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Elias C. Tonias · Constantine N. Tonias Geometric Procedures for Civil Engineers Geometric Procedures for Civil Engineers Elias C. Tonias (cid:129) Constantine N. Tonias Geometric Procedures for Civil Engineers Elias C. Tonias Constantine N. Tonias Tonias Engineers The CEDRA Corporation Pittsford , NY , USA Pittsford , NY , USA ISBN 978-3-319-24293-4 ISBN 978-3-319-24295-8 (eBook) DOI 10.1007/978-3-319-24295-8 Library of Congress Control Number: 2016934071 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfi lms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. T he use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specifi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland This book is dedicated to all those civil engineers of the 1950s, 1960s, and 1970s who labored endlessly to develop electronic computing software under primitive computer operating systems and hardware for the benefi t not only of their business entities but of the civil engineering profession above all. The authors wish to acknowledge the contributions of their friends and colleagues, too numerous to list, in the civil engineering staff of various civil engineering institutions, the U.S. Army Corps of Engineers, the early members of the Civil Engineering Program Applications (CEPA), the staff of the American Society of Civil Engineers (ASCE), and the International Business Machines (IBM) Corporation. Pref ace I t is said that next to the Holy Bible, Euclid’s E lements and related books are the most p ublished books. Most of these books on geometry concentrate on the theoretical aspects of geometry and various geometric constructions. Geometric Procedures for Civil Engineers has been written not only for civil engineers but also for those who are interested in the preparation and use of computer-aided solutions to most common geometric constructions. Computer-aided drafting (CAD) programs have removed the tedious aspects of fi nding the solution for geometric construction problems, but they also have removed from the user’s mind the geometric principles and methodology of the solution of said constructions. The actual product of this book is an extensive set of computer procedures that can be used to solve individual geometric constructions, and even to help create a customized geometric design system or supplement an existing system with specialized add-on procedures. The description of the various computer procedures includes a presentation of the geometric principles of the solution, and a general textual description of the computer steps to accomplish the solution as part of the book. The source code in generic Visual Basic format of the procedure and of a test program is included as part of an accompanying compact disc (CD). Although this book is not intended as an instructional book, its contents do provide solutions to numerous geometric construction problems that could act as a supplement to an instructional book. A basic knowledge of Euclidean Geometry is assumed. The book is comprised of three parts. Part 1 is composed of the fi rst three chapters and concentrates on certain historic and rudimental aspects of geometry. Chapter 1 presents an historic synopsis of geometry from its early years through the Hellenistic period, the Middle Ages, the Renaissance years, and up to the current computer revolution. The intent of this chapter is twofold, (a) to familiarize the reader in the evolution of geometry from its infancy to its present state (early twenty-fi rst century), and (b) to acquaint the modern engineer and computer-aided design/drafting (CAD) user of the “good old days” prior to the introduction of and during the early years of not too long ago of the computer revolution. Several photographs of computer hardware have been included. In the early years of the computer there was a distinct appearance among the computers of the various manufacturers, whereas nowadays they all seem to look alike. The reader should note that the names of the various Greek mathematicians and geometers are written in this book in their Greek form and not in their Latinized form such as Apollonios rather than Apollonius. It seems unnecessary to transliterate from one language into a second via a third language. Chapter 2 presents the defi nition of the basic geometric features such as points, lines, and curves and classifi cation thereof, as well as of certain terms and conventions as used in the book. Chapter 3 presents various rudimental procedures referred to as support procedures that are used extensively by the construction procedures of the subsequent chapters. These support procedures are assembled in the following groups that: vii viii Preface • Compute basic and inverse trigonometric functions (see the commentary after the last bullet). • Provide for the conversion of various forms of angles and directions, such as from degrees in decimal form to azimuths in radians, and vice versa, as well as the extraction of degrees, minutes, and seconds from a text string. • Compute arc lengths, distances, and directions between points. • Compute angles under a variety of input data conditions. • P rovide for miscellaneous operations including the computation of elevations, Lagrangian interpolation for polynomial curve generation, handling of strings of geometric features, and stationing of distances in the US Customary and SI formats. • C onvert a polyline or polygon comprised of a series of chords into a series of lines and circular arcs if certain chords are small enough to constitute a circular arc defi ned by a center point, two endpoints, and a radius. • Convert a distance value into a US Customary or SI station format. In the infancy years of electronic computing, the trigonometric functions were not a staple item of the programming languages as they are today. Numerous discussions were held regarding methodology, precision and speed until the mid-1960s. Some