TL;DR: In this article, it was shown that any two bipartite quadrangulations of any closed surface are transformed into each other by two kinds of transformations, called the diagonal slide and the diagonal rotation, up to homeomorphism, if they have the same and sufficiently large number of vertices.
TL;DR: A new method is introduced, which reduces the set-up and measuring time for squareness error by applying four volumetric diagonal measurements instead of six face diagonal measurements, which makes it possible to use a single properly sized artefact, which fits the machine's rectangular volume.
Abstract: The face diagonal method for squareness error measurement is already well established. Because it is easy to apply, this method is widely used in CMMs (coordinate measuring machines). This paper introduces a new method, which reduces the set-up and measuring time for squareness error by applying four volumetric diagonal measurements instead of six face diagonal measurements. The method also makes it possible to use a single properly sized artefact, which fits the machine’s rectangular volume. The choice of artefacts used in the method is wide, and they need not be precalibrated before measurements.
TL;DR: In this article, a truss structure for use in outer space comprises a plurality of module units and can be deployed into a structure having a flat surface or a curved surface, each module unit comprises members, two pairs of flexible chords and four diagonal members.
Abstract: A truss structure for use in outer space comprises a plurality of module units and can be deployed into a structure having a flat surface or a curved surface. Each module unit comprises members, two pairs of flexible chords and four diagonal members. The members are arranged along the edges of an imaginary rectangular parallelepiped. The flexible chords of the first pair are stretched in that face of the imaginary parallelepiped which defines the flat or curved face, along the diagonals of this face. The flexible chords of the second pair are stretched in that face of the imaginary parallelepiped which opposes said face, along the diagonals of this face. Each diagonal member is provided in the corresponding one of the other four faces of the imaginary parallelepiped, along one diagonal of this face, extending at right angles to the diagonal member provided in the opposite parallel face. As the diagonal members are elongated, the module unit is folded, thus slackening the chords and positioning the members and diagonal members substantially parallel to one another. As the diagonal members are contracted, the module unit is deployed.
TL;DR: It is shown that any two outer-triangulations on the same closed surface can be transformed into each other by a sequence of diagonal flips, up to isotopy, if they have a sufficiently large and equal number of vertices.