Generation of elastic geodesic gridshells with anisotropic cross sections

Geodesic gridshells are shell structures made of continuous elements following geodesic lines. Their properties ease the use of beams with anisotropic cross-sections by avoiding bending about their strong axis. However, such bending may arise when flattening arbitrary geodesic grids, which forbids their initial assembly on the ground. This study provides a process to design elastic geodesic gridshells, that is, gridshells that minimise bending moments in both formed and near-flat configurations. The generation process first brings a target geodesic network onto a plane by maintaining arc lengths. The flat mesh is then relaxed to minimise its main curvatures and hence bending moments in its members. The result is an elastic geodesic gridshell that can be assembled flat on the ground and then lifted up into its target surface. The method is applied to the design of six geodesic gridshells made of reclaimed skis.

Lifts of connections to the bundle of (1,1) type tensor frames

In this paper we consider the bundle of (1,1) type tensor frames over a smooth manifold, define the horizontal and complete lifts of symmetric linear connection into this bundle. Also we study the properties of the geodesic lines corresponding to the complete lift of the linear connection and investigate the relations between Sasaki metric and lifted connections on the bundle of (1,1) type tensor frames.

The Dаrbоuх Theory and Geodesics in the Metric Space with Torsion

The geometry of n Yn space is generated congruently together by the metric tensor and the torsion tensor. In the presented article has been obtained an analog of the Dаrbоuх theory in the n Yn space, also studied the deduction of the equation of the geodesic lines on the hypersurface that embedded in such spaces, showed that in the n Yn space the structure of the curvature tensor has special features and for curvature tensor obtained Ricci - Jacobi identity. We establish that the equations of the geodesics have additional summands, which are caused by the presence of torsion in the space. In n Yn space, the variation of the length of the geodesic lines is proportional to the product of metric and torsion tensors gijSjpk. We have introduced the second fundamental tensor παβ for the hypersurface n Yn-1 and established its structure, which is fundamentally different from the case of the Riemannian spaces with zero torsion. Furthermore, the results on the structure of the curvature tensor have been obtained.

Lines on the surface in the quasi-hiperbolic space 11^S1/3

Quasi-hyperbolic spaces are projective spaces with decaying abso­lute. This work is a continuation of the author's work [7], in which surfac­es in one of these spaces are examined by methods of external forms and a moving frame. The semi-Chebyshev and Chebyshev net­works of lines on the surface in quasi-hyperbolic space are considered. In this pa­per we use the definition of parallel transfer adopted in [6]. By analogy with Euclidean geometry, the semi-Chebyshev network of lines on the surface is the network in which the tangents to the lines of one family are carried parallel along the lines of another family. A Che­byshev network is a network in which tangents to the lines of each family are carried parallel along the lines of another family. We proved three theorems. In Theorem 1, we obtain a natural equa­tion for non-geodesic lines that are part of a conjugate semi-Chebyshev network on the surface so that tangents to lines of another family are transferred in parallel along them. In Theorem 2, the natural equation of non-geodesic lines in the Chebyshev network is obtained. In Theorem 3 we prove that conjugate Chebyshev networks, one family of which is nei­ther geodesic lines, nor Euclidean sections, exist on surfaces with the lati­tude of four functions of one argument.

Conjugate words and intersections of geodesics in ℍ2

We investigate intersections of geodesic lines in [Formula: see text] and in an associated tree [Formula: see text], proving the following result. Let [Formula: see text] be a punctured hyperbolic torus and let [Formula: see text] be a closed geodesic in [Formula: see text]. Any edge of any triangle formed by distinct geodesic lines in the preimage of [Formula: see text] in [Formula: see text] is shorter than [Formula: see text]. However, a similar result does not hold in the tree T. Let [Formula: see text] be a reduced and cyclically reduced word in [Formula: see text]. We construct several examples of triangles in [Formula: see text] formed by distinct axes in [Formula: see text] stabilized by conjugates of [Formula: see text] such that an edge in those triangles is longer than [Formula: see text]. We also prove that if [Formula: see text] overlaps two of its conjugates in such a way that the overlaps cover all of [Formula: see text] and the overlaps do not intersect, then there exists a decomposition [Formula: see text], with [Formula: see text] a terminal subword of [Formula: see text] and [Formula: see text] an initial subword of [Formula: see text].