Table Of ContentElias 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
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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.
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imply, even in the absence of a specifi c statement, that such names are exempt from the relevant protective laws and
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express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.
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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 writ-
ten 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 draft-
ing (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 prin-
ciples 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 prob-
lems 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 com-
puter 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 cen-
ter 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 vari-
ous input requirements. The procedures addressing these intersections are grouped into inter-
sections 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 intersec-
tion 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 pro-
cedures, 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 con-
sidered 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 proce-
dure 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 introduc-
tion of tangent lines to spirals. A discussion of four other spiral types and a comparison with
the highway transition spiral are also provided.
