# Plane Networks and their Applications

• Kai Borre
Book

1. Front Matter
Pages i-xiii
2. Kai Borre
Pages 1-23
3. Kai Borre
Pages 25-54
4. Kai Borre
Pages 55-100
5. Kai Borre
Pages 101-126
6. Kai Borre
Pages 127-138
7. Kai Borre
Pages 139-151
8. Back Matter
Pages 153-170

### Introduction

Surely most geodesists have been occupied by seeking optimal shapes of a net­ work. I'm no exception. This book contains the more fruitful results on the topic. No matter how you choose to understand the adjective "optimal," it is no doubt useful as a beginning to understand error propagation in various types of net­ works. Basically, geodesists are familar with the actual, discrete network. So this book brings together some elementary means of analyzing networks with a few hundred points. The effectofchanging boundary conditions is especially studied. The variance propagation in the network is derived from covariance matrices. During a symposium in Oxford in 1973 geodesists were asking: Is it possible to create a special theory for geodetic networks? The key is that geodetic networks share a fundamental characteristic: The connections are local. Observations are taken between neighbors. The underlying graph has no edges connecting distant points. And we can obtain stable information about the global problem for the whole network by solving a simpler problem for a local neighborhood within the network. This bookalso deals with networktheory in acontinuousmode. When the num­ ber of points becomes very large, it is natural to look for a substitute for the dis­ crete method. The fruitful transition from discreteness to continuum is to let the distance between points tend to zero and at the same time boundcertain functions. A major step is to redefine the weights for all observationsas weightperunitarea.

### Keywords

Algebra Eigenvector Interpolation MATLAB Matrix Operator applied math/network theory calculus linear algebra mechanical engineering model partial differential equation pdes

#### Authors and affiliations

• Kai Borre
• 1
1. 1.Danish GPS CenterAalborg UniversityAalborg ØDenmark

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4612-0165-6
• Copyright Information Birkhäuser Boston 2001
• Publisher Name Birkhäuser, Boston, MA
• eBook Packages
• Print ISBN 978-1-4612-6642-6
• Online ISBN 978-1-4612-0165-6
• Buy this book on publisher's site