PLS Mastery

Chapter 1: Introduction to Surveying

Definition for Surveying

• The science, art, and technology involved in determining the relative positions of points on, above, or below the Earth’s surface.
• A field that encompasses various methods for measuring and gathering data about the physical environment, processing this information, and delivering diverse outputs to a broad range of users.

What do we do in SURVEYING?

Earth Measurements

• Distances: Includes horizontal, slope, and vertical measurements.
• Angles: Covers horizontal, vertical, and zenith angles.
• Relative Positioning: Determining the location of points on, above, or below the Earth’s surface.
• Position Calculation: Achieved through mathematical processing of field measurements.
• Final Output: Provides positional data in the form of:
  • Maps – Available in digital and paper formats.
  • Coordinates – Precise numerical representation of locations.
  • Features – Physical and environmental attributes of surveyed areas.

Why Surveying is Important

Role of Surveying in Civil Engineering

• The planning and design of buildings, parks, roads, bridges, tunnels, sewers, and other infrastructure rely on data gathered through surveying.
• During construction, projects of any scale are built based on reference points and alignments established by surveying.
• Therefore, surveying is a fundamental requirement for all civil engineering projects.

Types of Surveying

Types of Surveying

• Land, Boundary, and Cadastral Surveying
  Focuses on marking and mapping property boundaries, as well as defining land ownership rights.
• Construction and Route Surveying
  Involves surveying tasks for planning, designing, executing, and monitoring construction projects such as roads, bridges, buildings, and infrastructure networks.
• Quantity Surveying
  Includes fieldwork and calculations for determining project quantities, such as earthworks (cut and fill) and construction materials.
• Plane Surveying (Covered in This Course)
  Suitable for projects over small areas where the Earth’s curvature is negligible and can be ignored.
• Geodetic Surveying
  A branch of geodesy that deals with large-scale projects where Earth’s curvature must be considered.
• Topographic Surveying
  Focuses on identifying locations and elevations of natural and man-made features for map creation.

Course Organization and Contents

Fundamentals of Surveying

Basics of Surveying

• Arithmetic and Measurements: Understanding percentages, ratios, and proportions in surveying calculations.
• Geometry: Types of angles, properties of polygons, triangles, and circles.
• Angles in Surveying: Concepts of bearings and azimuths in line direction measurements.
• Theory of Errors: Definitions, types, sources of errors, and precision measurement techniques.

Measurements and Field Practice

• Distance Measurement: Measuring slope, horizontal, and vertical distances using tapes, electronic distance measurement (EDM) instruments, and leveling methods while accounting for errors.
• Direction and Angle Measurement: Utilizing theodolites and total stations for precise angle and direction measurements.

Applications

• Topographic Surveying and Mapping: Understanding contour lines, their characteristics, contour map construction, and interpretation.
• Traversing: Applications of open and closed traverses, latitude and departure calculations, error detection, and correction techniques.
• Areas and Volumes: Determining traverse areas, profiles, and cross-sections using various methods, and calculating earthwork volumes using grids or contour maps.

Chapter 2: Basic Surveying Math

Subjects

Essential Mathematical Concepts in Surveying

• Percentages – Fundamental calculations for scaling and comparisons in surveying.
• Pythagorean Theorem – Essential for distance and height calculations in right-angled triangles.
• Ratio and Proportion – Used for scaling and proportional measurements.
• Types of Angles – Understanding different angle classifications and their applications in surveying.
• Polygons – Properties and applications of multi-sided shapes in surveying calculations.
• Triangles – Classification and properties of triangles relevant to land measurements.
• Rectangles and Squares – Basic geometric shapes used in area and boundary calculations.
• Trapezoids – Properties and applications in land area computation.
• Circles and Arc Length – Calculating the length of arcs and circular measurements.
• Perimeter – Determining the total boundary length of different geometric shapes.
• Solution of Oblique Triangles – Applying trigonometric principles to solve non-right-angled triangles.

