Plane Networks and their Applications

1 Introduction.- 1.1 Description of the Discrete Model.- 1.1.1 Free-Free Network.- 1.1.2 Fixed-Free Network.- 1.1.3 Fixed-Fixed Network.- 1.2 Description of the Continuous Model.- 1.2.1 A to AT: Integration by Parts.- 1.2.2 Green’s Function.- 1.2.3 Change of Boundary Conditions.- 1.2.4 Modified Green Function.- 1.2.5 Green’s Function as a Formal Covariance Function.- 1.3 Variance Propagation.- 2 Discrete Approach.- 2.1 Motivation for the Study.- 2.2 Basic Matrix of Leveling.- 2.2.1 Eigenvectors and Eigenvalues.- 2.2.2 Pseudoinverse.- 2.2.3 Singular Value Decomposition.- 2.2.4 Two-Dimensional Networks.- 2.3 Regular Traverse.- 2.3.1 Random Errors in the Regular Traverse.- 2.3.2 Systematic Errors in the Regular Traverse.- 2.4 Varying the Boundary Conditions.- 2.4.1 Straight Line.- 2.4.2 Circumference of a Circle.- 2.5 Variance Propagation.- 2.6 Asymptotic Behavior of the Node Variance.- 2.7 On the Smoothness and Roughness of the Eigenvectors.- 2.8 Green’s Formula for Plane Trigonometric Networks.- 3 Continuous Approach.- 3.1 Leveling Networks.- 3.1.1 Single Triangle.- 3.1.2 Entire Network.- 3.2 Advanced Error Analysis.- 3.2.1 Green’s Function for the Unit Circle.- 3.2.2 Green’s Function for the Ellipse.- 3.2.3 Green’s Function for the Annulus.- 3.3 Plane Elastic Continuous Networks: A Heuristic Exposition.- 3.4 Distance Networks.- 3.4.1 Single Triangle.- 3.4.2 Distance Network.- 3.4.3 Azimuth Networks.- 3.4.4 Combined Distance and Azimuth Networks.- 3.5 Estimates of the Weighted Square Sum of Residuals: the Korn Inequality.- 4 Networks with Relative Observations.- 4.1 Dealing with Relative Observations.- 4.2 Fundamental Solution.- 4.3 Solution of the Boundary Value Problem.- 5 Spectrum.- 5.1 Spectral Density of the Discrete Laplacian.- 5.2 Spectral Distribution Function N(?).- 5.3 Additional Remarks on the Spectral Properties of Geodetic Networks.- 6 Simple Applications.- 6.1 Stiffness Matrix in Practice.- 6.2 Displacement Functions given a Priori.- 6.3 Merging of Digitized Maps.- 6.4 Interpolation of Discrete Vector Field: Cubic Splines.- Author Index.