This paper presents a new methodology of using tetrahedron configurations to determine the forward kinematics of 6-dof parallel manipulators. The approach is introduced in the form of the Tetrahedron Proposition and Theorem using the special characteristics of tetrahedral geometry which are then applied to obtaining the kinematic solutions of two 6-dof parallel manipulators such as the Hunt-Pri...

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern– Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than o...

In this paper we formulate an integrable model on the simple cubic lattice. The N – valued spin variables of the model belong to edges of the lattice. The Boltzmann weights of the model obey the vertex type Tetrahedron Equation. In the thermodynamic limit our model is equivalent to the Bazhanov – Baxter Model. In the case when N = 2 we reproduce the Korepanov’s and Hietarinta’s solutions of the...

In this paper, we propose a sparse spectral-Galerkin approximation scheme for solving the second-order partial differential equations on an arbitrary tetrahedron. Generalized Koornwinder polynomials are introduced reference tetrahedron as basis functions with their various recurrence relations and differentiation properties being explored. The method leads to well-conditioned linear systems who...

Institute of Engineering Cybernetics Academy of Sciences of Republic Belarus Surganova 6, 220012 Minsk, Belarus {tuzikov,sheynin,vasiliev}@mpen.bas-net.by Received April 10, 2002 It is well known that such important geometric characteristics of 3D bodies as volume and orientation can be defined in terms of moments. A moment computation depends greatly on a body representation. The most popular ...

There is a revolution in the ability to analyze gene expression of single cells in a tissue. To understand this data we must comprehend how cells are distributed in a high-dimensional gene expression space. One open question is whether cell types form discrete clusters or whether gene expression forms a continuum of states. If such a continuum exists, what is its geometry? Recent theory on evol...