Basic Principles of Surveying and Mathematics

AMRC 2012 MODULE 1 Basic Principles of Surveying and Mathematics CONTENTS Overview... 1-1 Objectives... 1-1 Procedures... 1-1 1.1 Surveying Defined... 1-3 1.2 Types of Surveys... 1-5 1.3 Precision and Accuracy... 1-7 1.4 Mathematics in Surveying... 1-9 1.5 Angles... 1-11 1.5.1 Addition of Angular Measurements... 1-11 1.5.2 Subtraction of Angular Measurements... 1-12 1.5.3 Multiplication of Angular Measurements... 1-14 1.5.4 Division of Angular Measurements... 1-14 1.5.5 Decimal Angles... 1-15 1.6 Bearings... 1-17 1.6.1 Azimuth Method... 1-17 1.6.2 Quadrant Method... 1-19 1.7 Basic Elements of Trigonometry... 1-23 1.8 Self-Test... 1-29 WPC #27362 07/09

Module 1 Basic Principles of Surveying and Mathematics Overview Module 1 provides an introduction to the general terminology used for surveying work including categories of surveys, basic reference systems, and a discussion of precision and accuracy. Surveying, and the interpretation of survey work, is fundamentally an exercise of the application of basic mathematical and geometric principles. This module explores the basic principles and calculations which will apply to the work in subsequent modules of this course. Objectives Upon completion of this module, the student will be able to: define surveying and discuss its relevance to land development and construction identify the types of surveying typically employed in the industry add, subtract, multiply, and divide angles expressed in degrees, minutes, and seconds, as well as degree and decimals of a degree determine the trigonometric functions of an angle, using function tables or a scientific calculator calculate the lengths of the sides of a triangle, using available angles and distances discuss azimuth and quadrant bearings calculate the azimuth and quadrant bearings of lines given the deflection angle between lines. Procedures Study the module materials and make notes as required. Perform the self-test on these principles and review the course materials in such a manner as to be able to successfully complete similar questions upon examination. WPC #27362 07/09 AMRC 2012 1-1

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SECTION 1.1 Surveying Defined Surveying is the science and art of making measurements for the purpose of determining the relative positions of points on, above, or below the earth s surface. Surveying operations extend over land and water areas and space. Surveys which cover only a small area where curvature of the earth need not be taken into account are called plane surveys. Those surveys covering large areas where curvature must be taken into account are called geodetic surveys. There are two phases of operations in surveying, the field survey phase and the office reduction phase. The field survey phase involves the collection and recording of field measurement or layout data during the field survey at the job site. The office phase involves the reduction, analysis, and adjustment of the field data back at the survey office to ensure it meets the requirements of the job specifications. The data is then presented in either a report form or more commonly represented in the form of a map or a plan. Maps and plans are available in both digital and hard copy format. WPC #27362 07/09 AMRC 2012 1-3

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SECTION 1.2 Types of Surveys Surveys are varied and while clear distinction cannot be made, generally, several categories can be established to cover most survey operations. Keep in mind that different agencies may utilize different names for certain survey types always ensure that you and anyone you are discussing specific survey works with have common definitions in mind. Aerial Surveys are operations for the procurement of aerial photography and the preparation of the display of information from such photography through digital means or maps and mosaics. Construction Surveys are surveys conducted for the layout of engineering works such as buildings, sewers, water lines, and other infrastructure. Geodetic Surveys entail surveys where the shape of the earth must be taken into account. An example of a geodetic survey would be the establishment of the survey control for the Alex Fraser Bridge. Geographical Information System (GIS) Surveys are used to compare geographical attributes such as size, location, area, length, etc. of surveyed features in order to perform an analysis on relationships. An example would be a GIS query on the listing of private property owners within 1 kilometre of a proposed highway right of way of 200 metres width or summarize the amount of money spent on maintenance for a particular length of road. Global Positioning Surveys (GPS) are surveys which utilize the Navstar or Glasnost satellite positioning systems. Satellites orbiting the earth at about 22 000 km send out radio signals to surveyors with receivers that calculate their position using trigonometric principles ranging from 100 metres accuracy to sub-centimetre accuracy anywhere on the earth. Hydrographic Surveys are operations necessary to map shorelines of bodies of water; to chart the bottom of streams, lakes, harbours, and coastal waters. Land Surveying (cadastral) is the determination of land or property boundaries and corners and the preparation of legal documentation such as the plans of such land or property in compliance with federal or provincial laws. Mine Surveys are made to determine the position of underground and surface works of mines. WPC #27362 07/09 AMRC 2012 1-5

