The Determination of Attitudes of Planar Structures By Stereographic Projection and Spherical Trigonometry

1976 ◽  
Author(s):  
J S O Lau ◽  
J E Gale
Keyword(s):  
Stereographic Projection ◽  
Spherical Trigonometry ◽  
Planar Structures

The determination of true orientations of fractures in rock cores

1983 ◽  
Vol 20 (2) ◽  
pp. 221-227 ◽  
Author(s):  
J. S. O. Lau
Keyword(s):  
Stereographic Projection ◽  
Management Program ◽  
Reference Line ◽  
Spherical Trigonometry ◽  
Mathematical Methods ◽  
Analytical Geometry ◽  
Deep Boreholes ◽  
Nuclear Fuel Waste ◽  
Apparent Orientation

Determination of the true orientations of fractures in diamond drill cores obtained from deep boreholes in plutonic bodies is an essential requirement of the geoscience component of the Canadian Nuclear Fuel Waste Management Program. A reference line can be painted on the entire length of the rock core, indicating the orientation of the core, and the apparent orientation of the fracture can be measured from this reference line. This paper describes three methods that have been developed to convert the apparent orientation to true orientation, namely, stereographic projection, spherical trigonometry, and analytical geometry. The results obtained from these techniques were compared to assess their relative accuracy. Whereas the graphical method is more readily adaptable for use in the field, the mathematical methods can be computer-programmed and the programs GEOCORE and ORIENTC are available from the Geological Survey of Canada to facilitate the calculation of large volumes of data. Keywords: true orientation, fracture, rock core, stereographic projection, spherical trigonometry, analytical geometry.

Stereographic Projection

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Stereographic Projection ◽  
Spherical Trigonometry ◽  
Arbitrary Triangle ◽  
Cesàro Method

This chapter deals with stereographic projection, which is superior to other projections of the sphere because of its angle-preserving and circle-preserving properties; the first property gave instrument makers a huge advantage and the second provides clear astronomical advantages. The earliest text on stereographic projection is Ptolemy's Planisphere, in which he explains how to use stereographic projection to solve problems involving rising times, suggesting that the astrolabe may have existed already. After providing an overview of the astrolabe, an instrument for solving astronomical problems, the chapter considers how stereographic projection is used in solving triangles. It then describes the Cesàro method, named after Giuseppe Cesàro, that uses stereographic projection to project an arbitrary triangle ABC onto a plane. It also examines B. M. Brown's complaint against Cesàro's approach to spherical trigonometry.

Note on the determination of the optic axial angle of a crystal in thin-section by the Mallard-Becke method

1913 ◽  
Vol 16 (77) ◽  
pp. 348-351 ◽  
Author(s):  
Harvey Collingridge
Keyword(s):  
Thin Section ◽  
Stereographic Projection ◽  
Original Method ◽  
Graphic Method ◽  
Camera Lucida ◽  
Axial Angle ◽  
Linear Distance

Mallard's original method was based on the measurement of the linear distance, as determined by an eyepiece-micrometer, between the optic axes in a section of the crystal at right angles to the acute biseetrix viewed in convergent light.Professor F. Becke implored on this method by utilizing sections which were not at right angles to the acute hissetrix, but in which both optic axes were visible in the field. He projected both axes by means of an Abbe camera lucida on to a revolving drawing-table, and by means of the Mallard equation plotted the axes on a stereographic projection and thus obtained the optic axial angle, the angles of course being corrected for refraction to the true angles in the crystal section. Professor Becke subsequently, by utilizing the Biot-Fresnel law, formulated a graphic method of obtaining the optic axial angle from a section in which only one axis was visible.

Least-squares Analysis of Fabric Data: A Note on Conical, Cylindroidal and Near–cylindroidal Folds

1971 ◽  
Vol 8 (6) ◽  
pp. 694-697 ◽  
Author(s):  
C. S. Venkitasubramanyan
Keyword(s):  
Least Squares ◽  
Stereographic Projection ◽  
Rotation Axis ◽  
Computational Procedure ◽  
Apical Angle ◽  
Small Circle ◽  
Best Fitting ◽  
Fabric Diagrams ◽  
The Mean