Percentages

Understanding Percentages

The term “percent” originates from a Latin word meaning “by the hundred.” The “%” symbol represents percent. A fraction with 100 as the denominator expresses a percentage.

Examples:

1. Converting Fractions or Decimals to Percent:
   • 0.005=51000=51000×100%=0.5%0.005 = \frac{5}{1000} = \frac{5}{1000} \times 100\% = 0.5\%0.005=10005​=10005​×100%=0.5%
2. Converting Percent to Fraction or Decimal:
   • 0.7%=0.7100=71000=0.0070.7\% = \frac{0.7}{100} = \frac{7}{1000} = 0.0070.7%=1000.7​=10007​=0.007
   • 150%=150100=1.50150\% = \frac{150}{100} = 1.50150%=100150​=1.50

More Examples:

• Finding a percentage of a number:
  • 1%1\%1% of 7823 → 7823×1100=78.237823 \times \frac{1}{100} = 78.237823×1001​=78.23
  • 68%68\%68% of 300 → 300×68100=204300 \times \frac{68}{100} = 204300×10068​=204
• Finding what percentage one number is of another:
  • 15 is what percent of 750?
    15750×100%=2%\frac{15}{750} \times 100\% = 2\%75015​×100%=2%
• Finding the whole when a percentage of it is known:
  • If 30 is 75% of a number Y, find Y:
    30=75100×Y30 = \frac{75}{100} \times Y30=10075​×Y
    Y=40Y = 40Y=40

Ratio and Proportion

Understanding Ratios and Proportions

Ratios:

A ratio is a comparison between two quantities. It can be written in different formats, such as:

• 2 to 5
• 2 : 5
• 2 ÷ 5
• 2 / 5

In surveying, ratios are often expressed in the form of:

• 1 to x
• 1 : x
• 1 ÷ x
• 1/x

Proportions:

A proportion represents the equality between two ratios. For example, the proportion:

• 2:5 = 4:10
• 2 ÷ 5 = 4 ÷ 10
• 2/5 = 4/10

Geometry: Angles

Polygons

Polygons and Interior Angles

• A polygon is a closed shape formed by straight lines lying in the same plane.
• The sum of the interior angles of a polygon is determined by the formula:
  Sum of Interior Angles=(n−2)×180∘\text{Sum of Interior Angles} = (n – 2) \times 180^\circSum of Interior Angles=(n−2)×180∘
  where n represents the number of sides.

Examples:

• A triangle (3 sides) → (3−2)×180∘=180∘(3 – 2) \times 180^\circ = 180^\circ(3−2)×180∘=180∘
• A rectangle (4 sides) → (4−2)×180∘=360∘(4 – 2) \times 180^\circ = 360^\circ(4−2)×180∘=360∘
• A pentagon (5 sides) → (5−2)×180∘=540∘(5 – 2) \times 180^\circ = 540^\circ(5−2)×180∘=540∘

Triangles (smallest polygon)

Exercise:
Find the measure of angle x

Triangles (smallest polygon)

Exercise:
Which of the following triangles are always similar?

Exercise Solution:

Given:

• A triangle with sides 5, 6, and 10.
• A similar triangle where the shortest side is 15.

Since the triangles are similar, their sides are proportional. The ratio of similarity is:155=3\frac{15}{5} = 3515​=3

Thus, multiplying the other sides by 3:6×3=18,10×3=306 \times 3 = 18, \quad 10 \times 3 = 306×3=18,10×3=30

Answer: The longest side of the similar triangle is 30.

True/False Question:

Statement: Similar triangles are exactly the same shape and size.
Answer: False – Similar triangles have the same shape but can have different sizes.

Solution to Right Triangles

The Pythagorean Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.c2=a2+b2c^2 = a^2 + b^2c2=a2+b2

where:

• c = Hypotenuse (longest side)
• a, b = Other two sides

Rearranging for Side Calculation:

• If c is known: a2=c2−b2a^2 = c^2 – b^2a2=c2−b2 b2=c2−a2b^2 = c^2 – a^2b2=c2−a2

This theorem is widely used in surveying, engineering, and construction for distance and angle calculations.

