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Controlling curvature in the Minkowski plane
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# Controlling curvature in the Minkowski plane

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Published by Dept. of Mathematics, University of Texas at Arlington in Arlington .
Written in English

### Subjects:

• Normed linear spaces,
• Curves, Plane

## Book details:

Edition Notes

Includes bibliographical references (leaf 4).

The Physical Object ID Numbers Statement Mostafa Ghandehari. Series TR / UTA Department of Mathematics -- #321., Technical report (University of Texas at Arlington. Dept. of Mathematics) -- no. 321. Contributions University of Texas at Arlington. Dept. of Mathematics. Pagination 4 leaves ; Open Library OL17460804M OCLC/WorldCa 37558239

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A geometric proof for the following problem is given: Let E be the unit circle in a Minkowski plane. Let C be any continuously differentiable closed curve with length l(C) (measured in Minkowski.   AbstractThis article is motivated by a problem posed by David A. Singer in and by the classical Euler elastic curves. We study spacelike and timelike curves in the Lorentz-Minkowski plane 𝕃2 whose curvature is expressed in terms of the Lorentzian pseudodistance to fixed geodesics. In this way, we get a complete description of all the elastic curves in 𝕃2 and Author: Ildefonso Castro, Ildefonso Castro-Infantes, Jesús Castro-Infantes. RESTRICTED CURVATURE IN THE MINKOWSKI PLANE MOSTAFAGHANDEHARI Abstract. A geometric proof for the following problem is given: Let E be the unit circle in a Minkowski plane. Let C be any continuously differentiable closed curve with length l(C) (measured in Minkowski metric). Assume |κ e(C,.)| kκ e(E,.) and κ e(E,.) denote Euclidean curvatures. Flow by curvature is theL2gradient ﬂow of the length First variation formula for length dL(α) = ∂L(γ(t) + α(t)) ∂. = Z. γ. ds The negative gradient vector ﬁeld, with respect to theL2metric on curves is ∇L = kN Flow by curvatureis thecurve shortening ﬂow. It shortens length as ‘quickly as possible’.File Size: 1MB.

There is a third way to obtain curvature in the Euclidean plane, which will later give birth to a certain curvature type in the normed plane (namely, the circular curvature). Actually, we obtain the curvature radius of a curve in each point. Let be a smooth curve parametrized by the arc length, Cited by: and introduced pseudosphere (a sphere with negative curvature). The results on hyperbolic geometry started to occur frequently. In , H. Minkowski reformulated the famous A. Einstein’s paper from and introduced space-time. Pavel Chalmovianský (KAGDM FMFI UK) Geometry of Minkowski Space Bratislava, 3 / and show how the curvature and the torsions of the four-dimensional path of the particle contain information on the electromagnetic ﬁeld acting on the particle. 1 Introduction Our aim in this paper is to look at some aspects of the diﬀerential geometry of curves in Minkowski space . The article is organized as follows. In Sec-File Size: KB. The Minkowski metric η is the metric tensor of Minkowski space. It is a pseudo-Euclidean metric, or more generally a constant pseudo-Riemannian metric in Cartesian coordinates. As such it is a nondegenerate symmetric bilinear form, a type (0, 2) tensor.

Controlling curvature in the Minkowski plane by Mostafa Ghandehari 1 edition - first published in The normal curvature, k n, is the curvature of the curve projected onto the plane containing the curve's tangent T and the surface normal u; the geodesic curvature, k g, is the curvature of the curve projected onto the surface's tangent plane; and the geodesic torsion (or relative torsion), τ r, measures the rate of change of the surface normal around the curve's tangent. 2. Curves in Minkowski space 53 3. Surfaces in Minkowski space 72 4. Spacelike surfaces with constant mean curvature 91 5. Elliptic equations on cmc spacelike surfaces 99 References The title of this work is motivated by the book of M. P. do Carmo, Diﬀerential Geometry of Curves and Surfaces (), and its origin was a mini-course given by the.   Motivated by the classical Euler elastic curves, David A. Singer posed in the problem of determining a plane curve whose curvature is given in terms of its position.