 1 and 2, we observe the following: (i) While the curve with the lowest bending energy in Figure 5(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 Figure 5(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 Figure 5(b) is 63% lower than that of the curve with the lowest bending energy in Figure 5(a).
These results suggest that our method can produce better interpolants through a more convenient and shorter algebraic computation than mapping.

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 the and -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  1 Hermite data-set in R 3 .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 arclength.
The work reported in this paper raises further questions: first, as shown in Section 4, by using different parameterizations to solve the  1 Hermite interpolation problem satisfying a single data-set  1  , 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 in R 2 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  1 interpolation, but this time with MPH curves?These could also be the nice themes for further study.

Figure 1 :
Figure 1: (a) A PH-preserving surface parameterized by P, derived in Example 3, when 0 ≤  ≤ 1; the blue curve on the surface is the image of (), shown in (b).

Table 1 :
Comparison of the bending energies and arc-length of the 3 important interpolants shown in Figure 5(a).

Table 2 :
Comparison of the bending energies and arc-lengths of the 3 important interpolants shown in Figure 5(b).