Solution of Oblique Triangles

Solution of Oblique Triangles

Solution of Oblique Triangles

Solution to Oblique Triangles

Rectangles and Squares

A rectangle is a four-sided polygon whose angles are right angles. A
square is a rectangle whose sides are equal

Trapezoid (Trapezium)

A trapezoid is a four-sided polygon that has two parallel sides and two
nonparallel sides.

Parallelograms and Rhombi

A parallelogram is a four-sided polygon with pairs of parallel sides. A
rhombus (plural – rhombi) is its special case with all sides equal
(equilateral).

Circles

A circle is a closed plane curve, all points on which are equidistant from
a point within called the center.

Circumference

It is the perimeter of a circle or an ellipse.
The sum of the lengths of the sides of a polygon is called the perimeter
of the polygon.

Chapter 3: Basics of Surveying

Drawing Scale

-A map cannot be the same size as the area it represents.
-A scaled drawing of a building maintains the same shape as the actual structure it represents but differs in size.

Common conventional scales

Types of Map Scale

1. Verbal Scale
   Describes the relationship between map distance and ground distance using words.
   Example: “One centimeter represents one kilometer” means 1 cm on the map equals 1 km in reality.
2. Representative Fraction Scale (Ratio Scale)
   Expresses the ratio of map distance to ground distance in numerical form, such as 1:X or 1/X.
3. Graphic (Bar) Scale
   A visual scale represented as a line marked with distances, allowing users to measure directly using a ruler.

Representative Fraction

RF is the ratio of map distance to ground distance.
Properties

• Uses the ( : ) or ( / ) ratio formats;
• has 1 as the map distance, and
• has no units
• In other words RF is the map distance divided by the ground
  distance in the same units.
• Examples
• 1 : 100
• 1 / 150

• If the ratio has units, it must be converted and removed
  Example 1
  Write the scale 1 cm to 1 m in RF format.
  1 cm to 1 m = 1 cm : 1 m
  = 1 cm : 100 cm
  = 1 : 100
  Example 2
  Simplify the scale 5 cm : 2 km to RF format
  5 cm : 2 km = 5 cm : 2000 m
  = 5 cm : 200,000 cm
  = 5 : 200,000
  = 1 : 40,000

To determine the scale of the drawing do the following:
1 Drawing distance
—– = —————————
X Ground distance
a. Pick an actual distance from the drawing. Take the 200 m distance
(why?)
b. Use a ruler to measure the corresponding drawing distance in
millimeters, it was found = 50 mm.
c. 1: X = drawing distance : ground distance ( a proportion )
drawing distance : ground distance
50 mm : 200 m
50 mm : 200000 mm
50 : 200000
1 : 4000
d. Scale is 1 : 4000

Convert map distances to ground distances
If the map scale is 1 : X
1 Map distance
—– = ————————-
X Ground distance
Ground distance = Map distance * X
Example
A particular map shows a scale of 1 : 5000. What is the
ground distance if the map distance is 8 cm?
Ground distance = 8 cm * 5000 = 40000 cm = 400 m

Convert ground distances to map distances
If the scale is 1 : x
1 Map distance
—– = ————————-
X Ground distance
Map distance = Ground distance / X
Example
A particular map shows a scale of 1 cm : 5 km, What is the
map distance (in cm) if the ground distance is 14 km?
Unify the scale: 1 cm : 5 km means the scale is 1 cm : 500,000 cm or
1:500,000
Map distance = 14 km * 100,000 / 500,000 = 2.8 cm

Graphic (Bar) Scale

Converting Between Scale Types

Finding area measurement

Finding Area Measurement from a Map and Scale

• Area must be expressed in square units, such as cm², km², etc.
• Squared conversion factors must be used when calculating the area from map measurements.

