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3 Rectangular Coordinate System and Graphs
06022_CH03_123-154.QXP 10/29/10 10:56 AM Page 123 3 Rectangular Coordinate System and Graphs In This Chapter A Bit of History Every student of mathematics pays the French mathematician René Descartes (1596–1650) hom- 3.1 The Rectangular Coordinate System age whenever he or she sketches a graph. Descartes is consid- ered the inventor of analytic geometry, which is the melding 3.2 Circles and Graphs of algebra and geometry—at the time thought to be completely 3.3 Equations of Lines unrelated fields of mathematics. In analytic geometry an equa- 3.4 Variation tion involving two variables could be interpreted as a graph in Chapter 3 Review Exercises a two-dimensional coordinate system embedded in a plane. The rectangular or Cartesian coordinate system is named in his honor. The basic tenets of analytic geometry were set forth in La Géométrie, published in 1637. The invention of the Cartesian plane and rectangular coordinates contributed significantly to the subsequent development of calculus by its co-inventors Isaac Newton (1643–1727) and Gottfried Wilhelm Leibniz (1646–1716). René Descartes was also a scientist and wrote on optics, astronomy, and meteorology. But beyond his contributions to mathematics and science, Descartes is also remembered for his impact on philosophy. Indeed, he is often called the father of modern philosophy and his book Meditations on First Philosophy continues to be required reading to this day at some universities. His famous phrase cogito ergo sum (I think, there- fore I am) appears in his Discourse on the Method and Principles of Philosophy. Although he claimed to be a fervent In Section 3.3 we will see that parallel lines Catholic, the Church was suspicious of Descartes’philosophy have the same slope. -
Evaluation of Recent Global Geopotential Models Based on GPS/Levelling Data Over Afyonkarahisar (Turkey)
Scientific Research and Essays Vol. 5(5), pp. 484-493, 4 March, 2010 Available online at http://www.academicjournals.org/SRE ISSN 1992-2248 © 2010 Academic Journals Full Length Research Paper Evaluation of recent global geopotential models based on GPS/levelling data over Afyonkarahisar (Turkey) Ibrahim Yilmaz1*, Mustafa Yilmaz2, Mevlüt Güllü1 and Bayram Turgut1 1Department of Geodesy and Photogrammetry, Faculty of Engineering, Kocatepe University of Afyonkarahisar, Ahmet Necdet Sezer Campus Gazlıgöl Yolu, 03200, Afyonkarahisar, Turkey. 2Directorship of Afyonkarahisar, Osmangazi Electricity Distribution Inc., Yuzbası Agah Cd. No: 24, 03200, Afyonkarahisar, Turkey. Accepted 20 January, 2010 This study presents the evaluation of the global geo-potential models EGM96, EIGEN-5C, EGM2008(360) and EGM2008 by comparing model based on geoid heights to the GPS/levelling based on geoid heights over Afyonkarahisar study area in order to find the geopotential model that best fits the study area to be used in a further geoid determination at regional and national scales. The study area consists of 313 control points that belong to the Turkish National Triangulation Network, covering a rough area. The geoid height residuals are investigated by standard deviation value after fitting tilt at discrete points, and height-dependent evaluations have been performed. The evaluation results revealed that EGM2008 fits best to the GPS/levelling based on geoid heights than the other models with significant improvements in the study area. Key words: Geopotential model, GPS/levelling, geoid height, EGM96, EIGEN-5C, EGM2008(360), EGM2008. INTRODUCTION Most geodetic applications like determining the topogra- evaluation studies of EGM2008 have been coordinated phic heights or sea depths require the geoid as a by the joint working group (JWG) between the Inter- corresponding reference surface. -
Meyers Height 1
