World Geodetic System 1984
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Chapter 11. Three Dimensional Analytic Geometry and Vectors
Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1. -
Radar Back-Scattering from Non-Spherical Scatterers
REPORT OF INVESTIGATION NO. 28 STATE OF ILLINOIS WILLIAM G. STRATION, Governor DEPARTMENT OF REGISTRATION AND EDUCATION VERA M. BINKS, Director RADAR BACK-SCATTERING FROM NON-SPHERICAL SCATTERERS PART 1 CROSS-SECTIONS OF CONDUCTING PROLATES AND SPHEROIDAL FUNCTIONS PART 11 CROSS-SECTIONS FROM NON-SPHERICAL RAINDROPS BY Prem N. Mathur and Eugam A. Mueller STATE WATER SURVEY DIVISION A. M. BUSWELL, Chief URBANA. ILLINOIS (Printed by authority of State of Illinois} REPORT OF INVESTIGATION NO. 28 1955 STATE OF ILLINOIS WILLIAM G. STRATTON, Governor DEPARTMENT OF REGISTRATION AND EDUCATION VERA M. BINKS, Director RADAR BACK-SCATTERING FROM NON-SPHERICAL SCATTERERS PART 1 CROSS-SECTIONS OF CONDUCTING PROLATES AND SPHEROIDAL FUNCTIONS PART 11 CROSS-SECTIONS FROM NON-SPHERICAL RAINDROPS BY Prem N. Mathur and Eugene A. Mueller STATE WATER SURVEY DIVISION A. M. BUSWELL, Chief URBANA, ILLINOIS (Printed by authority of State of Illinois) Definitions of Terms Part I semi minor axis of spheroid semi major axis of spheroid wavelength of incident field = a measure of size of particle prolate spheroidal coordinates = eccentricity of ellipse = angular spheroidal functions = Legendse polynomials = expansion coefficients = radial spheroidal functions = spherical Bessel functions = electric field vector = magnetic field vector = back scattering cross section = geometric back scattering cross section Part II = semi minor axis of spheroid = semi major axis of spheroid = Poynting vector = measure of size of spheroid = wavelength of the radiation = back scattering -
John Ellipsoid 5.1 John Ellipsoid
CSE 599: Interplay between Convex Optimization and Geometry Winter 2018 Lecture 5: John Ellipsoid Lecturer: Yin Tat Lee Disclaimer: Please tell me any mistake you noticed. The algorithm at the end is unpublished. Feel free to contact me for collaboration. 5.1 John Ellipsoid In the last lecture, we discussed that any convex set is very close to anp ellipsoid in probabilistic sense. More precisely, after renormalization by covariance matrix, we have kxk2 = n ± Θ(1) with high probability. In this lecture, we will talk about how convex set is close to an ellipsoid in a strict sense. If the convex set is isotropic, it is close to a sphere as follows: Theorem 5.1.1. Let K be a convex body in Rn in isotropic position. Then, rn + 1 B ⊆ K ⊆ pn(n + 1)B : n n n Roughly speaking, this says that any convex set can be approximated by an ellipsoid by a n factor. This result has a lot of applications. Although the bound is tight, making a body isotropic is pretty time- consuming. In fact, making a body isotropic is the current bottleneck for obtaining faster algorithm for sampling in convex sets. Currently, it can only be done in O∗(n4) membership oracle plus O∗(n5) total time. Problem 5.1.2. Find a faster algorithm to approximate the covariance matrix of a convex set. In this lecture, we consider another popular position of a convex set called John position and its correspond- ing ellipsoid is called John ellipsoid. Definition 5.1.3. Given a convex set K. -
State Plane Coordinate System
Wisconsin Coordinate Reference Systems Second Edition Published 2009 by the State Cartographer’s Office Wisconsin Coordinate Reference Systems Second Edition Wisconsin State Cartographer’s Offi ce — Madison, WI Copyright © 2015 Board of Regents of the University of Wisconsin System About the State Cartographer’s Offi ce Operating from the University of Wisconsin-Madison campus since 1974, the State Cartographer’s Offi ce (SCO) provides direct assistance to the state’s professional mapping, surveying, and GIS/ LIS communities through print and Web publications, presentations, and educational workshops. Our staff work closely with regional and national professional organizations on a wide range of initia- tives that promote and support geospatial information technologies and standards. Additionally, we serve as liaisons between the many private and public organizations that produce geospatial data in Wisconsin. State Cartographer’s Offi ce 384 Science Hall 550 North Park St. Madison, WI 53706 E-mail: [email protected] Phone: (608) 262-3065 Web: www.sco.wisc.edu Disclaimer The contents of the Wisconsin Coordinate Reference Systems (2nd edition) handbook are made available by the Wisconsin State Cartographer’s offi ce at the University of Wisconsin-Madison (Uni- versity) for the convenience of the reader. This handbook is provided on an “as is” basis without any warranties of any kind. While every possible effort has been made to ensure the accuracy of information contained in this handbook, the University assumes no responsibilities for any damages or other liability whatsoever (including any consequential damages) resulting from your selection or use of the contents provided in this handbook. Revisions Wisconsin Coordinate Reference Systems (2nd edition) is a digital publication, and as such, we occasionally make minor revisions to this document. -