Example:
A rectangular property measures 3 cm × 4 cm on a map with a scale of 1:24,000.
The area in ground units is:3×24,000×4×24,000=12×24,00023 \times 24,000 \times 4 \times 24,000 = 12 \times 24,000^23×24,000×4×24,000=12×24,0002 12×576,000,000=6,912,000,000 cm2=691,200 m212 \times 576,000,000 = 6,912,000,000 \text{ cm}^2 = 691,200 \text{ m}^212×576,000,000=6,912,000,000 cm2=691,200 m2

Large Scale vs. Small Scale Maps

• Small-scale maps cover large areas with less detail.
• Large-scale maps cover smaller areas with more detail.

For example, in a set of maps:

• A 1:100,000 map has the smallest scale.
• A 1:25,000 map has the largest scale.

Angle Measurement Systems

The following angle unit systems will be discussed in this
chapter.

The circumference of a circle is 2π times the radius r. Therefore, the number of arcs of length r that fit along the circumference of a circle is 2π.

Write the measure of the indicated angles in radians as a
ratio of   (all are regular polygons)

Sexagesimal System

This is the most common system for measuring angles. In this system:

• A full circle is divided into 360 degrees (°).
• 1 degree (°) = 60 minutes (‘).
• 1 minute (‘) = 60 seconds (“).
• Example: 36 degrees, 24 minutes, and 52 seconds is written as 36° 24′ 52″.

Conversion Rules

1. Converting Degrees-Minutes-Seconds (DMS) to Decimal Degrees (DD):

Formula:Decimal Degrees=Degrees+Minutes60+Seconds3600\text{Decimal Degrees} = \text{Degrees} + \frac{\text{Minutes}}{60} + \frac{\text{Seconds}}{3600}Decimal Degrees=Degrees+60Minutes​+3600Seconds​

Example: Convert 37° 45′ 17″ to decimal degrees:37+4560+173600=37+0.75+0.00472222=37.75472222°37 + \frac{45}{60} + \frac{17}{3600} = 37 + 0.75 + 0.00472222 = 37.75472222°37+6045​+360017​=37+0.75+0.00472222=37.75472222°

2. Converting Decimal Degrees (DD) to Degrees-Minutes-Seconds (DMS):

Steps:

• Degrees: Integer part of the decimal degree.
• Minutes: Multiply the decimal part by 60, take the integer.
• Seconds: Multiply the remaining decimal by 60.

Example: Convert 37.75472222° to DMS:

• Degrees = 37°
• Minutes = (37.75472222 – 37) × 60 = 45.28333333 → 45′
• Seconds = (45.28333333 – 45) × 60 = 17″

Result: 37.75472222°=37°45′17″\text{Result: } 37.75472222° = 37° 45′ 17″Result: 37.75472222°=37°45′17″

In DMS format, only seconds can have a decimal portion.

Centesimal System

• Also known as the metric system of angular measurement.
• A full circle is divided into 400 grads (g).
• Each grad is subdivided into:
  • Centigrads (c) – equivalent to centesimal minutes.
  • Centicentigrads (cc) – equivalent to centesimal seconds.

Example: Convert 36.123450 g to grads, centigrads, and centicentigrads:36.123450g=36g12c34.5cc36.123450 g = 36 g 12 c 34.5 cc36.123450g=36g12c34.5cc

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Mil System

• Primarily used in military and artillery calculations.
• A full circle is divided into 6400 mils (mil).

Conversion between Measures of Angles

The presented systems are:

Radians and degrees

• 2  rad = 360°
•   rad = 180°
  Dividing both sides   gives
  1 rad = 180°/  = 57.2957795…°
  Or dividing both sides of previous equation by 180°
  1° =  /180° rad = 0.01745329… rad

Radians and grads

• 2  rad = 400 g
•   rad = 200 g
  Dividing both sides   gives
  1 rad = 200 g /  = 63.6619772… g
  Or dividing both sides of previous equation by 200 g
  1 g =  /200 g rad = 0.01570796326… rad

Angles in Surveying

In surveying, angles are measured between a line and a reference plane or a reference line to determine direction, elevation, or position.