University of Connecticut DigitalCommons@UConn Peer-reviewed Articles 12-1-2004 What Does Height Really Mean? Part I: Introduction Thomas H. Meyer University of Connecticut, [email protected] Daniel R. Roman National Geodetic Survey David B. Zilkoski National Geodetic Survey Follow this and additional works at: http://digitalcommons.uconn.edu/thmeyer_articles Recommended Citation Meyer, Thomas H.; Roman, Daniel R.; and Zilkoski, David B., "What Does Height Really Mean? Part I: Introduction" (2004). Peer- reviewed Articles. Paper 2. http://digitalcommons.uconn.edu/thmeyer_articles/2 This Article is brought to you for free and open access by DigitalCommons@UConn. It has been accepted for inclusion in Peer-reviewed Articles by an authorized administrator of DigitalCommons@UConn. For more information, please contact [email protected]. Land Information Science What does height really mean? Part I: Introduction Thomas H. Meyer, Daniel R. Roman, David B. Zilkoski ABSTRACT: This is the first paper in a four-part series considering the fundamental question, “what does the word height really mean?” National Geodetic Survey (NGS) is embarking on a height mod- ernization program in which, in the future, it will not be necessary for NGS to create new or maintain old orthometric height benchmarks. In their stead, NGS will publish measured ellipsoid heights and computed Helmert orthometric heights for survey markers. Consequently, practicing surveyors will soon be confronted with coping with these changes and the differences between these types of height. Indeed, although “height’” is a commonly used word, an exact definition of it can be difficult to find. These articles will explore the various meanings of height as used in surveying and geodesy and pres- ent a precise definition that is based on the physics of gravitational potential, along with current best practices for using survey-grade GPS equipment for height measurement. -
THE EARTH's GRAVITY OUTLINE the Earth's Gravitational Field
GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY OUTLINE The Earth's gravitational field 2 Newton's law of gravitation: Fgrav = GMm=r ; Gravitational field = gravitational acceleration g; gravitational potential, equipotential surfaces. g for a non–rotating spherically symmetric Earth; Effects of rotation and ellipticity – variation with latitude, the reference ellipsoid and International Gravity Formula; Effects of elevation and topography, intervening rock, density inhomogeneities, tides. The geoid: equipotential mean–sea–level surface on which g = IGF value. Gravity surveys Measurement: gravity units, gravimeters, survey procedures; the geoid; satellite altimetry. Gravity corrections – latitude, elevation, Bouguer, terrain, drift; Interpretation of gravity anomalies: regional–residual separation; regional variations and deep (crust, mantle) structure; local variations and shallow density anomalies; Examples of Bouguer gravity anomalies. Isostasy Mechanism: level of compensation; Pratt and Airy models; mountain roots; Isostasy and free–air gravity, examples of isostatic balance and isostatic anomalies. Background reading: Fowler §5.1–5.6; Lowrie §2.2–2.6; Kearey & Vine §2.11. GEOPHYSICS (08/430/0012) THE EARTH'S GRAVITY FIELD Newton's law of gravitation is: ¯ GMm F = r2 11 2 2 1 3 2 where the Gravitational Constant G = 6:673 10− Nm kg− (kg− m s− ). ¢ The field strength of the Earth's gravitational field is defined as the gravitational force acting on unit mass. From Newton's third¯ law of mechanics, F = ma, it follows that gravitational force per unit mass = gravitational acceleration g. g is approximately 9:8m/s2 at the surface of the Earth. A related concept is gravitational potential: the gravitational potential V at a point P is the work done against gravity in ¯ P bringing unit mass from infinity to P. -
Analysis of Graviresponse and Biological Effects of Vertical and Horizontal Clinorotation in Arabidopsis Thaliana Root Tip
plants Article Analysis of Graviresponse and Biological Effects of Vertical and Horizontal Clinorotation in Arabidopsis thaliana Root Tip Alicia Villacampa 1 , Ludovico Sora 1,2 , Raúl Herranz 1 , Francisco-Javier Medina 1 and Malgorzata Ciska 1,* 1 Centro de Investigaciones Biológicas Margarita Salas-CSIC, Ramiro de Maeztu 9, 28040 Madrid, Spain; [email protected] (A.V.); [email protected] (L.S.); [email protected] (R.H.); [email protected] (F.