Geodesy Methods Over Time
Geodesy methods over time GISC3325 - Lecture 4 Astronomic Latitude/Longitude • Given knowledge of the positions of an astronomic body e.g. Sun or Polaris and our time we can determine our location in terms of astronomic latitude and longitude. US Meridian Triangulation This map shows the first project undertaken by the founding Superintendent of the Survey of the Coast Ferdinand Hassler. Triangulation • Method of indirect measurement. • Angles measured at all nodes. • Scaled provided by one or more precisely measured base lines. • First attributed to Gemma Frisius in the 16th century in the Netherlands. Early surveying instruments Left is a Quadrant for angle measurements, below is how baseline lengths were measured. A non-spherical Earth • Willebrod Snell van Royen (Snellius) did the first triangulation project for the purpose of determining the radius of the earth from measurement of a meridian arc. • Snellius was also credited with the law of refraction and incidence in optics. • He also devised the solution of the resection problem. At point P observations are made to known points A, B and C. We solve for P. Jean Picard’s Meridian Arc • Measured meridian arc through Paris between Malvoisine and Amiens using triangulation network. • First to use a telescope with cross hairs as part of the quadrant. • Value obtained used by Newton to verify his law of gravitation. Ellipsoid Earth Model • On an expedition J.D. Cassini discovered that a one-second pendulum regulated at Paris needed to be shortened to regain a one-second oscillation. • Pendulum measurements are effected by gravity! Newton • Newton used measurements of Picard and Halley and the laws of gravitation to postulate a rotational ellipsoid as the equilibrium figure for a homogeneous, fluid, rotating Earth. -
Ts 144 031 V12.3.0 (2015-07)
ETSI TS 1144 031 V12.3.0 (201515-07) TECHNICAL SPECIFICATION Digital cellular telecocommunications system (Phahase 2+); Locatcation Services (LCS); Mobile Station (MS) - SeServing Mobile Location Centntre (SMLC) Radio Resosource LCS Protocol (RRLP) (3GPP TS 44.0.031 version 12.3.0 Release 12) R GLOBAL SYSTTEM FOR MOBILE COMMUNUNICATIONS 3GPP TS 44.031 version 12.3.0 Release 12 1 ETSI TS 144 031 V12.3.0 (2015-07) Reference RTS/TSGG-0244031vc30 Keywords GSM ETSI 650 Route des Lucioles F-06921 Sophia Antipolis Cedex - FRANCE Tel.: +33 4 92 94 42 00 Fax: +33 4 93 65 47 16 Siret N° 348 623 562 00017 - NAF 742 C Association à but non lucratif enregistrée à la Sous-Préfecture de Grasse (06) N° 7803/88 Important notice The present document can be downloaded from: http://www.etsi.org/standards-search The present document may be made available in electronic versions and/or in print. The content of any electronic and/or print versions of the present document shall not be modified without the prior written authorization of ETSI. In case of any existing or perceived difference in contents between such versions and/or in print, the only prevailing document is the print of the Portable Document Format (PDF) version kept on a specific network drive within ETSI Secretariat. Users of the present document should be aware that the document may be subject to revision or change of status. Information on the current status of this and other ETSI documents is available at http://portal.etsi.org/tb/status/status.asp If you find errors in the present document, please send your comment to one of the following services: https://portal.etsi.org/People/CommiteeSupportStaff.aspx Copyright Notification No part may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and microfilm except as authorized by written permission of ETSI. -
Introduction to Aberrations OPTI 518 Lecture 13