Components of an Angle
In surveying, three components need to be known to describe an
angle. These parts are:

1. Reference or starting line.
2. Sense of turning, and
3. Magnitude (value of the angle).
   For surveying works three types of angles are
   measured, mainly: horizontal, vertical, and zenith
   angles.

Horizontal Angles
Reference Line: Horizontal directions use a meridian as the reference
line.

• Meridians are lines, on the mean surface of the earth, that join the
  north and south poles.

Horizontal Angles: Reference Line
For plane surveying, a rectangular coordinate system is defined as follows:

• origin on the point,
• the Y-axis pointing to north direction (the meridian
  passing through the point),
• X-axis pointing to east direction, while
• Z-axis points towards the up
  direction which is the direction of
  gravity at this point.
  This coordinate system is called
  Northing-Easting-Up instead of X-Y-Z
  and is abbreviated as ENU.
  The North-South axis of the ENU system
  (or the meridian) is the reference for
  horizontal directions which are measured
  on the EN horizontal plane.

Horizontal Angles: Summary

• Are measured in the horizontal plane
• The left-most line is called Backsight
• The right-most from the backsight is called Foresight
• The intersection of the two lines is usually the point at which an instrument
  is set to measure the angle – it is called the Station.

Horizontal Angles: Azimuth
It represents the orientation (direction) of objects in the horizontal
plane and is measured clockwise from the north (meridian) and must
be in the range: 0 <   < 360 .

Horizontal Angles: Back azimuth of a Line
Every line has a forward and a backward azimuths;   12 and   21
respectively. The back azimuth of a line 1-2 is the azimuth of line 2-1.

Horizontal Angles: Back azimuth of a Line
The back azimuth of a line may be found by either of two ways:
Adding 180° to the azimuth of the line if the azimuth is less than 180°
or by subtracting 180° from the azimuth of the line if the azimuth is
greater than 180°. Referring to following figure:

• If   12 < 180°, then   21 =   12 + 180°,
• If   12 > 180°, then   21 =   12 – 180°,
  Example
• for azimuth 40°, the back azimuth = 40° + 180° = 220°
• for azimuth 140°, the back azimuth = 140° + 180° = 320°
• for azimuth 220°, the back azimuth = 220° – 180° = 40°
• for azimuth 320°, the back azimuth = 320° – 180° = 140°

Horizontal Angles: Bearing

• The bearing of a line is the horizontal acute angle between the
  meridian and the line with values in the range 0 <   < 90 .
• Because the bearing of a line cannot exceed 90°, the full horizontal
  circle is divided into four quadrants;
  northeast (NE), southeast (SE), southwest (SW), and northwest (NW).
• Bearing is measured clockwise or counterclockwise from the north
  or south end of the meridian and is always accompanied by letters
  that locate the quadrant in which the line falls (NE, NW, SE, or SW).

Horizontal Angles: Bearing

• The bearing of a line is the horizontal acute angle between the
  meridian and the line with values in the range 0 <   < 90 .
• Because the bearing of a line cannot exceed 90°, the full horizontal
  circle is divided into four quadrants;
  northeast (NE), southeast (SE), southwest (SW), and northwest (NW).
• Bearing is measured clockwise or counterclockwise from the north
  or south end of the meridian and is always accompanied by letters
  that locate the quadrant in which the line falls (NE, NW, SE, or SW).

Horizontal Angles: Back Bearing
To reverse a bearing … Reverse the direction letters.

Horizontal Angles: Converting Bearings to Azimuths
The conversion is quadrant dependent as in the following table:

Horizontal Angles: Converting Bearings to Azimuths
Example
Convert the following bearings to azimuths and back azimuths.

