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Excerpt from The Transition Spiral: And Its Introduction to Railway Curves With Field Exercises in Construction and Alignment The aim of this handbook is to afford concise and complete demonstration of the introduction of transition curves, or spirals, in the various argument problems that accrue in connection with the construction or improvement of railway curves. Now the extent to which such problems can be satisfactorily essayed depends upon the mathematical development of the form of transition adopted, and the facility with which this development admits utilisation in the field - the effective compromise of mathematical truth with practical precision. Hence, besides fulfilling its objective function, the ideal transition curve must be easy and flexible in construction; a curve with simple, definite relations to main curves and straights; and, above all, one which can be introduced without the aid of special tables, such as hamper the construction of American tabular spirals. The curve most readily amenable to these requirements is to be found in the clothoid = m the spiral advanced and investigated in this connection by Mr. James Glover, M.A., A.M.I.C.E., in a paper read to the Institution of Civil Engineers. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works."
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Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space.