Introduction to aberrations OPTI 518 Lecture 13 Prof. Jose Sasian OPTI 518 Topics • Aspheric surfaces • Stop shifting • Field curve concept Prof. Jose Sasian OPTI 518 Aspheric Surfaces • Meaning not spherical • Conic surfaces: Sphere, prolate ellipsoid, hyperboloid, paraboloid, oblate ellipsoid or spheroid • Cartesian Ovals • Polynomial surfaces • Infinite possibilities for an aspheric surface • Ray tracing for quadric surfaces uses closed formulas; for other surfaces iterative algorithms are used Prof. Jose Sasian OPTI 518 Aspheric surfaces The concept of the sag of a surface 2 cS 4 6 8 10 ZS ASASASAS4 6 8 10 ... 1 1 K ( c2 1 ) 2 S Sxy222 K 2 K is the conic constant K=0, sphere K=-1, parabola C is 1/r where r is the radius of curvature; K is the K<-1, hyperola conic constant (the eccentricity squared); -1<K<0, prolate ellipsoid A’s are aspheric coefficients K>0, oblate ellipsoid Prof. Jose Sasian OPTI 518 Conic surfaces focal properties • Focal points for Ellipsoid case mirrors Hyperboloid case Oblate ellipsoid • Focal points of lenses K n2 Hecht-Zajac Optics Prof. Jose Sasian OPTI 518 Refraction at a spherical surface Aspheric surface description 2 cS 4 6 8 10 ZS ASASASAS4 6 8 10 ... 1 1 K ( c2 1 ) 2 S 1122 2222 22 y x Aicrasphe y 1 x 4 K y Zx 28rr Prof. Jose Sasian OTI 518P Cartesian Ovals ln l'n' Cte . Prof. Jose Sasian OPTI 518 Aspheric cap Aspheric surface The aspheric surface can be thought of as comprising a base sphere and an aspheric cap Cap Spherical base surface Prof. -
Part V: the Global Positioning System ______
PART V: THE GLOBAL POSITIONING SYSTEM ______________________________________________________________________________ 5.1 Background The Global Positioning System (GPS) is a satellite based, passive, three dimensional navigational system operated and maintained by the Department of Defense (DOD) having the primary purpose of supporting tactical and strategic military operations. Like many systems initially designed for military purposes, GPS has been found to be an indispensable tool for many civilian applications, not the least of which are surveying and mapping uses. There are currently three general modes that GPS users have adopted: absolute, differential and relative. Absolute GPS can best be described by a single user occupying a single point with a single receiver. Typically a lower grade receiver using only the coarse acquisition code generated by the satellites is used and errors can approach the 100m range. While absolute GPS will not support typical MDOT survey requirements it may be very useful in reconnaissance work. Differential GPS or DGPS employs a base receiver transmitting differential corrections to a roving receiver. It, too, only makes use of the coarse acquisition code. Accuracies are typically in the sub- meter range. DGPS may be of use in certain mapping applications such as topographic or hydrographic surveys. DGPS should not be confused with Real Time Kinematic or RTK GPS surveying. Relative GPS surveying employs multiple receivers simultaneously observing multiple points and makes use of carrier phase measurements. Relative positioning is less concerned with the absolute positions of the occupied points than with the relative vector (dX, dY, dZ) between them. 5.2 GPS Segments The Global Positioning System is made of three segments: the Space Segment, the Control Segment and the User Segment. -
Reference Systems for Surveying and Mapping Lecture Notes
Delft University of Technology Reference Systems for Surveying and Mapping Lecture notes Hans van der Marel ii The front cover shows the NAP (Amsterdam Ordnance Datum) ”datum point” at the Stopera, Amsterdam (picture M.M.Minderhoud, Wikipedia/Michiel1972). H. van der Marel Lecture notes on Reference Systems for Surveying and Mapping: CTB3310 Surveying and Mapping CTB3425 Monitoring and Stability of Dikes and Embankments CIE4606 Geodesy and Remote Sensing CIE4614 Land Surveying and Civil Infrastructure February 2020 Publisher: Faculty of Civil Engineering and Geosciences Delft University of Technology P.O. Box 5048 Stevinweg 1 2628 CN Delft The Netherlands Copyright ©20142020 by H. van der Marel The content in these lecture notes, except for material credited to third parties, is licensed under a Creative Commons AttributionsNonCommercialSharedAlike 4.0 International License (CC BYNCSA). Third party material is shared under its own license and attribution. The text has been type set using the MikTex 2.9 implementation of LATEX. Graphs and diagrams were produced, if not mentioned otherwise, with Matlab and Inkscape. Preface This reader on reference systems for surveying and mapping has been initially compiled for the course Surveying and Mapping (CTB3310) in the 3rd year of the BScprogram for Civil Engineering. The reader is aimed at students at the end of their BSc program or at the start of their MSc program, and is used in several courses at Delft University of Technology. With the advent of the Global Positioning System (GPS) technology in mobile (smart) phones and other navigational devices almost anyone, anywhere on Earth, and at any time, can determine a three–dimensional position accurate to a few meters. -
