Coordinates in Geodesy

1. Introduction.- 2. General Fundamentals of Surface Coordinates.- 2.1 Fundamentals of the Theory of Surfaces.- 2.1.1 Rudiments.- 2.1.2 First Fundamental Form.- 2.1.3 Covariant and Contravariant Bases.- 2.1.4 Equations of Gauss and Weingarten.- 2.1.5 Covariant Derivatives of Surface Vectors.- 2.1.6 Measures of the Curvature of Surface Curves and Surfaces.- 2.1.7 Normal and Principal Curvatures of Surfaces.- 2.1.8 Surface Curves with Given Geodesic Curvature.- 2.1.9 Geodesic Lines.- 2.1.10 Geodesic Surface Coordinates.- 2.1.11 Special Studies of Geodesic Polar Coordinates.- 2.1.12 Riemannian Normal Coordinates.- 2.1.13 Isothermal Surface Coordinates.- 2.1.14 Special Studies of Isothermal Surface Coordinates.- 2.2 Fundamentals of Complex Analysis.- 2.2.1 Preliminary Remarks.- 2.2.2 Functions of a Complex Variable.- 2.2.3 Differentiation and Integration of Analytic Functions.- 2.2.4 Power Series of Analytic Functions.- 3. Representing the Transformation Equations Between Surface Coordinates by Power Series.- 3.1 Constructing Surface Coordinates.- 3.2 Representing Power Series.- 3.3 Transformations Between Geodesic Polar Coordinates and Arbitrary Surface Coordinates.- 3.3.1 General Transformation Equations.- 3.3.2 Calculating Small Geodesic Triangles.- 3.4 Transformations Between Geodesic Parallel Coordinates and Arbitrary Surface Coordinates.- 3.4.1 Indirect Representation by Power Series.- 3.4.2 Direct Representation by Power Series.- 3.5 Transformations Between Isothermal Surface Coordinates and Arbitrary Surface Coordinates.- 3.5.1 General Transformation Equations.- 3.5.2 Transformations Between Two Isothermal Coordinate Systems.- 4. Surface Coordinates on Ellipsoids of Revolution.- 4.1 Preliminary Remarks.- 4.2 Ellipsoids of Revolution and Their Representation Using Geographic Coordinates.- 4.3 Transformations Between Geodesic Polar Coordinates and Geographic Coordinates.- 4.3.1 Transforming the Coordinates.- 4.3.2 Transforming the Metric Tensor.- 4.3.3 Tangent Vectors and Azimuths of Geodesic r-Lines.- 4.3.4 Transformations Between the Arc Length of a Meridian and the Ellipsoidal Latitude.- 4.4 Transformations Between Soldner’s Parallel Coordinates and Geographic Coordinates.- 4.4.1 Transforming the Coordinates.- 4.4.2 Transforming the Metric Tensor.- 4.4.3 Meridian Convergence.- 4.5 Defining Isothermal Surface Coordinates in the Geographic Coordinate System.- 4.6 Transformations Between Isothermal Geographic Coordinates and Geographic Coordinates.- 4.6.1 Preliminary Remarks.- 4.6.2 Isothermal Latitude.- 4.6.3 Isothermal Longitude.- 4.7 Transformations Between Gaussian Isothermal Coordinates and Geographic Coordinates.- 4.7.1 Directly Transforming the Coordinates.- 4.7.2 Indirectly Transforming the Coordinates.- 4.7.3 Transforming the Metric Tensor.- 4.7.4 Meridian Convergence.- 4.8 Transformations Between Gaussian Isothermal Coordinates and Geodesic Polar Coordinates.- 4.8.1 Directly Transforming the Coordinates.- 4.8.2 Tangent Vectors and Direction Angles of Geodesic r-Lines.- 4.8.3 Coordinate Transformation by Reducing Directions and Distances.- 4.9 Transformations Between Two Systems of Gaussian Isothermal Coordinates.- 4.9.1 Indirectly Transforming the Coordinates.- 4.9.2 Directly Transforming the Coordinates.- 4.10 Transformations Between Isothermal Stereographic Coordinates and Geographic Coordinates.- 4.10.1 Transforming the Coordinates.- 4.10.2 Transforming the Metric Tensor.- 5. Three-Dimensional Coordinates.- 5.1 Preliminary Remarks.- 5.2 Fundamentals of Three-Dimensional Euclidean Geometry.- 5.2.1 Coordinate Transformations.- 5.2.2 Representing Transformations Between Three-Dimensional Curvilinear and Cartesian Coordinates by Power Series.- 5.2.3 Space Curves.- 5.3 Surface-Normal Coordinates.- 5.3.1 General Fundamentals.- 5.3.2 Representing Transformations Between Surface-Normal Coordinates and Cartesian Coordinates by Power Series.- 5.3.3 Transformations Between Three-Dimensional Polar Coordinates and Polar Coordinates on the Reference Surface.- 5.4 Geodetic Coordinates.- 5.4.1 Preliminary Remarks.- 5.4.2 Geographically Geodetic Coordinates.- 5.4.3 Gaussian Geodetic Coordinates.- 5.4.4 Transformations Between Ellipsoidal Polar Coordinates and Polar Coordinates on the Reference Ellipsoid.- 5.5 Transformations Between Geographically Geodetic Coordinates.- 5.5.1 Fundamentals.- 5.5.2 Transformations Between Concentrically Geodetic Coordinate Systems.- 5.5.3 Transformations Between Arbitrary Geodetic Systems Based on Central Transformation Parameters.- 5.5.4 Transformations Between Arbitrary Geodetic Systems Based on Local Transformation Parameters.- 5.5.5 Determining Transformation Parameters.- 5.5.6 Determining Mean Reference Ellipsoids.