In Coordinates in Geodesy definitions and transformations are treated based on the general principles of differential geometry for surfaces and three-dimensional Euclidean space, strictly using the tensor calculus. The broad approach applying general concepts of constructing and transforming coordinates allows clearly arranged solutions for all geodetic applications. Moreover, the great number of examples given in this book explain in detail the principles of coordinates in geodetic surveying using ellipsoids of revolution as reference surfaces.

Inhaltsangabe1. 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 Coor