اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدDiscrete mKdV and Discrete Sine-Gordon Flows on Discrete Space Curves
133
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric condition. The curve is reconstructed from the deformation data by using the Sym-Tafel formula. The isoperimetric equidistant deformation of the space curves does not preserve the torsion in general. However, it is possible to construct the torsion-preserving deformation by tuning the deformation parameters. Further, it is also possible to make an arbitrary choice of the deformation described by the discrete mKdV equation or by the discrete sine-Gordon equation at each step. We finally show that the discrete deformation of discrete space curves yields the discrete K-surfaces.
قيم البحث
اقرأ أيضاً
The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrodinger equation (NLS
). In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schrodinger equation (dNLS). We also present explicit formulas for both NLS and dNLS flows in terms of the $tau$ function of the 2-component KP hierarchy.
A linkage mechanism consists of rigid bodies assembled by joints which can be used to translate and transfer motion from one form in one place to another. In this paper, we are particularly interested in a family of spacial linkage mechanisms which c
onsist of $n$-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous threefold symmetric Bricard6R linkage also known as the Kaleidocycle, which exhibits a characteristic turning over motion. We can model such a linkage as a discrete closed curve in $mathbb{R}^3$ with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. We describe certain motions of this object that are governed by the semi-discrete mKdV equations, where infinitesimally the motion of each vertex is confined in the osculating plane.
We study deformations of plane curves in the similarity geometry. It is known that continuous deformations of smooth curves are described by the Burgers hierarchy. In this paper, we formulate the discrete deformation of discrete plane curves describe
d by the discrete Burgers hierarchy as isogonal deformations. We also construct explicit formulas for the curve deformations by using the solution of linear diffusion differential/difference equations.
After characterizing the integrable discrete analogue of the Eulers elastica, we focus our attention on the problem of approximating a given discrete planar curve by an appropriate discrete Eulers elastica. We carry out the fairing process via a $L^2
!$-distance minimization to avoid the numerical instabilities. The optimization problem is solved via a gradient-driven optimization method (IPOPT). This problem is non-convex and the result strongly depends on the initial guess, so that we use a discrete analogue of the algorithm provided by Brander et al., which gives an initial guess to the optimization method.
The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrodinger equation. In
this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schrodinger equation. We also present explicit formulas for both smooth and discrete curves in terms of $tau$ functions of the two-component KP hierarchy.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