Plane Surveys define most of the survey work carried out on a daily basis. The shape of the earth is not taken into account during a plane survey rather the earth is considered flat. This type of survey is adequate for surveys covering a small area, for example, a legal survey or a site survey. Remote Sensing is a type of survey using satellites or specially equipped aircraft to obtain an image of the ground using electromagnetic energy. Sensors aboard the aircraft or satellite filter out specific electromagnetic wavelengths resulting in an image that can be used to analyse changes such as the effect of lava flow from a volcano, changes in the ocean temperature, or forest fire hot spots. Route Surveys are made for the purpose of engineering projects associated with transportation and communication. They include railways, highways, forest roads, mining access roads, pipelines, and transmission lines. Site Surveys are typically made to obtain data for the preparation of plans or maps showing the existing features and structures of an area or site, typically so that new features or structures may be planned. Topographic Surveys are those surveys made to obtain data for the preparation of topographic maps or plans where the presentation of landform information with graphic representation of elevation is required. 1-6 AMRC 2012 WPC #27362 07/09

SECTION 1.3 Precision and Accuracy Precision refers to the repeatability of a measurement. Measurements can be precise without being accurate and vice versa. The precision of a survey refers to the care and refinement to which a survey or survey measurement is made. Accuracy refers to the difference between the final measured value of a quantity and its true value. It does follow that higher precision will yield results of higher accuracy. The two terms, however, have different meanings. Figure 1.1 Figure 1.1 illustrates the difference between accuracy and precision. The centre of the target represents the true value for a set of eight observations. The first target shows an uneven pattern of observations scattered around the true value. The average of the observations is close to the true value. These observations are accurate but not precise. The second illustration shows a small cluster of observations at the edge of the target. The observations are precise but not accurate. The mean of these observations is not close to the true value. The observations in the third illustration are tightly clustered around the true value. The mean is close to the true value. These observations are accurate and precise. 1 1 Davis, Raymond E., Francis S. Foote, James M. Anderson, & Edward M. Mikhail. (1981). Surveying: Theory and Practice. McGraw Hill WPC #27362 07/09 AMRC 2012 1-7

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SECTION 1.4 Mathematics in Surveying Survey work, and the interpretation of survey data, is fundamentally an exercise in the application of basic mathematical and geometric principles. To understand survey work, including data collection, legal plan information, and site layout functions, a clear understanding of angles and their measurement is needed, as well as line bearings and direction. This module covers the basic calculations involved in the use of angle data and distance measurements. WPC #27362 07/09 AMRC 2012 1-9

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SECTION 1.5 Angles Angles are usually measured in degrees, minutes, and seconds. There are 360 degrees to a full circle. A degree is divided into 60 minutes and a minute is divided into 60 seconds. Abbreviations for twenty degrees is written as 20. Thirty minutes is written as 30' and twenty seconds as 20". Addition and subtraction of angles is slightly more difficult than numbers and decimals because they are not in the decimal system. 1.5.1 Addition of Angular Measurements Example Add 45 31' 45 31' 14 10' 14 10' 120 20' 120 20' 179 61' 1 1 01' Answer 180 01' 1. Add up minute column. 2. Convert minute column to degrees and minutes if the total is 60 minutes or greater. 3. Carry degrees to degree column. 4. Add up the degree column. WPC #27362 07/09 AMRC 2012 1-11

Example Add 22 42' 45" 22 42' 45" 15 28' 32" 15 28' 32" 32 19' 56" 32 19' 56" 69 89' 133" 2' 2' 13" 91' 13" 1 1 31' Answer 70 31' 13" 1. Add up seconds column. 2. Convert seconds column to minutes and seconds if the total is 60 seconds or greater. 3. Carry minutes to minute column. 4. Add up minute column. 5. Convert minute column to degrees and minutes if the total is 60 minutes or greater. 6. Carry degrees to degree column. 7. Add up the degree column. 1.5.2 Subtraction of Angular Measurements Example Subtract 140 55' 30 45' Answer 110 10' 1. Subtract the minute column. 2. Subtract the degree column. 1-12 AMRC 2012 WPC #27362 07/09

Example 180 00' 179 60' Subtract 30 45' 30 45' Answer 149 15' 1. Subtract the minute column. Borrow 1 or 60 minutes from the degree column if the value of the minuend is less than the subtrahend. 2. Subtract the degree column. Example 188 15' 187 75' Subtract 135 27' 135 27' Answer 52 48' Example 140 55' 42" Subtract 65 25' 32" Answer 75 30' 10" 1. Subtract the seconds column. 2. Subtract the minute column. 3. Subtract the degree column. Example 26 27' 38" 26 26' 98" Subtract 13 13' 47" 13 13' 47" Answer 13 13' 51" 1. Subtract the seconds column. Borrow 1' or 60 seconds from the minute column if the value of the minuend is less than the subtrahend. 2. Subtract the minute column. 3. Subtract the degree column. WPC #27362 07/09 AMRC 2012 1-13

