• Title/Summary/Keyword: H. Minkowski

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Interactive Control of Geometric Shape Morphing based on Minkowski Sum (민코프스키 덧셈 연산에 근거한 기하 도형의 모핑 제어 방법)

  • Lee, J.-H.;Lee, J. Y.;Kim, H.;Kim, H. S.
    • Korean Journal of Computational Design and Engineering
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    • v.7 no.4
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    • pp.269-279
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    • 2002
  • Geometric shape morphing is an interesting geometric operation that interpolates two geometric shapes to generate in-betweens. It is well known that Minkowski operations can be used to test and build collision-free motion paths and to modify shapes in digital image processing. In this paper, we present a new geometric modeling technique to control the morphing on geometric shapes based on Minkowski sum. The basic idea develops from the linear interpolation on two geometric shapes where the traditional algebraic sum is replaced by Minkowski sum. We extend this scheme into a Bezier-like control structure with multiple control shapes, which enables the interactive control over the intermediate shapes during the morphing sequence as in the traditional CAGD curve/surface editing. Moreover, we apply the theory of blossoming to our control structure, whereby our control structure becomes even more flexible and general. In this paper, we present mathematical models of control structure, their properties, and computational issues with examples.

TIMELIKE TUBULAR SURFACES OF WEINGARTEN TYPES AND LINEAR WEINGARTEN TYPES IN MINKOWSKI 3-SPACE

  • Chenghong He;He-jun Sun
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.2
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    • pp.401-419
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    • 2024
  • Let K, H, KII and HII be the Gaussian curvature, the mean curvature, the second Gaussian curvature and the second mean curvature of a timelike tubular surface Tγ(α) with the radius γ along a timelike curve α(s) in Minkowski 3-space E31. We prove that Tγ(α) must be a (K, H)-Weingarten surface and a (K, H)-linear Weingarten surface. We also show that Tγ(α) is (X, Y)-Weingarten type if and only if its central curve is a circle or a helix, where (X, Y) ∈ {(K, KII), (K, HII), (H, KII), (H, HII), (KII , HII)}. Furthermore, we prove that there exist no timelike tubular surfaces of (X, Y)-linear Weingarten type, (X, Y, Z)-linear Weingarten type and (K, H, KII, HII)-linear Weingarten type along a timelike curve in E31, where (X, Y, Z) ∈ {(K, H, KII), (K, H, HII), (K, KII, HII), (H, KII, HII)}.

ON NULL SCROLLS SATISFYING THE CONDITION ${\triangle}$H = AH

  • Pak, Jin-Suk;Yoon, Dae-Won
    • Communications of the Korean Mathematical Society
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    • v.15 no.3
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    • pp.533-540
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    • 2000
  • In the present paper, we study a non-degenrate ruled surface along a null curve in a 3-dimensional Minkowski space E31, which is called a null scroll, an investigate some characterizations of null scrolls satisfying the condition H=AH, A Mat(3, ), where denotes the Laplacian of the surface with respect to the induced metric, H the mean curvature vector and Mat(3, ) the set of 3$\times$3-real matrices.

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On the History of the Birth of Finsler Geometry at Göttingen (괴팅겐에서 핀슬러 기하가 탄생한 역사)

  • Won, Dae Yeon
    • Journal for History of Mathematics
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    • v.28 no.3
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    • pp.133-149
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    • 2015
  • Arrivals of Hilbert and Minkowski at $G\ddot{o}ttingen$ put mathematical science there in full flourish. They further extended its strong mathematical tradition of Gauss and Riemann. Though Riemann envisioned Finsler metric and gave an example of it in his inaugural lecture of 1854, Finsler geometry was officially named after Minkowski's academic grandson Finsler. His tool to generalize Riemannian geometry was the calculus of variations of which his advisor $Carath\acute{e}odory$ was a master. Another $G\ddot{o}ttingen$ graduate Busemann regraded Finsler geometry as a special case of geometry of metric spaces. He was a student of Courant who was a student of Hilbert. These figures all at $G\ddot{o}ttingen$ created and developed Finsler geometry in its early stages. In this paper, we investigate history of works on Finsler geometry contributed by these frontiers.

THE m-TH ROOT FINSLER METRICS ADMITTING (α, β)-TYPES

  • Kim, Byung-Doo;Park, Ha-Yong
    • Bulletin of the Korean Mathematical Society
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    • v.41 no.1
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    • pp.45-52
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    • 2004
  • The theory of m-th root metric has been developed by H. Shimada [8], and applied to the biology [1] as an ecological metric. The purpose of this paper is to introduce the m-th root Finsler metrics which admit ($\alpha,\;\beta$)-types. Especially in cases of m = 3, 4, we give the condition for Finsler spaces with such metrics to be locally Minkowski spaces.

POSITION VECTORS OF A SPACELIKE W-CURVE IN MINKOWSKI SPACE 𝔼13

  • Ilarslan, Kazim;Boyacioglu, Ozgur
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.3
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    • pp.429-438
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    • 2007
  • In this paper, we study the position vectors of a spacelike W-curve (or a helix), i.e., curve with constant curvatures, with spacelike, timelike and null principal normal in the Minkowski 3-space $\mathbb{E}_1^3$. We give some characterizations for spacelike W - curves whose image lies on the pseudohyperbolical space $\mathbb{H}_0^2$ and Lorentzian sphere $\mathbb{S}_1^2$ by using the positions vectors of the curve.

ANOTHER CHARACTERIZATION OF ROUND SPHERES

  • Lee, Seung-Won;Koh, Sung-Eun
    • Bulletin of the Korean Mathematical Society
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    • v.36 no.4
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    • pp.701-706
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    • 1999
  • A characterization of geodesic spheres in the simply connected space forms in terms of the ratio of the Gauss-Kronecker curvature and the (usual) mean curvature is given: An immersion of n dimensional compact oriented manifold without boundary into the n + 1 dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilicimmersion if the mean curvature $H_1$ does not vanish and the ratio $H_n$/$H_1$ of the Gauss-Kronecker curvature $H_n$ and $H_1$ is constant.

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A CHARACTERIZATION OF HYPERBOLIC SPACES

  • Kim, Dong-Soo;Kim, Young Ho;Lee, Jae Won
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.4
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    • pp.1103-1107
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    • 2018
  • Let M be a complete spacelike hypersurface in the (n + 1)-dimensional Minkowski space ${\mathbb{L}}^{n+1}$. Suppose that every unit speed curve X(s) on M satisfies ${\langle}X^{\prime\prime}(s),X^{\prime\prime}s){\rangle}{\geq}-1/r^2$ and there exists a point $p{\in}M$ such that for every unit speed geodesic X(s) of M through the point p, ${\langle}X^{\prime\prime}(s),X^{\prime\prime}s){\rangle}=-1/r^2$ holds. Then, we show that up to isometries of ${\mathbb{L}}^{n+1}$, M is the hyperbolic space $H^n(r)$.