Geodetic Position Computations
GEODETIC POSITION COMPUTATIONS E. J. KRAKIWSKY D. B. THOMSON February 1974 TECHNICALLECTURE NOTES REPORT NO.NO. 21739 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. GEODETIC POSITION COMPUTATIONS E.J. Krakiwsky D.B. Thomson Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton. N .B. Canada E3B5A3 February 197 4 Latest Reprinting December 1995 PREFACE The purpose of these notes is to give the theory and use of some methods of computing the geodetic positions of points on a reference ellipsoid and on the terrain. Justification for the first three sections o{ these lecture notes, which are concerned with the classical problem of "cCDputation of geodetic positions on the surface of an ellipsoid" is not easy to come by. It can onl.y be stated that the attempt has been to produce a self contained package , cont8.i.ning the complete development of same representative methods that exist in the literature. The last section is an introduction to three dimensional computation methods , and is offered as an alternative to the classical approach. Several problems, and their respective solutions, are presented. The approach t~en herein is to perform complete derivations, thus stqing awrq f'rcm the practice of giving a list of for11111lae to use in the solution of' a problem. -
Models for Earth and Maps
Earth Models and Maps James R. Clynch, Naval Postgraduate School, 2002 I. Earth Models Maps are just a model of the world, or a small part of it. This is true if the model is a globe of the entire world, a paper chart of a harbor or a digital database of streets in San Francisco. A model of the earth is needed to convert measurements made on the curved earth to maps or databases. Each model has advantages and disadvantages. Each is usually in error at some level of accuracy. Some of these error are due to the nature of the model, not the measurements used to make the model. Three are three common models of the earth, the spherical (or globe) model, the ellipsoidal model, and the real earth model. The spherical model is the form encountered in elementary discussions. It is quite good for some approximations. The world is approximately a sphere. The sphere is the shape that minimizes the potential energy of the gravitational attraction of all the little mass elements for each other. The direction of gravity is toward the center of the earth. This is how we define down. It is the direction that a string takes when a weight is at one end - that is a plumb bob. A spirit level will define the horizontal which is perpendicular to up-down. The ellipsoidal model is a better representation of the earth because the earth rotates. This generates other forces on the mass elements and distorts the shape. The minimum energy form is now an ellipse rotated about the polar axis. -
Geodetic Control
FGDC-STD-014.4-2008 Geographic Information Framework Data Content Standard Part 4: Geodetic Control May 2008 Federal Geographic Data Committee Established by Office of Management and Budget Circular A-16, the Federal Geographic Data Committee (FGDC) promotes the coordinated development, use, sharing, and dissemination of geographic data. The FGDC is composed of representatives from the Departments of Agriculture, Commerce, Defense, Education, Energy, Health and Human Services, Homeland Security, Housing and Urban Development, the Interior, Justice, Labor, State, and Transportation, the Treasury, and Veteran Affairs; the Environmental Protection Agency; the Federal Communications Commission; the General Services Administration; the Library of Congress; the National Aeronautics and Space Administration; the National Archives and Records Administration; the National Science Foundation; the Nuclear Regulatory Commission; the Office of Personnel Management; the Small Business Administration; the Smithsonian Institution; the Social Security Administration; the Tennessee Valley Authority; and the U.S. Agency for International Development. Additional Federal agencies participate on FGDC subcommittees and working groups. The Department of the Interior chairs the committee. FGDC subcommittees work on issues related to data categories coordinated under the circular. Subcommittees establish and implement standards for data content, quality, and transfer; encourage the exchange of information and the transfer of data; and organize the collection of