In this paper, we present a method of $ G^2 $ Hermite interpolation of convolution curves of regular plane curves and ellipses. We show that our approximant is also a $ C^1 $ Hermite interpolation of the convolution curve. This method yields a polynomial curve if the trajectory curve is a polynomial curve. Our approximation method is applied to two previous numerical examples. The results of our method are compared with those of previous methods, and the merits and demerits are analyzed. Compared with previous methods, the merits of our method are that the approximant is $ G^2 $ and $ C^1 $ Hermite interpolation, and the degree of the approximant or the required number of segments of the approximant within error tolerances is small.
Citation: Young Joon Ahn. An approximation method for convolution curves of regular curves and ellipses[J]. AIMS Mathematics, 2024, 9(12): 34606-34617. doi: 10.3934/math.20241648
In this paper, we present a method of $ G^2 $ Hermite interpolation of convolution curves of regular plane curves and ellipses. We show that our approximant is also a $ C^1 $ Hermite interpolation of the convolution curve. This method yields a polynomial curve if the trajectory curve is a polynomial curve. Our approximation method is applied to two previous numerical examples. The results of our method are compared with those of previous methods, and the merits and demerits are analyzed. Compared with previous methods, the merits of our method are that the approximant is $ G^2 $ and $ C^1 $ Hermite interpolation, and the degree of the approximant or the required number of segments of the approximant within error tolerances is small.
| [1] |
I.-K. Lee, M.-S. Kim, G. Elber, Polynomial/rational approximation of Minkowski sum boundary curves, Graphical Models and Image Processing, 60 (1998), 136–165. https://doi.org/10.1006/gmip.1998.0464 doi: 10.1006/gmip.1998.0464
|
| [2] |
R. Yamada, T. Tsuji, T. Hiramitsu, H. Seki, T. Nishimura, Y. Suzuki, et al., Fast and precise approximation of minkowski sum of two rotational ellipsoids with a superellipsoid, Vis. Comput., 40 (2024), 4609–4621. https://doi.org/10.1007/s00371-024-03445-9 doi: 10.1007/s00371-024-03445-9
|
| [3] |
C. L. Bajaj, M.-S. Kim, Generation of configuration space obstacles: The case of moving algebraic curves, Algorithmica, 4 (1989), 157–172. https://doi.org/10.1007/BF01553884 doi: 10.1007/BF01553884
|
| [4] |
M. Kohler, M. Spreng, Fast computation of the C-space of convex 2D algebraic objects, Int. J. Robot. Res., 14 (1995), 590–608. https://doi.org/10.1177/027836499501400605 doi: 10.1177/027836499501400605
|
| [5] |
S. Ruan, G. S. Chirikjian, Closed-form Minkowski sums of convex bodies with smooth positively curved boundaries, Comput.-Aided Design, 143 (2022), 103133. https://doi.org/10.1016/j.cad.2021.103133 doi: 10.1016/j.cad.2021.103133
|
| [6] |
J. Vršek, M. Lávička, On convolutions of algebraic curves, J. Symb. Comput., 45 (2010), 657–676. https://doi.org/10.1016/j.jsc.2010.02.001 doi: 10.1016/j.jsc.2010.02.001
|
| [7] |
J. Vršek, M. Lávička, Exploring hypersurfaces with offset-like convolutions, Comput. Aided Geom. D., 29 (2012), 676–690. https://doi.org/10.1016/j.cagd.2012.07.002 doi: 10.1016/j.cagd.2012.07.002
|
| [8] |
Y. J. Ahn, C. Hoffmann, Y. S. Kim, Curvature-continuous offset approximation based on circle approximation using quadratic Bézier biarcs, Comput.-Aided Design, 43 (2011), 1011–1017. https://doi.org/10.1016/j.cad.2011.04.005 doi: 10.1016/j.cad.2011.04.005
|
| [9] |
R. Lee, Y. J. Ahn, Geometric shape analysis for convolution curve of two compatible quadratic Bézier curves, J. Comput. Appl. Math., 288 (2015), 141–150. https://doi.org/10.1016/j.cam.2015.04.012 doi: 10.1016/j.cam.2015.04.012
|
| [10] |
I.-K. Lee, M.-S. Kim, G. Elber, Planar curve offset based on circle approximation, Comput.-Aided Design, 28 (1996), 617–630. https://doi.org/10.1016/0010-4485(95)00078-X doi: 10.1016/0010-4485(95)00078-X
