September  2019, 24(9): 4815-4826. doi: 10.3934/dcdsb.2019032

Comparing motion of curves and hypersurfaces in $ \mathbb{R}^m $

1. 

Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Prague, 12000, Czech Republic

2. 

Faculty of Mathematics and Physics, Kanazawa University, Kakuma, Kanazawa, 920-1192, Japan

* Corresponding author: Jiří Minarčík

Received  January 2018 Revised  June 2018 Published  February 2019

Fund Project: The first and third author were partly supported by the project No. 15-27178A of Ministry of Health of the Czech Republic, the first author was partly supported by the Foundation Nadání Josefa, Marie a Zdeňky Hlávkových, by the Study at KU Scholarship from Kanazawa University and by the project of the Student Grant Agency of the Czech Technical University in Prague No. SGS17/194/OHK4/3T/14, and the second author was partly supported by JSPS KAKENHI Grant Numbers 15H03632 and 16H03953

This article aims to contribute to the understanding of the curvature flow of curves in a higher-dimensional space. Evolution of curves in $ \mathbb{R}^m $ by their curvature is compared to the motion of hypersurfaces with constrained normal velocity. The special case of shrinking hyperspheres is further analyzed both theoretically and numerically by means of a semi-discrete scheme with discretization based on osculating circles. Computational examples of evolving spherical curves are provided along with the measurement of the experimental order of convergence.

Citation: Jiří Minarčík, Masato Kimura, Michal Beneš. Comparing motion of curves and hypersurfaces in $ \mathbb{R}^m $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4815-4826. doi: 10.3934/dcdsb.2019032
References:
[1]

S. J. Altschuler, Singularities for the curve shortening flow for space curves, Journal of Differential Geometry, 34 (1991), 491-514.  doi: 10.4310/jdg/1214447218.  Google Scholar

[2]

S. J. Altschuler and M. A. Grayson, Shortening space curves and flow through singularities, Journal of Differential Geometry, 35 (1992), 283-298.  doi: 10.4310/jdg/1214448076.  Google Scholar

[3]

L. Ambrosio and H. M. Soner, A level set approach to the evolution of surfaces of any codimension, Journal of Differential Geometry, 43 (1996), 693-737.  doi: 10.4310/jdg/1214458529.  Google Scholar

[4]

M. BenešM. KimuraP. PaušD. ŠevčovičT. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 3 (2008), 509-523.   Google Scholar

[5]

P. BurchardL. T. ChengB. Merriman and S. Osher, Motion of curves in three spatial dimensions using a level set approach, Journal of Computational Physics, 170 (2001), 720-741.  doi: 10.1006/jcph.2001.6758.  Google Scholar

[6]

J. Christiansen, Numerical solution of ordinary simultaneous differential equations of the 1st order using a method for automatic step change, Numerische Mathematik, 14 (1970), 317-324.  doi: 10.1007/BF02165587.  Google Scholar

[7]

K. Corrales, Non existence of type Ⅱ singularities for embedded and unknotted space curves, preprint, arXiv: 1605.03100v1, 2016. Google Scholar

[8]

G. Dziuk, Convergence of a semi-discrete scheme for the curve shortening flow, Mathematical Models and Methods in Applied Sciences, 4 (1994), 589-606.  doi: 10.1142/S0218202594000339.  Google Scholar

[9] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.   Google Scholar
[10]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, Journal of Differential Geometry, 23 (1986), 69-96.  doi: 10.4310/jdg/1214439902.  Google Scholar

[11]

M. Grayson, The heat equation shrinks embedded plane curves to round points, Journal of Differential Geometry, 26 (1987), 285-314.  doi: 10.4310/jdg/1214441371.  Google Scholar

[12]

S. He, Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow, preprint, arXiv: 1209.5146v1, 2012. Google Scholar

[13]

M. Holodniok, A. Klíč, M. Kubíček and M. Marek, Methods of Analysis of Nonlinear Dynamical Models, Academia Praha, 1986. Google Scholar

[14]

G. Huisken, Flow by mean curvature of convex surfaces into spheres, Journal of Differential Geometry, 20 (1984), 237-266.  doi: 10.4310/jdg/1214438998.  Google Scholar

[15]

G. Khan, A condition ensuring spatial curves develop type-Ⅱ singularities under curve shortening flow, preprint, arXiv: 1209.4072v3, 2015. Google Scholar

[16]

M. Kimura, Geometry of hypersurfaces and moving hypersurfaces in $\mathbb{R}^m$ for the study of moving boundary problems, Topics on Partial Differential Equations, Jindřich Nečas Center for Mathematical Modeling, Lecture notes, 4 (2008), 39–93.  Google Scholar

[17]

M. KolářM. Beneš and D. Ševčovič, Area preserving geodesic curvature driven flow of closed curves on a surface, Discrete and Continuous Dynamical Systems - Series B, 22 (2017), 3671-3689.  doi: 10.3934/dcdsb.2017148.  Google Scholar

[18]

K. Mikula, Image processing with partial differential equations, Nato Science Series, 75 (2002), 283-322.   Google Scholar

[19]

