C¹ HERMITE INTERPOLATION WITH MPH CURVES USING PH-MPH TRANSITIVE MAPPINGS

Abstract

We introduce polynomial PH-MPH transitive mappings which transform planar PH curves to MPH curves in ${\mathbb{R}}^{2,1}$, and prove that parameterizations of Enneper surfaces of the 1st and the 2nd kind and conjugates of Enneper surfaces of the 2nd kind are PH-MPH transitive. We show how to solve $C^1$ Hermite interpolation problems in ${\mathbb{R}}^{2,1}$, for an admissible $C^1$ Hermite data-set, by using the parametrization of Enneper surfaces of the 1st kind. We also show that we can obtain interpolants for at least some inadmissible data-sets by using MPH biarcs on Enneper surfaces of the 1st kind.

FIGURE 1. Enneper surfaces over the intervals -1 ≤ u ≤ 1and -1 ≤ v ≤ 1: (a) Ω₁; (b) Σ₁; (c) Σ^∗₁. These surfaces arerespectively parameterized by Ψ_k, Φ_k and Φ^∗ _k, when k = 1.

FIGURE 2. r(t) from Example 3.1, over the interval 0 ≤ t ≤ 1,and images of the Enneper surfaces produced by the mappingsΨ₁, Φ₁ and Φ^∗₁ in ℝ^2,1: (a) r(t), (b) Ψ₁(r(t)), (c) Φ₁(r(t)), and(d) Φ^∗₁(r(t)).

FIGURE 3. Projection of the region of $V^{s}_{1}$ into the x₁x₂-plane, for a fixed value of v₁₃, when $H^{s}_{C^{1}}$ is regular: (a) π(D₁); (b) π(D₂). The green region represents π($D^{c}_{1}$ ∩ $D^{c}_{2}$).

FIGURE 4. MPH interpolants satisfying $H^{s}_{C^{1}}$ and $H^{\tilde{s}}_{C^{1}}$ in Example 3.2: (a) four MPH interpolants satisfying $H^{s}_{C^{1}}$ on Ω_k, obtained using Ψ_k when u₁ = -0.2761423750; (b) four MPHinterpolants satisfying $H^{s}_{C^{1}}$ on Ω_k, obtained using Ψ_k whenu1 = -1.707106781; (c) four MPH interpolants satisfying $H^{\tilde{s}}_{C^{1}}$ on Ω_k, obtained using Ψ_k when u₁ = 0.177124344; and (d)four MPH interpolants satisfying $H^{\tilde{s}}_{C^{1}}$ on Ω_k, obtained using Ψ_k when u₁ = 2.822875656. The interpolants drawn in red areobtained by applying the mapping Ψ_k to the correspondingred interpolants in Figure 5, which are those with the lowestbending energy.

FIGURE 5. PH quintic interpolants in ℝ² satisfying $H^{p1}_{C^{1}}$ and $H^{p2}_{C^{1}}$ in Example 3.2: (a) four PH quintic interpolants satisfying $H^{p1}_{C^{1}}$ in ℝ² when u₁ = -0.2761423750; (b) four PH quinticinterpolants satisfying $H^{p1}_{C^{1}}$ in ℝ² when u₁ = -1.707106781; (c) four PH quintic interpolants satisfying $H^{p2}_{C^{1}}$ in ℝ² when u₁ = 0.177124344; and (d) four PH quintic interpolants sat-isfying $H^{p2}_{C^{1}}$ in ℝ² when u₁ = 2.822875656. The interpolantsdrawn in red are those with the lowest bending energy for thatvalue of u₁.

FIGURE 6. MPH interpolants which pass through the junction-point p∗ = ($\frac{\sqrt{3}}{\sqrt{3}+1}$, $\frac{1}{\sqrt{3}+1}$, 0 ) with C¹ continuity, and satisfy $H^{s}_{C^{1}}$ in Example 3.3: (a) the best-shaped MPH in-terpolant satisfying $H^{s}_{C^{1}}$ ; (b) 16 MPH interpolants on two Enneper surfaces of the 1st kind satisfying $H^{s}_{C^{1}}$ containing the best-shaped MPH interpolant. The interpolants satisfying $H^{1}_{C^{1}}$ are drown in blue, and these satisfying $H^{2}_{C^{1}}$ in red.

TABLE 1. Comparison of the bending energies and arc-lengths of the interpolants shown in Fig. 5.