of the trigonometric procedures presented in this publication are for historic purposes, although some of them are still in use today. Part 2 is composed of the next fi ve chapters that include various procedures that perform the geometric constructions summarized below. A knowledge of these procedures is considered essential in following the various construction procedures of the subsequent chapters. These procedures may also be used in the programming of entirely new construction procedures that may not be available in this book. In general, in this book, and unless otherwise qualifi ed, the term “curve” in singular or plural form refers to a circular arc. Spirals, parabolas, and ellipses are addressed as such unless it is evident that the term curve or curves pertains to one of them. The various procedures that are presented in the fi rst two parts of this book may be used to easily develop solutions to problems that may not have been addressed in this book. Similarly, the procedures of the geometric constructions as presented herein may be modifi ed to refl ect individual desires, ambitions, and goals. Chapter 4 contains procedures that create and manipulate points and lines between two points, lines comprised of a series of concatenated two-point lines referred to as polylines and poly- gons. A polyline comprising a closed fi gure does not constitute a polygon unless specifi cally is so made to be. These procedures are assembled in the following groups that: • I ntroduce points along a circular arc given a plus arc distance from its start point and an offset distance therefrom, or locating the midpoint of an arc, as well as locating the PC and PT and center point of an arc tangent to two given lines, or fi nding the PI when knowing the PC, the PT, and the center point of the arc. • I ntroduce points along a line or arc given a distance along the line or arc from the start point and a normal offset to the left or right of the line or arc. • Project points on a two-point line, a polyline or circular arc. Such projections may or may not involve stationing. The procedure projecting a point on a polyline provides a multitude of user options. Preface ix • Determine whether a point lies within the extent of a line or circular arc, and rotate and/or scale points about a pivot point. • Compute the coeffi cients of the implicit form of a straight line. Chapter 5 presents procedures that create circular curve features that are tangent to a given line or circular arc, as well as create such features that are not tangent to any other feature. These procedures generate individual curves under a variety of conditions of given curve element composition including length of arc, chord, tangent or middle ordinate, central angle, or pass- ing through a given point. Compound and reverse curves are treated separately in Chap. 8 . Also included in this chapter are procedures that provide for the conversion of curves into polyline for graphic presentation or other purposes. Chapter 6 is dedicated to various geometric intersections of lines and circular arcs under various input requirements. The procedures addressing these intersections are grouped into intersections of (a) lines with lines, (b) lines with arcs, and (c) arcs with arcs. These procedures are followed by a fourth group of fi ve generic intersection procedures that provide for the intersection of two strings of concatenated series of lines and arcs, again under various input and/or desirable results. Chapter 7 presents procedures that construct tangent lines to circular curves and circular arcs tangent to other circular arcs under various conditions. These procedures are divided into four groups: (a) tangent lines to circular arcs, (b) circular arcs to lines, (c) circular arcs tangent to lines and other circular arcs, and (d) circular arcs tangent to other circular arcs. Lines tangent to spirals are addressed in Chap. 9 , and tangent to parabolic curves are addressed in Chap. 1 1 . To construct a circular arc tangent to a spiral or parabolic curve at a specifi c point, a user needs to locate the point on the pertinent curve and determine the instantaneous radial and therefore the normal direction. Depending on the construction problem and on which parameters have been specifi ed as being known, there could be more than one solution to a problem. The procedures, unless otherwise noted, compute all potential solutions and provide them to the end user for the desirable selection. Chapter 8 concentrates on the creation of compound and reversed curves. The procedures of this chapter are grouped into three sets or groups depending on what information is to be considered as given, and which features of the curves may vary or fl oat. The fi rst set contains eight procedures that construct pairs of compound curves under different given input requirements, while the second set contains fi ve procedures that construct three compound curves, and the third set contains but one procedure that constructs a multicentric set of compound curves. Generally the combinations of given requirements are the radii, central angles, short and long tangent lengths, and a point through which a curve has to pass through. The back and forward tangents could be fi xed or fl oating as could be the start and endpoint of the curves. The fourth set contains nine procedures that construct pairs of reversed curves. The multicentric procedure of the third group could include one or more reversed curves. Part 3 is comprised of the remaining four chapters and addresses transition spiral construction problems and composite construction design problems. Chapter 9 is dedicated to the establishment of highway transition spirals; the intersection of spirals with lines, curves and other spirals; the projection of points on spirals; and the introduction of tangent lines to spirals. A discussion of four other spiral types and a comparison with the highway transition spiral are also provided.