-J.M.) 2 Department of Aerospace Science and Technology, Politecnico di Milano, Via La Masa 34, 20156 Milano, Italy * Correspondence: [email protected]; Tel.: +34-91-837-3112 (ext. 4260); Fax: +34-91-536-0432 Abstract: Clinorotation was the first method designed to simulate microgravity on ground and it remains the most common and accessible simulation procedure. However, different experimental set- tings, namely angular velocity, sample orientation, and distance to the rotation center produce different responses in seedlings. Here, we compare A. thaliana root responses to the two most commonly used velocities, as examples of slow and fast clinorotation, and to vertical and horizontal clinorotation. We investigate their impact on the three stages of gravitropism: statolith sedimentation, asymmetrical auxin distribution, and differential elongation. We also investigate the statocyte ultrastructure by electron microscopy. Horizontal slow clinorotation induces changes in the statocyte ultrastructure related to a stress response and internalization of the PIN-FORMED 2 (PIN2) auxin transporter in the lower endodermis, probably due to enhanced mechano-stimulation. Additionally, fast clinorotation, Citation: Villacampa, A.; Sora, L.; as predicted, is only suitable within a very limited radius from the clinorotation center and triggers Herranz, R.; Medina, F.-J.; Ciska, M. -
Datums in Texas NGS: Welcome to Geodesy
Datums in Texas NGS: Welcome to Geodesy Geodesy is the science of measuring and monitoring the size and shape of the Earth and the location of points on its surface. NOAA's National Geodetic Survey (NGS) is responsible for the development and maintenance of a national geodetic data system that is used for navigation, communication systems, and mapping and charting. ln this subject, you will find three sections devoted to learning about geodesy: an online tutorial, an educational roadmap to resources, and formal lesson plans. The Geodesy Tutorial is an overview of the history, essential elements, and modern methods of geodesy. The tutorial is content rich and easy to understand. lt is made up of 10 chapters or pages (plus a reference page) that can be read in sequence by clicking on the arrows at the top or bottom of each chapter page. The tutorial includes many illustrations and interactive graphics to visually enhance the text. The Roadmap to Resources complements the information in the tutorial. The roadmap directs you to specific geodetic data offered by NOS and NOAA. The Lesson Plans integrate information presented in the tutorial with data offerings from the roadmap. These lesson plans have been developed for students in grades 9-12 and focus on the importance of geodesy and its practical application, including what a datum is, how a datum of reference points may be used to describe a location, and how geodesy is used to measure movement in the Earth's crust from seismic activity. Members of a 1922 geodetic suruey expedition. Until recent advances in satellite technology, namely the creation of the Global Positioning Sysfem (GPS), geodetic surveying was an arduous fask besf suited to individuals with strong constitutions, and a sense of adventure. -
The Evolution of Earth Gravitational Models Used in Astrodynamics
JEROME R. VETTER THE EVOLUTION OF EARTH GRAVITATIONAL MODELS USED IN ASTRODYNAMICS Earth gravitational models derived from the earliest ground-based tracking systems used for Sputnik and the Transit Navy Navigation Satellite System have evolved to models that use data from the Joint United States-French Ocean Topography Experiment Satellite (Topex/Poseidon) and the Global Positioning System of satellites. This article summarizes the history of the tracking and instrumentation systems used, discusses the limitations and constraints of these systems, and reviews past and current techniques for estimating gravity and processing large batches of diverse data types. Current models continue to be improved; the latest model improvements and plans for future systems are discussed. Contemporary gravitational models used within the astrodynamics community are described, and their performance is compared numerically. The use of these models for solid Earth geophysics, space geophysics, oceanography, geology, and related Earth science disciplines becomes particularly attractive as the statistical confidence of the models improves and as the models are validated over certain spatial resolutions of the geodetic spectrum. INTRODUCTION Before the development of satellite technology, the Earth orbit. Of these, five were still orbiting the Earth techniques used to observe the Earth's gravitational field when the satellites of the Transit Navy Navigational Sat were restricted to terrestrial gravimetry. Measurements of ellite System (NNSS) were launched starting in 1960. The gravity were adequate only over sparse areas of the Sputniks were all launched into near-critical orbit incli world. Moreover, because gravity profiles over the nations of about 65°. (The critical inclination is defined oceans were inadequate, the gravity field could not be as that inclination, 1= 63 °26', where gravitational pertur meaningfully estimated. -
