Athanasios Dermanis

Coordinates and Reference Systems

Ziti Publications, Thessaloniki 2005

Chapter 1: Introduction

1.1  Space and Time in Physical Sciences

1.2  The Measurement of Time

1.3  Measurements in Space. From Geometry to Geodesy

1.4  Measurements in Space. Shape and Position

1.5  Evolution of Mathematical Models for the Shape of the Erath and its Gravity Field

1.6  Geodesy in the Space Age

Chapter 2: Fundamental Concepts

2.1  Geometric and Analytical Methods in Mathematics

2.2  Coordinate Systems and Local Vectorial Reference Systems

2.3  Coordinate and Reference Systems in Euclidean Space

2.4  Relation Between Two Different Reference Systems

2.5  Description of the Rotation Matrix. Rotations Around the Axes

2.6  Geometric Interpretation of the Angles of Rotation Around the Axes

2.7  Complex Numbers and Rotation in the Plane

Chapter 3: Systems of Curvilinear Coordinates

3.1  General Characteristics

3.2  Spherical Coordinates

3.3  Cylindrical Coordinates

3.4  Geodetic Coordinates

3.5  Ellipsoidal Coordinates

3.6  integrals Using Curvilinear Coordinates

Appendix 3a: Relation Between Cartesian and Geodetic Coordinates

Chapter 4: Reference Systems in Motion

4.1  Inertial and Accelerated Reference Systems 

4.2  Pseuodoforces in an Accelerated Reference System

4.3  The Rotational Motion of a Rigid Body

4.4  The Vector of Instantaneous Rotational Velocity

4.5  The Euler Kinematic Equations

4.6  Reference Systems for Deformable Bodies

Chapter 5: The Gravity Field of the Earth

5.1  Gravitational Force and Potential

5.2  Centrifugal Force and Gravity

5.3  Geometric Characteristics of the Gravity Field

5.4  Determination of the Gravity Field

5.5  The Local Astronomic Reference System

5.6  The Local Geodetic Reference System

5.7  Reduction from A Reference System Related to the Rotation Axis of the Earth to an Earth-Fixed Reference System

5.8  Reduction from the Local Astronomic to the Local Geodetic Reference System

5.9  Height Systems

Chapter 6: Geodetic Datum and Networks

6.1  Geodetic Coordinates and the Geodetic Datum

6.2  Cartographic Projections

6.3  The Problem of Scale Variation

6.4  Determination of the Parameters of Transformation from one Reference System to Another

6.5  The Space-Time Reference System for a Network of Control Points

Chapter 7: Reference Systems for the Rotating Earth

7.1  Precession-Nutation, Diurnal Rotation and Polar Motion

7.2  Reference Systems With the 3rd Axis in the Direction of the Rotation Vector

7.3  The Rotation of the Earth

Appendix 7Š: Determination of the Non-Rotating Origin (NRO)

Chapter 8: Systems of Time

8.1  Newtonian and Relativistic Time

8.2  Sidereal and Universal Time

8.3  Dynamic Time

8.4  Atomic Time

8.5  The Role of the Theory of Relativity

Chapter 9: Reference Systems for the Description of Directions in Astronomy

9.1  Horizontal System, Hour Angle System, Right Ascension System and Ecliptic System

9.2  Transformation Relations between Astronomical Systems

9.3  Classical Description of the Rotation of the Earth in Astronomy

9.4  Astrogeodetic Methods of Longitude and Latitude Determination

Appendix 9a: Spherical Trigonometry

Chapter 10: The Realization of Reference Systems by The IERS

10.1.  Definition and Realization of Reference Systems by the IERS

10.2.  Computation of Precession-Nutation

10.3.  Computation of Polar Motion

10.4.  Computation of Diurnal Rotation

10.5.  Computations for the Classical Astronomical Description

10.6.  Subroutines for the Computation of the Transformation from the Celestial to the Terrestrial Reference System

Chapter 11: Reference Systems in Satellite Geodesy

11.1.  Keplerís Laws

11.2.  Satellite Motion in the Orbital Plane

11.3.  The Keplerian Ellipse in Space

11.4.  Description of Satellite Orbits for the Global Positioning System

Appendix 11Š: The World Geodetic System WGS 84

Chapter 12: Reference Systems and Linear Equations

12.1.  Systems of Linear Equations

12.2.  Geometric Determination of the Least-Squares and Minimum-Norm Solutions

12.3.  Change of the Reference System in the Space of the Unknowns and in the Space of the Observables

12.4.  Singular Value Decomposition

12.5.  The Role of the Reference System of the Euclidean Space in the Linear Equations of the Analysis of Control Networks

Chapter 13: Reference Systems in Function Spaces

13.1.  Non Orthogonal Bases in Euclidean Space

13.2.  Spaces of Polynomial Functions

13.3.  Best Approximation of a Function by Polynomials

13.4.  Expansion of a Function in Fourier Series

13.5.  Spherical Harmonics

Chapter 14: Curvilinear Coordinates and Tensor Analysis

14.1.  Vectors and Linear Forms

14.2.  Tensors

14.3.  Inner Product and the Metric Tensors

14.4.  Contraction

14.5.  Tensor Component Transformation

14.6.  Symmetric and Antisymmetric Tensors

14.7.  The Mixed Vector Product and the Volume Tensor

14.8.  The Exterior Vector Product

14.9.  Correspondence Between Vectors and Linear Antisymmetric Forms (Volume Duality)

Chapter 15:. Differential Calculus and Vector Analysis Using Curvilinear Coordinates

15.1.  Tensor Fields and Differential Calculus in Euclidean Space

15.2.  Derivative of a Vector along a Curve

15.3.  Differentiation along a Curve and with Respect to a Vector

15.4.  Covariant Derivatives and Connection

15.5.  Differential Calculus

15.6.  Applications to Orthogonal Curvilinear Coordinates