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. This concise, fast-paced text introduces the concepts and applications behind plane networks. Currently, there is nothing in book form dealing with the topics covered in this work. The presentation unfolds in a systematic, user-friendly style and goes from the basics to cutting-edge research. Key features include: * presentation of the basics required: fundamental material from linear algebra and differential equations * examination of classical mathematical tools for analyzing discrete networks, followed by a well-developed theory, which is the continuous analogue of a discrete network * transition from the discrete to the continuous case, described via finite elements; Ch. 3 involves an analysis of linear operators, variational calculus, boundary value problems for PDEs, and Green's functions; Green's functions are the continuous analogue of the discrete error covariance functions, and form the basis for all types of error prediction * numerous examples and illustrations * techniques applied to leveling and other observation types of networks in one and two dimensions * several different applications of the continuous theory * practical problems, supported by MATLAB files, underscore the continuous theory; additional material can be downloaded from the author's website at www.kom.auc.dk/~borre/network * bibliography of recent results and index Plane Networks and their Applications is aimed at applied mathematicians, mechanical engineers, geodesists and graduate students, and should be an excellent text for self-study, classroom, or reference Front Matter....Pages i-xiii Introduction....Pages 1-23 Discrete Approach....Pages 25-54 Continuous Approach....Pages 55-100 Networks with Relative Observations....Pages 101-126 Spectrum....Pages 127-138 Simple Applications....Pages 139-151 Back Matter....Pages 153-170