Vertical Angles

• Plumb Line: A vertical line extending from any point on Earth toward the Earth’s center, following the direction of gravity at the observer’s location.
• Zenith: The point on the celestial sphere directly above the observer, where the plumb line intersects.
• Nadir: The point on the celestial sphere directly below the observer, opposite to the zenith.

Vertical Angles

• A vertical angle is defined as the angle between the horizontal
  plane and the line of object as seen in the figure below.
• A zero vertical angle means line AB is in the horizontal plane

Vertical Angles

• Above Horizon: 0° to +90°
• Below Horizon: 0° to -90°
• Angle of Elevation: The vertical angle measured upward from the horizontal. It is positive.
• Angle of Depression: The vertical angle measured downward from the horizontal. It is negative.

Zenith Angles

• Measured from the up direction downward to the line,
• A zero zenith angle means vertical direction (pointing upward),
• Range: 0  to 180 
• 0  to 90 : above horizon
• 90  to 180 : below horizon

Measurements and Errors

Measurements in Surveying

Definition of a Measurement

A measurement is an observation made to determine an unknown quantity.

Characteristics of Measurements

• No measurement is exact.
• Every measurement contains errors.
• The true value of a measurement is never known.
• The exact size of the errors present is always unknown.

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Types of Measurements

Direct Measurements

• Made by directly applying an instrument to the unknown quantity and observing its value.
• Example:
  • Measuring the distance between two points using a graduated tape.
  • Measuring an angle using a theodolite or total station.

Indirect Measurements

• Used when direct measurement is not possible or practical.
• The quantity is determined mathematically using direct measurements.
• Example:
  • Measuring line lengths and angles to compute station coordinates.

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Sources of Errors

1. Instrument Errors

• Caused by imperfections in the construction or adjustment of measuring instruments.
• Example: A tape with unevenly spaced divisions.

2. Environmental Errors

• Caused by changes in atmospheric conditions.
• Example: Temperature variations affecting tape length.

3. Human Errors

• Caused by limitations in human senses and conditions affecting performance.
• Example: Difficulty in reading a micrometer accurately.

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Types of Errors

1. Blunders (Gross Errors)

• Caused by human mistakes such as incorrect readings or recording errors.
• Example: Writing 27.55 instead of 25.75.

2. Systematic Errors

• Follow a predictable law and can be corrected.
• Example: Using a tape that is 10 cm short.

3. Random Errors

• Cannot be systematically predicted or corrected.
• Example: Small errors in reading a graduated scale.

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Precision vs. Accuracy

Precision

• Degree of consistency between repeated measurements.
• Depends on:
  • Instrument sensitivity
  • Observer’s skill

Accuracy

• Closeness of a measurement to the true value (which is never exactly known).
• Accuracy is always an estimate since the true value is unknown.

Accuracy of Angle Measurement

Importance of Centering & Leveling

• Theodolites and total stations must be properly centered over a point and leveled.
• This ensures accurate horizontal and vertical angle readings.

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Relationship Between Angular & Linear Precision

The formula for arc length, radius, and angle:S=r⋅θS = r \cdot \thetaS=r⋅θ

With an angular error dθd\thetadθ, the linear equivalent is:dS=r⋅dθdS = r \cdot d\thetadS=r⋅dθ

Examples of Angular Error Conversions

At a 100 m sighting distance:

• 20″ (arc seconds) → 10 mm error
• 10″ → 5 mm error
• 5″ → 2.5 mm error
• 1″ → 0.5 mm error

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Choosing the Right Instrument

• If a 5 mm tolerance is required at 100 m, a 10″ theodolite or total station is necessary.
• If the maximum distance is 50 m, a 20″ instrument is sufficient.
• High-precision instruments (5″, 1″) are only needed for control networks or high-accuracy projects like:
  • Dams
  • Nuclear Power Plants

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Understanding Instrument Accuracy

• Minimum reading ≠ Accuracy
• Always check the manufacturer’s specifications for actual accuracy, not just resolution.