A cylinder and a plane may be considered as special limiting cases for a right circular cone as the semi-apical angle approaches 0° and 90° respectively (Loudon 1964, Kelley 1966). If these forms are viewed as surfaces generated by an array of lines in space, the rotation axis for the array (the axis of the "cone") can be determined from the orientations of the surface-generating lines by a single computational procedure, using least-squares criterion. The mean angle between the rotation axis and the surface-generating lines will be the semi-apical angle of the cone. However, if this method for determination of the semi-apical angle of the cone, and therefore the best-fitting small circle, is extended to fabric diagrams, in which an array of lines may only statistically describe a great circle or small circle on a stereographic projection, ambiguities arise in certain cases and the semi-apical angle obtained may not be the true semi-apical angle. The difficulty arises because the poles to foliation surfaces are arbitrarily assigned "senses".

scholarly journals Detection of Acetonitrile and Chloroform Using Structures on the Base of Porous Silicon

2019 ◽  
pp. 89
Author(s):  
S.M. Manakov ◽  
M.K. Ibraimov ◽  
Ye. Sagidolda ◽  
Sh.A. Zhumatova ◽  
M.B. Darmenkulova
Keyword(s):  
Porous Silicon ◽  
Organic Compounds ◽  
Electrochemical Etching ◽  
Scanning Probe ◽  
Sensitive Material ◽  
Nanoporous Silicon ◽  
Planar Structures ◽  
Probe Microscope ◽  
Adsorption Phenomena

In this work porous silicon samples obtained by electrochemical etching were investigated. Using scanning probe microscope the morphology of porous silicon samples was studied. To determine the thickness of the porous layer and the pore diameter, micrographs were obtained using a scanning electron microscope. The dimensions of the nanocrystallites were determined from the Raman spectra. For the detection of vapors of organic compounds, planar structures were used. The results of the study confirmed the possibility of using nanoporous silicon as a sensitive material for the determination of acetonitrile and chloroform vapors. It is shown that the adsorption phenomena in porous silicon depend on its structure and morphology. It is established that the humidity of the air when detecting the vapors of organic compounds under investigation has a significant effect on the sensitivity. It is also shown that such structures can be used as instruments for measuring air humidity.

Wire-medium loaded planar structures: modal analysis, near fields, and radiation features

2016 ◽  
Vol 8 (4-5) ◽  
pp. 713-722 ◽  
Author(s):  
Davide Comite ◽  
Paolo Baccarelli ◽  
Paolo Burghignoli ◽  
Alessandro Galli
Keyword(s):  
Simulation Software ◽  
Planar Structures ◽  
Transverse Resonance ◽  
Tem Mode ◽  
Field Matching ◽  
Field Radiation ◽  
Vertically Aligned ◽  
Transverse Electromagnetic ◽  
Spurious Radiation

A novel transmission-line model is used for the analysis of planar structures, including wire-medium (WM) slabs with vertically aligned wires. The network formalism allows for an effective determination of the relevant spectral Green's functions, of the modal dispersion equation via transverse resonance, as well as of the far-field radiation pattern produced by simple sources via reciprocity, as opposed to the more cumbersome field-matching approach. Numerical results, validated also against state-of-the-art simulation software, confirm the accuracy and effectiveness of the proposed approach. In particular, modal and radiation features are presented for a class of leaky-wave antennas based on planar WM loaded configurations covered by partially reflecting screens, for which leaky unimodal regimes are achieved by minimizing spurious radiation from the quasi-transverse electromagnetic (TEM) mode.

Great Circles and Rhumb Lines on the Complex Plane

2017 ◽  
Vol 70 (3) ◽  
pp. 618-627
Author(s):  
Robin G. Stuart
Keyword(s):  
Complex Plane ◽  
Stereographic Projection ◽  
Riemann Sphere ◽  
Complex Numbers ◽  
Spherical Trigonometry ◽  
Numerical Examples ◽  
Computer Evaluation

Mapping points on the Riemann sphere to points on the plane of complex numbers by stereographic projection has been shown to offer a number of advantages when applied to problems in navigation traditionally handled using spherical trigonometry. Here it is shown that the same approach can be used for problems involving great circles and/or rhumb lines and it results in simple, compact expressions suitable for efficient computer evaluation. Worked numerical examples are given and the values obtained are compared to standard references.

A new method for determining the normals to planar structures and their trace directions in transmission electron microscopy

1994 ◽  
Vol 27 (5) ◽  
pp. 762-766 ◽  
Author(s):  
Q. Liu
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Electron Microscope ◽  
Transmission Electron Microscope ◽  
New Method ◽  
Planar Structures ◽  
Transmission Electron

With aid of a transmission-electron-microscope (TEM) double-tilt holder, a method for determining the normals to planar structures and their traces in a TEM is developed. This method is considered to be simple and convenient when compared with other methods. The accuracy of the method for the determination of both the normals to planar structures and their traces is within 2°.

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