Geometry of Horizontal Curves

Geometric Design Solution of Horizontal Curves
Horizontal Curve Design and Layout
Objectives:
1. Become familiar with the geometry of horizontal curves.
2. Design a curve and then lay it out.
Overview:
There are numerous construction applications for horizontal curves. A curve gives a
smooth transition from one line of direction to another. The most common
applications are roads, railroads, sidewalks, water channels, and curb returns for
drainage purposes.
The radius is the most important parameter of a curve. It defines how sharp or flat
the curve is. The tangent into and out of a curve is perpendicular to the radius. The
central angle determines the length of the curve. The chord is a straight line
between the two ends of a curve. The central angle is equal to the deflection angle
formed by the intersection of the two tangents (prove this to yourself if you haven’t
already).
The five variables highlighted above are all mathematically related. When two of
these five variables are known, the other three can be calculated using geometry and
trigonometry. Once a curve is defined by the radius and central angle, any point on
the curve can be located. Its relationship with all of the other points on the curve and
its location can be calculated. Some of the points that are important in describing a
curve are the point of curvature (PC) (also called the beginning of curve or BC), the
point of tangency (PT) (also called the end of curve or EC), and the point of
intersection of the two tangents (PI). (See the Figure below.) If an angle is
described at PI, it is the deflection angle of the intersection of the two tangents and is
equal to I (or .).
All data for a horizontal curve can be calculated from this basic curve information,
including the area enclosed or excluded by the curve. This means that from any point
on the curve or tangent, the location of equidistant points along the curve (i.e., every
50 or 100 feet for example), or any other point on the curve can be computed by
determining a distance and a deflection angle.
If the coordinates of the basic curve elements are known, the position of every point
is fixed within the coordinate system. This fact makes curve layout easier because the
curve can be laid out from any point within the coordinate system and not just from
points on the curve or tangent.
Horizontal curve definitions
Where:
I or . = Intersection or Central Angle
R = Radius of the curve
T = Tangent
L = Curve Length
LC = Long Chord
E = External distance
M = Mid-Ordinate
I (sometimes defined as .), R, T, L, and LC are the five basic variables, of which two
must be known (usually these are I and R).
Degree of Curve
The degree of curve (D), which can be related to the radius (R) is used by many
highway agencies to define the sharpness of a curve, especially on curves with large
radii. Highway design usually employs the arc definition of the degree of curve.
Railroad design often uses the chord definition of the degree of curve.


L = 100.00'
RRDaLC
LRR.5Dc.5DcLC = 100.00'
Dc
Arc definition: Chord definition:

Central angle subtended by a Central angle subtended by a
100.00 ft. arc 100.00 ft. chord.
L = 100.00 ft. L = 100.. ft.
L.C. = 99.. ft. LC = 100.00 ft.
Formulas
The following formulas assume that I (or .) and R are known:
Tangent distance = T
Curve Length = L
Chord Length = LC
External Distance = E
Mid Distance = M
Central Angles = IX

Laying Out a Curve
1. Curves are staked out using deflection angles and sub-chords
2. Stations for a curve are measured along the arc of the curve.
3. Measured values in curve staking
tangent
PT
PI
PC
tangent
Long Chords

Short Chords
Laying out a horizontal curve
Moving Up on a Curve
1. To move up on a curve and retain the original field notes, occupy any station and
back sight to any other station with the angle set to the total deflections of the
stations sighted.
2. Make sure the deflections are on the correct side of 0..
3. Continue to deflect angles as per the notes.

 PC (Original Total Station Location)
Theodolite
moved to C

Procedure:
1. Calculate all other basic variables based on a curve defined by: D=20., I =33.69.,
and PI @ 2+45.15.
2. Calculate the deflection angles and long chords for the following stations: PC; 2 +
00; 2 + 50; 3 + 00; PT.
3. Go to the specified location and set up over a convenient point.
4. Lay out the curve based on your calculations, making sure not to set your arc up
so that it goes into a tree or a building.

Field Book:
1. Show in tabular form the deflection angles and chords for the specified stations.
2. Plot the curve, including the location of each station.

Trigonometry Problems

103 Trigonomertic Problems
Contents

Preface vii
Acknowledgments ix
Abbreviations and Notation xi
1 Trigonometric Fundamentals 1
Definitions of Trigonometric Functions in Terms of Right Triangles 1
Think Within the Box 4
You’ve Got the Right Angle 6
Think Along the Unit Circle 10
Graphs of Trigonometric Functions 14
The Extended Law of Sines 18
Area and Ptolemy’s Theorem 19
Existence, Uniqueness, and Trigonometric Substitutions 23
Ceva’s Theorem 28
Think Outside the Box 33
Menelaus’s Theorem 33
The Law of Cosines 34
Stewart’s Theorem 35
Heron’s Formula and Brahmagupta’s Formula 37
Brocard Points 39
Vectors 41
The Dot Product and the Vector Form of the Law of Cosines 46
The Cauchy–Schwarz Inequality 47
Radians and an Important Limit 47
Constructing Sinusoidal Curves with a Straightedge 50
Three Dimensional Coordinate Systems 51
Traveling on Earth 55
Where Are You? 57
De Moivre’s Formula 58
2 Introductory Problems 63
3 Advanced Problems 73
4 Solutions to Introductory Problems 83
5 Solutions to Advanced Problems 125
Glossary 199
Further Reading 211


Complete Book in PDF Formate

Area of Triangle

Area=SQRT(s(s-a)(s-b)(s-c))
where s=(a+b+c)/2 or perimeter/2

Click for Excel File

The SUM of TWO SIDES MUST ADD UP TO BE GREATER THAN THE LENGTH OF THE REMAINING THIRD SIDE.
OR (with sides A,B,C):

• A + B > C,
• B + C> A,
• and A + C>B

So, can you have a triangle with sides 5, 12, 16?
How about 5, 12, 13? What does this look like?
Interestingly heron's formula also leads us to a result where we can calculate the height of a triangle. With base c and height h the height is,

h =

2

(SQRT(s(s-a)(s-b)(s-c))