Isonemal prefabrics with only parallel axes of symmetry

Isonemal weaving designs, introduced into mathematical literature by Gr\"unbaum and Shephard, were classified into thirty-nine infinite sets and a small number of exceptions by Richard Roth. This paper refines Roth's taxonomy for the first ten of these families in order to solve three problems, which designs exist in various sizes, which prefabrics can be doubled and remain isonemal, and which can be halved and remain isonemal.Comment: 25 page

Optical tracker having overlapping reticles on parallel axes Patent

Optical tracker with pair of FM reticles having patterns 90 deg out of phas

Hingeless helicopter rotor with improved stability

Improved stability was provided in a hingeless helicopter rotor by inclining the principal elastic flexural axes and coupling pitching of the rotor blade with the lead-lag bending of the blade. The primary elastic flex axes were inclined by constructing the blade of materials that display non-uniform stiffness, and the specification described various cross section distributions and the resulting inclined flex axes. Arrangements for varying the pitch of the rotor blade in a predetermined relationship with lead-lag bending of the blade, i.e., bending of the blade in a plane parallel to its plane of rotation were constructed

Interferometer mirror tilt correcting system

An interferometer is described, having several means for automatically adjusting the angular tilt of a reflecting surface in one of two paths to maintain the exit beams from the two paths parallel to each other. Three detectors at the output of the interferometer were disposed on mutually perpendicular axes which define a plane normal to the nominal exit beam axis. One detector at the origin of the axes was used as a reference for separate phase difference comparison with the outputs of the other two detectors on the X and Y axes to develop servo error signals

Modified Denavit-Hartenberg parameters for better location of joint axis systems in robot arms

The Denavit-Hartenberg parameters define the relative location of successive joint axis systems in a robot arm. A recent justifiable criticism is that one of these parameters becomes extremely large when two successive joints have near-parallel rotational axes. Geometrically, this parameter then locates a joint axis system at an excessive distance from the robot arm and, computationally, leads to an ill-conditioned transformation matrix. In this paper, a simple modification (which results from constraining a transverse vector between successive joint rotational axes to be normal to one of the rotational axes, instead of both) overcomes this criticism and favorably locates the joint axis system. An example is given for near-parallel rotational axes of the elbow and shoulder joints in a robot arm. The regular and modified parameters are extracted by an algebraic method with simulated measurement data. Unlike the modified parameters, extracted values of the regular parameters are very sensitive to measurement accuracy

Scattering of electromagnetic waves by many thin cylinders

Electromagnetic wave scattering by many parallel infinite cylinders is studied asymptotically as $a\to 0$. Here $a$ is the radius of the cylinders. It is assumed that the points $\hat{x}_m$ are distributed so that $\mathcal{N}(\Delta)=\frac{1}{a}\int_{\Delta}N(x)dx[1+o(1)],$ where $\mathcal{N}(\Delta)$ is the number of points $\hat{x}_m=(x_{m1},x_{m2})$ in an arbitrary open subset of the plane $xoy$, the axes of the cylinders are passing through points $\hat{x}_m$, these axes are parallel to the z-axis. The function $N(x)\geq 0$ is a given continuous function. An equation for the self-consistent (efficient) field is derived as $a\to 0$. The cylinders are assumed perfectly conducting. Formula is derived for the effective refraction coefficient in the medium in which many cylinders are distributed