A Hierarchy of Models, from Planetary to Mesoscale, Within a Single
A hierarchy of models, from planetary to mesoscale, within a single switchable numerical framework Nigel Wood, Andy White and Andrew Staniforth Dynamics Research, Met Office © Crown copyright Met Office Outline of the talk • Overview of Met Office’s Unified Model • Spherical Geopotential Approximation • Relaxing that approximation • A method of implementation • Alternative view of Shallow-Atmosphere Approximation • Application to departure points © Crown copyright Met Office Met Office’s Unified Model Unified Model (UM) in that single model for: Operational forecasts at ¾Mesoscale (resolution approx. 12km → 4km → 1km) ¾Global scale (resolution approx. 25km) Global and regional climate predictions (resolution approx. 100km, run for 10-100 years) + Research mode (1km - 10m) and single column model 20 years old this year © Crown copyright Met Office Timescales & applications © Crown copyright Met Office © Crown copyright Met Office UKV Domain 4x4 1.5x4 4x4 4x1.5 1.5x1.5 4x1.5 4x4 1.5x4 4x4 Variable 744(622) x 928( 810) points zone © Crown copyright Met Office The ‘Morpeth Flood’, 06/09/2008 1.5 km L70 Prototype UKV From 15 UTC 05/09 12 km 0600 UTC © Crown copyright Met Office 2009: Clark et al, Morpeth Flood UKV 4-20hr forecast 1.5km gridlength UK radar network convection permitting model © Crown copyright Met Office © Crown copyright Met Office © Crown copyright Met Office The underlying equations... c Crown Copyright Met Office © Crown copyright Met Office Traditional Spherically Based Equations Dru uv tan φ cpdθv ∂Π uw u − − (2Ω sin -
Geophysical Journal International
Geophysical Journal International Geophys. J. Int. (2015) 203, 1773–1786 doi: 10.1093/gji/ggv392 GJI Gravity, geodesy and tides GRACE time-variable gravity field recovery using an improved energy balance approach Kun Shang,1 Junyi Guo,1 C.K. Shum,1,2 Chunli Dai1 and Jia Luo3 1Division of Geodetic Science, School of Earth Sciences, The Ohio State University, 125 S. Oval Mall, Columbus, OH 43210, USA. E-mail: [email protected] 2Institute of Geodesy & Geophysics, Chinese Academy of Sciences, Wuhan, China 3School of Geodesy and Geomatics, Wuhan University, 129 Luoyu Road, Wuhan 430079, China Accepted 2015 September 14. Received 2015 September 12; in original form 2015 March 19 Downloaded from SUMMARY A new approach based on energy conservation principle for satellite gravimetry mission has been developed and yields more accurate estimation of in situ geopotential difference observ- ables using K-band ranging (KBR) measurements from the Gravity Recovery and Climate Experiment (GRACE) twin-satellite mission. This new approach preserves more gravity in- http://gji.oxfordjournals.org/ formation sensed by KBR range-rate measurements and reduces orbit error as compared to previous energy balance methods. Results from analysis of 11 yr of GRACE data indicated that the resulting geopotential difference estimates agree well with predicted values from of- ficial Level 2 solutions: with much higher correlation at 0.9, as compared to 0.5–0.8 reported by previous published energy balance studies. We demonstrate that our approach produced a comparable time-variable gravity solution with the Level 2 solutions. The regional GRACE temporal gravity solutions over Greenland reveals that a substantially higher temporal resolu- at Ohio State University Libraries on December 20, 2016 tion is achievable at 10-d sampling as compared to the official monthly solutions, but without the compromise of spatial resolution, nor the need to use regularization or post-processing. -
The Joint Gravity Model 3
