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

  • Kim, Gwangil (Department of Mathematics and RINS College of Natural Science GyeongSang National University) ;
  • Kong, Jae Hoon (Department of Mathematics GyeongSang National University) ;
  • Lee, Hyun Chol (Department of Mathematics GyeongSang National University)
  • Received : 2018.06.29
  • Accepted : 2018.08.02
  • Published : 2019.05.01

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.

Keywords

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FIGURE 1. Enneper surfaces over the intervals -1 ≤ u ≤ 1and -1 ≤ v ≤ 1: (a) Ω1; (b) Σ1; (c) Σ1. These surfaces arerespectively parameterized by Ψk, Φk and Φ k, when k = 1.

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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 Φ1 in ℝ2,1: (a) r(t), (b) Ψ1(r(t)), (c) Φ1(r(t)), and(d) Φ1(r(t)).

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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}$).

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

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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.

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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.

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References

  1. G. Albrecht and R. T. Farouki, Construction of $C^2$ Pythagorean-hodograph interpolating splines by the homotopy method, Adv. Comput. Math. 5 (1996), no. 4, 417-442. https://doi.org/10.1007/BF02124754
  2. H. I. Choi, S. W. Choi, and H. P. Moon, Mathematical theory of medial axis transform, Pacific J. Math. 181 (1997), no. 1, 57-88. https://doi.org/10.2140/pjm.1997.181.57
  3. H. I. Choi, D. S. Lee, and H. P. Moon, Clifford algebra, spin representation, and rational parameterization of curves and surfaces, Adv. Comput. Math. 17 (2002), no. 1-2, 5-48. https://doi.org/10.1023/A:1015294029079
  4. R. T. Farouki, Pythagorean-hodograph curves: algebra and geometry inseparable, Geometry and Computing, 1, Springer, Berlin, 2008.
  5. R. T. Farouki, C. Manni, M. L. Sampoli, and A. Sestini, Shape-preserving interpolation of spatial data by Pythagorean-hodograph quintic spline curves, IMA J. Numer. Anal. 35 (2015), no. 1, 478-498. https://doi.org/10.1093/imanum/drt072
  6. R. T. Farouki, C. Manni, and A. Sestini, Shape-preserving interpolation by $G^1$ and $G^2$ PH quintic splines, IMA J. Numer. Anal. 23 (2003), no. 2, 175-195. https://doi.org/10.1093/imanum/23.2.175
  7. R. T. Farouki and C. A. Neff, Hermite interpolation by Pythagorean hodograph quintics, Math. Comp. 64 (1995), no. 212, 1589-1609. https://doi.org/10.1090/S0025-5718-1995-1308452-6
  8. R. T. Farouki, F. Pelosi, M. L. Sampoli, and A. Sestini, Tensor-product surface patches with Pythagorean-hodograph isoparametric curves, IMA J. Numer. Anal. 36 (2016), no. 3, 1389-1409. https://doi.org/10.1093/imanum/drv025
  9. R. T. Farouki and T. Sakkalis, Pythagorean hodographs, IBM J. Res. Develop. 34 (1990), no. 5, 736-752. https://doi.org/10.1147/rd.345.0736
  10. R. T. Farouki and Z. Sir, Rational Pythagorean-hodograph space curves, Comput. Aided Geom. Design 28 (2011), no. 2, 75-88. https://doi.org/10.1016/j.cagd.2011.01.002
  11. Z. Habib and M. Sakai, $G^2$ Pythagorean hodograph quintic transition between two circles with shape control, Comput. Aided Geom. Design 24 (2007), no. 5, 252-266. https://doi.org/10.1016/j.cagd.2007.03.004
  12. G. Jaklic, J. Kozak, M. Krajnc, V. Vitrih, and E. Zagar, On interpolation by planar cubic $G^2$ Pythagorean-hodograph spline curves, Math. Comp. 79 (2010), no. 269, 305-326. https://doi.org/10.1090/S0025-5718-09-02298-4
  13. G.-I. Kim and S. Lee, Pythagorean-hodograph preserving mappings, J. Comput. Appl. Math. 216 (2008), no. 1, 217-226. https://doi.org/10.1016/j.cam.2007.04.026
  14. O. Kobayashi, Maximal surfaces in the 3-dimensional Minkowski space $L^3$, Tokyo J. Math. 6 (1983), no. 2, 297-309. https://doi.org/10.3836/tjm/1270213872
  15. J. H. Kong, S. P. Jeong, S. Lee, and G. Kim, $C^1$ Hermite interpolation with simple planar PH curves by speed reparametrization, Comput. Aided Geom. Design 25 (2008), no. 4-5, 214-229. https://doi.org/10.1016/j.cagd.2007.11.006
  16. J. H. Kong, H. C. Lee, and G. I. Kim, $C^1$ Hermite interpolation with PH curves by boundary data modification, J. Comput. Appl. Math. 248 (2013), 47-60. https://doi.org/10.1016/j.cam.2013.01.016
  17. J. H. Kong, S. Lee, and G. Kim, Minkowski Pythagorean-hodograph preserving mappings, J. Comput. Appl. Math. 308 (2016), 166-176. https://doi.org/10.1016/j.cam.2016.05.032
  18. J. Kosinka and B. Juttler, $C^1$ Hermite interpolation by Pythagorean hodograph quintics in Minkowski space, Adv. Comput. Math. 30 (2009), no. 2, 123-140. https://doi.org/10.1007/s10444-007-9059-y
  19. J. Kosinka and M. Lavicka, A unified Pythagorean hodograph approach to the medial axis transform and offset approximation, J. Comput. Appl. Math. 235 (2011), no. 12, 3413-3424. https://doi.org/10.1016/j.cam.2011.02.001
  20. J. Kosinka and M. Lavicka, Pythagorean hodograph curves: a survey of recent advances, J. Geom. Graph. 18 (2014), no. 1, 23-43.
  21. J. Kosinka and Z. Sir, $C^2$ Hermite interpolation by Minkowski Pythagorean hodograph curves and medial axis transform approximation, Comput. Aided Geom. Design 27 (2010), no. 8, 631-643. https://doi.org/10.1016/j.cagd.2010.04.005
  22. S. Lee, H. C. Lee, M. R. Lee, S. Jeong, and G. I. Kim, Hermite interpolation using Mobius transformations of planar Pythagorean-hodograph cubics, Abstr. Appl. Anal. 2012 (2012), Art. ID 560246, 15 pp.
  23. H. P. Moon, Minkowski Pythagorean hodographs, Comput. Aided Geom. Design 16 (1999), no. 8, 739-753. https://doi.org/10.1016/S0167-8396(99)00016-3
  24. F. Pelosi, R. T. Farouki, C. Manni, and A. Sestini, Geometric Hermite interpolation by spatial Pythagorean-hodograph cubics, Adv. Comput. Math. 22 (2005), no. 4, 325-352. https://doi.org/10.1007/s10444-003-2599-x
  25. Z. Sir and B. Juttler, Euclidean and Minkowski Pythagorean hodograph curves over planar cubics, Comput. Aided Geom. Design 22 (2005), no. 8, 753-770. https://doi.org/10.1016/j.cagd.2005.03.002