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

  • * Corresponding author: Jiří Minarčík

    * Corresponding author: Jiří Minarčík 

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

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  • 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.

    Mathematics Subject Classification: Primary: 53C44, 14Q05, 14J70; Secondary: 34K12, 35B51.


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  • 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 $5.0943 \cdot 10^{-5}$ $3.5888 \cdot 10^{-5}$ $2.4118 \cdot 10^{-5}$
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    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 $4.6211 \cdot 10^{-3}$ $4.6209 \cdot 10^{-3}$ $3.3186 \cdot 10^{-3}$
     | Show Table
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