T. Mura, Micromechanics of Defects in Solids, Springer Netherlands, 1987. Google Scholar

[20]

S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[21]

J. R. Rice and M. Mu, An experimental performance analysis for the rate of convergence of collocation on general domains, Numerical Methods for Partial Differential Equations, 5 (1989), 45-52.  doi: 10.1002/num.1690050105.  Google Scholar

[22] J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1996.   Google Scholar
[23]

A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.  Google Scholar

[24]

Y. Y. Yang and X. X. Jiao, Curve shortening flow in arbitrary dimensional Euclidian space, Acta Mathematica Sinica, 21 (2005), 715-722.  doi: 10.1007/s10114-004-0426-z.  Google Scholar

show all references

References:
[1]

S. J. Altschuler, Singularities for the curve shortening flow for space curves, Journal of Differential Geometry, 34 (1991), 491-514.  doi: 10.4310/jdg/1214447218.  Google Scholar

[2]

S. J. Altschuler and M. A. Grayson, Shortening space curves and flow through singularities, Journal of Differential Geometry, 35 (1992), 283-298.  doi: 10.4310/jdg/1214448076.  Google Scholar

[3]

L. Ambrosio and H. M. Soner, A level set approach to the evolution of surfaces of any codimension, Journal of Differential Geometry, 43 (1996), 693-737.  doi: 10.4310/jdg/1214458529.  Google Scholar

[4]

M. BenešM. KimuraP. PaušD. ŠevčovičT. Tsujikawa and S. Yazaki, Application of a curvature adjusted method in image segmentation, Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 3 (2008), 509-523.   Google Scholar

[5]

P. BurchardL. T. ChengB. Merriman and S. Osher, Motion of curves in three spatial dimensions using a level set approach, Journal of Computational Physics, 170 (2001), 720-741.  doi: 10.1006/jcph.2001.6758.  Google Scholar

[6]

J. Christiansen, Numerical solution of ordinary simultaneous differential equations of the 1st order using a method for automatic step change, Numerische Mathematik, 14 (1970), 317-324.  doi: 10.1007/BF02165587.  Google Scholar

[7]

K. Corrales, Non existence of type Ⅱ singularities for embedded and unknotted space curves, preprint, arXiv: 1605.03100v1, 2016. Google Scholar

[8]

G. Dziuk, Convergence of a semi-discrete scheme for the curve shortening flow, Mathematical Models and Methods in Applied Sciences, 4 (1994), 589-606.  doi: 10.1142/S0218202594000339.  Google Scholar

[9] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992.   Google Scholar
[10]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, Journal of Differential Geometry, 23 (1986), 69-96.  doi: 10.4310/jdg/1214439902.  Google Scholar

[11]

M. Grayson, The heat equation shrinks embedded plane curves to round points, Journal of Differential Geometry, 26 (1987), 285-314.  doi: 10.4310/jdg/1214441371.  Google Scholar

[12]

S. He, Distance comparison principle and Grayson type theorem in the three dimensional curve shortening flow, preprint, arXiv: 1209.5146v1, 2012. Google Scholar

[13]

M. Holodniok, A. Klíč, M. Kubíček and M. Marek, Methods of Analysis of Nonlinear Dynamical Models, Academia Praha, 1986. Google Scholar

[14]

G. Huisken, Flow by mean curvature of convex surfaces into spheres, Journal of Differential Geometry, 20 (1984), 237-266.  doi: 10.4310/jdg/1214438998.  Google Scholar

[15]

G. Khan, A condition ensuring spatial curves develop type-Ⅱ singularities under curve shortening flow, preprint, arXiv: 1209.4072v3, 2015. Google Scholar

[16]

M. Kimura, Geometry of hypersurfaces and moving hypersurfaces in $\mathbb{R}^m$ for the study of moving boundary problems, Topics on Partial Differential Equations, Jindřich Nečas Center for Mathematical Modeling, Lecture notes, 4 (2008), 39–93.  Google Scholar

[17]

M. KolářM. Beneš and D. Ševčovič, Area preserving geodesic curvature driven flow of closed curves on a surface, Discrete and Continuous Dynamical Systems - Series B, 22 (2017), 3671-3689.  doi: 10.3934/dcdsb.2017148.  Google Scholar

[18]

K. Mikula, Image processing with partial differential equations, Nato Science Series, 75 (2002), 283-322.   Google Scholar

[19]

T. Mura, Micromechanics of Defects in Solids, Springer Netherlands, 1987. Google Scholar

[20]

S. Osher and J. A. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[21]

J. R. Rice and M. Mu, An experimental performance analysis for the rate of convergence of collocation on general domains, Numerical Methods for Partial Differential Equations, 5 (1989), 45-52.  doi: 10.1002/num.1690050105.  Google Scholar

[22] J. A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, 1996.   Google Scholar
[23]

A. Visintin, Models of Phase Transitions, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4078-5.  Google Scholar

[24]

Y. Y. Yang and X. X. Jiao, Curve shortening flow in arbitrary dimensional Euclidian space, Acta Mathematica Sinica, 21 (2005), 715-722.  doi: 10.1007/s10114-004-0426-z.  Google Scholar

Figure 1.  Visualization for the definition of $ \mathbf{y}(u,t) $ and $ \bar{u}(t) $
Figure 2.  Visualization of the geometric quantities defined in the osculating plane
Figure 3.  Results of the numerical simulation from Example 1. The discretized curve is visualized at four different time levels along with the corresponding sphere
Figure 4.  Results of the numerical simulation from Example 2. The discretized curve is visualized at four different time levels along with the corresponding sphere
Table 1.  Results of the numerical computation from Example 1. The error measurements were taken during time interval $ [0,0.45] $
$N$ $\mathcal{E}_{\infty}(N)$ EOC$_{\infty}$ $\mathcal{E}_{1}(N)$ EOC1 $\mathcal{E}_{2}(N)$ EOC2
100 $1.0205 \cdot 10^{-2}$ 2.0283 $9.3770 \cdot 10^{-3}$ 2.0540 $6.4629 \cdot 10^{-3}$ 2.0794
200 $2.5018 \cdot 10^{-3}$ 1.9372 $2.2581 \cdot 10^{-3}$ 1.9449 $1.5292 \cdot 10^{-3}$ 1.9758
400 $6.5329 \cdot 10^{-4}$ 1.9746 $5.8649 \cdot 10^{-4}$ 2.0234 $3.8876 \cdot 10^{-4}$ 1.9924
800 $1.6623 \cdot 10^{-4}$ 1.7062 $1.4426 \cdot 10^{-4}$ 2.0071 $9.7702 \cdot 10^{-5}$ 2.0183
1600 $5.0943 \cdot 10^{-5}$ $3.5888 \cdot 10^{-5}$ $2.4118 \cdot 10^{-5}$
$N$ $\mathcal{E}_{\infty}(N)$ EOC$_{\infty}$ $\mathcal{E}_{1}(N)$ EOC1 $\mathcal{E}_{2}(N)$ EOC2
100 $1.0205 \cdot 10^{-2}$ 2.0283 $9.3770 \cdot 10^{-3}$ 2.0540 $6.4629 \cdot 10^{-3}$ 2.0794
200 $2.5018 \cdot 10^{-3}$ 1.9372 $2.2581 \cdot 10^{-3}$ 1.9449 $1.5292 \cdot 10^{-3}$ 1.9758
400 $6.5329 \cdot 10^{-4}$ 1.9746 $5.8649 \cdot 10^{-4}$ 2.0234 $3.8876 \cdot 10^{-4}$ 1.9924
800 $1.6623 \cdot 10^{-4}$ 1.7062 $1.4426 \cdot 10^{-4}$ 2.0071 $9.7702 \cdot 10^{-5}$ 2.0183
1600 $5.0943 \cdot 10^{-5}$ $3.5888 \cdot 10^{-5}$ $2.4118 \cdot 10^{-5}$
Table 2.  Results of the numerical computation from Example 2. The error measurements were taken during time interval $ [0,0.45] $
$N$ $\mathcal{E}_{\infty}(N)$EOC$_{\infty}$ $\mathcal{E}_{1}(N)$EOC$_1$ $\mathcal{E}_{2}(N)$EOC$_2$
400 $1.1538 \cdot 10^{-1}$1.9639 $1.1538 \cdot 10^{-1}$1.9640 $1.0340 \cdot 10^{-1}$2.2232
800 $2.9575 \cdot 10^{-2}$2.0276 $2.9574 \cdot 10^{-2}$2.0280 $2.2146 \cdot 10^{-2}$2.0975
1200 $1.2998 \cdot 10^{-2}$2.0218 $1.2996 \cdot 10^{-2}$2.0222 $9.4608 \cdot 10^{-3}$2.0547
1600 $7.2659 \cdot 10^{-3}$2.0281 $7.2636 \cdot 10^{-3}$2.0269 $5.2386 \cdot 10^{-3}$2.0459
2000 $4.6211 \cdot 10^{-3}$ $4.6209 \cdot 10^{-3}$ $3.3186 \cdot 10^{-3}$
$N$ $\mathcal{E}_{\infty}(N)$EOC$_{\infty}$ $\mathcal{E}_{1}(N)$EOC$_1$ $\mathcal{E}_{2}(N)$EOC$_2$
400 $1.1538 \cdot 10^{-1}$1.9639 $1.1538 \cdot 10^{-1}$1.9640 $1.0340 \cdot 10^{-1}$2.2232
800 $2.9575 \cdot 10^{-2}$2.0276 $2.9574 \cdot 10^{-2}$2.0280 $2.2146 \cdot 10^{-2}$2.0975
1200 $1.2998 \cdot 10^{-2}$2.0218 $1.2996 \cdot 10^{-2}$2.0222 $9.4608 \cdot 10^{-3}$2.0547
1600 $7.2659 \cdot 10^{-3}$2.0281 $7.2636 \cdot 10^{-3}$2.0269 $5.2386 \cdot 10^{-3}$2.0459
2000 $4.6211 \cdot 10^{-3}$ $4.6209 \cdot 10^{-3}$ $3.3186 \cdot 10^{-3}$
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