Related papers: Spherical indicatrices with the modified orthogona…
We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on $\mathbb S^3$ and to rectangular…
The article contributes to the theory of infinitesimal bendings of smooth surfaces in Euclidean 3-space. We derive a linear differential equation of the first order, which previously did not appear in the literature and which is satisfied…
We introduce circular evolutes and involutes of framed curves in the Euclidean space. Circular evolutes of framed curves stem from the curvature circles of Bishop directions and singular value sets of normal surfaces of Bishop directions.…
A hyperbolic framed curve is a smooth curve with a moving frame in hyperbolic 3-space. It may have singularities. By using this moving frame, we can investigate the differential geometry properties of curves, even at singular points. In…
In this paper we use the group of mirror symmetries for the construction of the vector fields on spheres.
We introduce canonical principal parameters on any strongly regular minimal surface in the three dimensional sphere and prove that any such a surface is determined up to a motion by its normal curvature function satisfying the Sinh-Poisson…
We consider mixed types of not only regular curves but also curves with singular points in the Lorentz-Minkowski 3-space. In order to consider mixed type of curves with singular points, we consider the lightcone frame and lightcone framed…
We analyze human poses and motion by introducing three sequences of easily calculated surface descriptors that are invariant under reparametrizations and Euclidean transformations. These descriptors are obtained by associating to each…
We study, theoretically and experimentally, a 1-parameter family of transformations and their limiting vector field on the space of plane polygons. These transformations are discrete analogs of completely integrable transformation on closed…
For sampling values along spherical Lissajous curves we establish a spectral interpolation and quadrature scheme on the sphere. We provide a mathematical analysis of spherical Lissajous curves and study the characteristic properties of…
The spherometer used for measuring radius of curvature of spherical surfaces is explicitly based on a geometric relation unique to circles and spheres. We present an alternate approach using coordinate geometry, which reproduces the…
In this paper we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This…
In this article, we study rectifying curves in arbitrary dimensional Euclidean space. A curve is said to be a rectifying curve if, in all points of the curve, the orthogonal complement of its normal vector contains a fixed point. We…
This paper deals with the study of some properties of immersed curves in the conformal sphere $\mathds{Q}_n$, viewed as a homogeneous space under the action of the M\"obius group. After an overview on general well-known facts, we briefly…
We explore geometric properties of circular analogues of tractrices and pseudospheres in $\mathbb R^3$.
A relationship between real, complex, and quaternionic vector fields on spheres is given by using a relationship between the corresponding standard inner products. The number of linearly independent complex vector fields on the standard…
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. Using the framework of Moebius…
Field-guided parametrization methods have proven effective for quad meshing of surfaces; these methods compute smooth cross fields to guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in…
Certain dissipative physical systems closely resemble Hamiltonian systems in $\mathbb{R}^{2n}$, but with the canonical equation for one of the variables in each conjugate pair rescaled by a real parameter. To generalise these dynamical…
The present article studies combinatorial tilings of Euclidean or spherical spaces by polytopes, serving two main purposes: first, to survey some of the main developments in combinatorial space tiling; and second, to highlight some new and…