|
| [11] |
S. W. Kim, S. C. Bae, Y. J. Ahn, An algorithm for $G^2$ offset approximation based on circle approximation by $G^2$ quadratic spline, Comput.-Aided Design, 73 (2016), 36–40. https://doi.org/10.1016/j.cad.2015.11.003 doi: 10.1016/j.cad.2015.11.003
|
| [12] |
Y. J. Ahn, C. M. Hoffmann, Approximate convolution with pairs of cubic Bézier LN curves, Comput. Aided Geom. D., 28 (2011), 357–367. https://doi.org/10.1016/j.cagd.2011.06.006 doi: 10.1016/j.cagd.2011.06.006
|
| [13] | B. Jüttler, Triangular Bézier surface patches with a linear normal vector field, In: The mathematics of surfaces VIII, information geometers, Winchester, 1998,431–446. |
| [14] |
B. Jüttler, M. L. Sampoli, Hermite interpolation by piecewise polynomial surfaces with rational offsets, Comput. Aided Geom. D., 17 (2000), 361–385. https://doi.org/10.1016/S0167-8396(00)00002-9 doi: 10.1016/S0167-8396(00)00002-9
|
| [15] | M. L. Sampoli, Computing the convolution and the Minkowski sum of surfaces, Proceedings of the 21st Spring Conference on Computer Graphics, New York: Association for Computing Machinery, 2005,111–117. https://doi.org/10.1145/1090122.1090142 |
| [16] |
M. L. Sampoli, M. Peternell, B. Jüttler, Rational surfaces with linear normals and their convolutions with rational surfaces, Comput. Aided Geom. D., 23 (2006), 179–192. https://doi.org/10.1016/j.cagd.2005.07.001 doi: 10.1016/j.cagd.2005.07.001
|
| [17] |
Y. J. Ahn, C. Hoffmann, Sequence of ${G}^n$ LN polynomial curves approximating circular arcs, J. Comput. Appl. Math., 341 (2018), 117–126. https://doi.org/10.1016/j.cam.2018.03.028 doi: 10.1016/j.cam.2018.03.028
|
| [18] |
R. T. Farouki, H. P. Moon, B. Ravani, Algorithms for Minkowski products and implicitly‐defined complex sets, Adv. Comput. Math., 13 (2000), 199–229. https://doi.org/10.1023/A:1018910412112 doi: 10.1023/A:1018910412112
|
| [19] |
S. W. Kim, R. Lee, Y. J. Ahn, A new method approximating offset curve by Bézier curve using parallel derivative curves, Comp. Appl. Math., 37 (2018), 2053–2064. https://doi.org/10.1007/s40314-017-0437-x doi: 10.1007/s40314-017-0437-x
|
| [20] |
Y. J. Ahn, $G^2/C^1$ Hermite interpolation of offset curves of parametric regular curves, AIMS Mathematics, 8 (2023), 31008–31021. https://doi.org/10.3934/math.20231587 doi: 10.3934/math.20231587
|
| [21] |
Y. J. Ahn, C. Hoffmann, Circle approximation using LN Bézier curves of even degree and its application, J. Math. Anal. Appl., 410 (2014), 257–266. https://doi.org/10.1016/j.jmaa.2013.07.079 doi: 10.1016/j.jmaa.2013.07.079
|
| [22] |
Y. J. Ahn, J. Cui, C. Hoffmann, Circle approximations on spheroids, J. Navigation, 64 (2011), 739–749. https://doi.org/10.1017/S0373463311000294 doi: 10.1017/S0373463311000294
|
| [23] |
H. M. Yoon, Y. J. Ahn, Circular arc approximation by hexic polynomial curves, Comp. Appl. Math., 42 (2023), 265. https://doi.org/10.1007/s40314-023-02315-9 doi: 10.1007/s40314-023-02315-9
|
| [24] |
M. Floater, An ${O}(h^2n)$ Hermite approximation for conic sections, Comput. Aided Geom. D., 14 (1997), 135–151. https://doi.org/10.1016/S0167-8396(96)00025-8 doi: 10.1016/S0167-8396(96)00025-8
|
| [25] |
Y. J. Ahn, A property of parametric polynomial approximants of half circles, Comput. Aided Geom. D., 87 (2021), 101990. https://doi.org/10.1016/j.cagd.2021.101990 doi: 10.1016/j.cagd.2021.101990
|
| [26] |
B.-G. Lee, Y. Park, J. Yoo, Application of Legendre-Bernstein basis transformations to degree elevation and degree reduction, Comput. Aided Geom. D., 19 (2002), 709–718. https://doi.org/10.1016/S0167-8396(02)00164-4 doi: 10.1016/S0167-8396(02)00164-4
|
Figures(2) / Tables(2)
Young Joon Ahn. An approximation method for convolution curves of regular curves and ellipses[J]. AIMS Mathematics, 2024, 9(12): 34606-34617. doi: 10.3934/math.20241648
DownLoad: