Showing papers by "Jun-ichi Inoguchi published in 2018"
TL;DR: In this paper, the authors determined magnetic curves corresponding to the Killing magnetic fields in the 3-dimensional homogeneous Riemannian space Sol3, and showed that these curves correspond to the same magnetic fields as those of the Earth's magnetic field.
Abstract: We determine magnetic curves corresponding to the Killing magnetic fields in the 3-dimensional homogenous Riemannian space Sol3.
17 citations
TL;DR: The result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.
8 citations
TL;DR: In this article, the authors studied contact magnetic curves in the unit tangent sphere bundle over the Euclidean plane, and obtained all contact magnetic curve which are slant, which is a special case of the slant magnetic curves.
8 citations
TL;DR: In this article, the authors considered a vector field ξ as a map from a Riemannian manifold into its tangent bundle endowed with the usual almost Kahlerian structure and found necessary and sufficient conditions for ξ to be a magnetic map with respect to ξ itself and the Kahler 2-form.
Abstract: In a previous paper, we introduced the notion of magnetic vector fields. More
precisely, we consider a vector field ξ as a map from a Riemannian manifold
into its tangent bundle endowed with the usual almost Kahlerian structure
and we find necessary and sufficient conditions for ξ to be a magnetic map
with respect to ξ itself and the Kahler 2-form. In this paper we give new
examples of magnetic vector fields.
3 citations
21 Nov 2018
2 citations
Posted Content•
TL;DR: This paper considers a class of plane curves called log-aesthetic curves and their generalization which is used in CAGD and proposes a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation.
Abstract: In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which is used in CAGD. We consider these curves in the context of similarity geometry and characterize them in terms of a "stationary" integrable flow on plane curves which is governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formalism developed here, we propose a discretization of both the curves and the associated variational principle which preserves the underlying integrable structure. We finally present an algorithm for the generation of discrete log-aesthetic curves for given ${\rm G}^1$ data. The computation time to generate discrete log-aesthetic curves is much shorter than that for numerical discretizations of log-aesthetic curves due to the avoidance of fine numerical integration to calculate their shapes. Instead, only coarse summation is required.
2 citations
TL;DR: In this article, the authors formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation, and construct an explicit formula for the corresponding motion of curves even though those solutions have singular points.
Abstract: We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the $\tau$ functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class give regular solutions to the defocusing mKdV equation. Some pictures illustrating typical dynamics of the curves are presented.