Journal of Geophysical Research Accepted for publication, 1996 The Joint Gravity Model 3 B. D. Tapley, M. M. Watkins,1 J. C. Ries, G. W. Davis,2 R. J. Eanes, S. R. Poole, H. J. Rim, B. E. Schutz, and C. K. Shum Center for Space Research, University of Texas at Austin R. S. Nerem,3 F. J. Lerch, and J. A. Marshall Space Geodesy Branch, NASA Goddard Space Flight Center, Greenbelt, Maryland S. M. Klosko, N. K. Pavlis, and R. G. Williamson Hughes STX Corporation, Lanham, Maryland Abstract. An improved Earth geopotential model, complete to spherical harmonic degree and order 70, has been determined by combining the Joint Gravity Model 1 (JGM 1) geopotential coef®cients, and their associated error covariance, with new information from SLR, DORIS, and GPS tracking of TOPEX/Poseidon, laser tracking of LAGEOS 1, LAGEOS 2, and Stella, and additional DORIS tracking of SPOT 2. The resulting ®eld, JGM 3, which has been adopted for the TOPEX/Poseidon altimeter data rerelease, yields improved orbit accuracies as demonstrated by better ®ts to withheld tracking data and substantially reduced geographically correlated orbit error. Methods for analyzing the performance of the gravity ®eld using high-precision tracking station positioning were applied. Geodetic results, including station coordinates and Earth orientation parameters, are signi®cantly improved with the JGM 3 model. Sea surface topography solutions from TOPEX/Poseidon altimetry indicate that the ocean geoid has been improved. Subset solutions performed by withholding either the GPS data or the SLR/DORIS data were computed to demonstrate the effect of these particular data sets on the gravity model used for TOPEX/Poseidon orbit determination. -
Vertical Datum Conversion Guidance
Guidance for Flood Risk Analysis and Mapping Vertical Datum Conversion May 2014 This guidance document supports effective and efficient implementation of flood risk analysis and mapping standards codified in the Federal Insurance and Mitigation Administration Policy FP 204- 07801. For more information, please visit the Federal Emergency Management Agency (FEMA) Guidelines and Standards for Flood Risk Analysis and Mapping webpage (http://www.fema.gov/guidelines-and- standards-flood-risk-analysis-and-mapping), which explains the policy, related guidance, technical references, and other information about the guidelines and standards process. Nothing in this guidance document is mandatory other than standards codified separately in the aforementioned Policy. Alternate approaches that comply with FEMA standards that effectively and efficiently support program objectives are also acceptable. Vertical Datum Conversion May 2014 Guidance Document 24 Page i Document History Affected Section or Date Description Subsection Initial version of new transformed guidance. The content was derived from the Guidelines and Specifications for Flood First Publication May 2014 Hazard Mapping Partners, Procedure Memoranda, and/or Operating Guidance documents. It has been reorganized and is being published separately from the standards. Vertical Datum Conversion May 2014 Guidance Document 24 Page ii Table of Contents 1.0 Overview .................................................................................................................................. -
Mind the Gap! a New Positioning Reference
A new positioning reference Why is the United States adopting NATRF2022? What are we doing about this in Canada? We want to hear from you! • The Canadian Geodetic Survey and the United States • Improved compatibility with Global Navigation • The Canadian Geodetic Survey is working closely National Geodetic Survey have collaborated for Satellite Systems (GNSS), such as GPS, is driving this with the United States National Geodetic Survey in • Send us your comments, the challenges you over a century to provide the fundamental reference change. The geometric reference frames currently defining reference frames to ensure they will also foresee, and any concerns to help inform our path Mind the gap! systems for latitude, longitude and height for their used in Canada and the United States, although be suitable for Canada. forward to either of these organizations: respective countries. compatible with each other, are offset by 2.2 m from • Geodetic agencies from across Canada are A new positioning reference the Earth’s geocentre, whereas GNSS are geocentric. - Canadian Geodetic Survey: nrcan. • Together our reference systems have evolved collaborating on reference system improvements geodeticinformation-informationgeodesique. to meet today’s world of GPS and geographical • Real-time decimetre-level accuracies directly from through the Canadian Geodetic Reference System [email protected] NATRF2022 information systems, while supporting legacy datums GNSS satellites are expected to be available soon. Committee, a working committee of the Canadian