Example 42 11' 15" 41 70' 75" 33 54' 43" 33 54' 43" Answer 8 16' 32" 1.5.3 Multiplication of Angular Measurements Multiply 27 47' 38" by 5 1. 38" 5 = 190" 03' 10" 2. 47' 5 = 235' 3 55' 3. 27 5 = 135 135 4. add columns 138 58' 10" Answer 1. Multiply seconds by multiplier and convert to minutes and seconds. 2. Multiply minutes by multiplier and convert to minutes and degrees. 3. Multiply degrees by multiplier. 4. Add up seconds, minute, and degree columns. 1.5.4 Division of Angular Measurements Divide 310 13' 12" by 6 1. 310 6 = 51 and 4 remainder (51 6 = 306) & (310 306 = 4) 4 ( 60) = 240' 2. 240' + 13' = 253' 253' 6 = 41' and 1' remainder 1' = 60" 3. 60" + 12" = 72" 72" 6 = 12" Answer 51 42' 12" 1-14 AMRC 2012 WPC #27362 07/09

1. Divide degrees by divisor and convert remainder to minutes. 2. Add remainder minutes to minutes and divide by divisor. Convert remainder minutes to seconds. 3. Add remainder seconds to seconds and divide by divisor. 4. Collect terms. 1.5.5 Decimal Angles An alternative to using degrees, minutes, and seconds throughout the calculation requires reduction to decimal values first, then addition, subtraction, multiplication, and division can be completed and the decimal answer converted to degrees, minutes, and seconds. For example, we would recalculate the division problem in 1.5.4 as follows. Divide 310 13' 12" by 6 1. Convert seconds to decimal minutes. 12" 60 = 0.2' Add to minutes 13' + 0.2' = 13.2' 2. Convert minutes to decimal degrees. 13.2' 60 = 0.22 Add to degrees 310 + 0.22 = 310.22 3. Divide by 6 310.22 6 = 51.7033 4. Convert answer to degrees, minutes, and seconds. 51 (degrees known from step #3). 0.7033 60 = 42.198' (42 minutes determined) 0.198' 60 = 11.88" (round to 12 seconds) Answer 51 42' 12" WPC #27362 07/09 AMRC 2012 1-15

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SECTION 1.6 Bearings Bearings or angular directional measurements are used to describe the direction of lines so that they can be retraced or drawn on plans. The bearing of a line is obtained by referring to previous surveys, by astronomic observations, or by magnetic compass. The direction of a line is defined by either one of two methods: azimuth or quadrant. 1.6.1 Azimuth Method The full circle is divided into 360 in a clockwise direction. North is 0 or 360, East is 90, South is 180, and West is 270. Figure 1.2 Azimuths are angles in a clockwise direction, starting from north. For example, if a line has an azimuth of 200, then the clockwise angle from north would be 200. Figure 1.3 illustrates how azimuth bearings are calculated when measuring the deflected angles of a series of lines. WPC #27362 07/09 AMRC 2012 1-17

Figure 1.3 Angle of Line AB right of north is 60E 60E Azimuth of Line AB is 60E Deflected angle (right) at B is 65E + 65E RT Azimuth of Line BC is 125E 125E Deflected angle (left) at C is 42E! 42E LT Azimuth of Line CD is 83E 83E Note: When computing azimuth: ADD angles deflected to the RIGHT (clockwise angles) SUBTRACT angles deflected to the LEFT (counterclockwise angles). It should also be noted that the bearing of any line will depend on the direction you are viewing the line from in other words, line CD in the previous example had an azimuth bearing of 83E when viewed from C to D, but the same line when viewed from D to C (line DC) has an azimuth 180E different, namely 263E. 1-18 AMRC 2012 WPC #27362 07/09

1.6.2 Quadrant Method The full circle is divided into quarters. Each quarter is divided into 0 to 90 with the bearings for each quarter shown in Figure 1.4. Figure 1.4 The Quadrant Method or Quadrantal Method measures bearings from (0 ) zero degrees to 90 East or West of North and (0 ) zero degrees to 90 East or West of South. For example, bearings in the Quadrant Method would be noted such as N 25 00' E, N 35 30' W, S 65 10' E, and S 15 00' W. When computing bearings by the Quadrant Method a diagram or sketch will indicate whether an angle is to be added or subtracted to obtain the correct direction. For example, see Figure 1.5. If a line has a bearing of S 70 12' E and a line deflects left from it 28 52', then its bearing is S 99 04' E. In the Quadrant Method, bearings do not exceed over 90 ; therefore, we must convert S 99 04' E to the correct quadrant by subtracting 99 04' from 180 to arrive at the bearing of N 80 56' E. WPC #27362 07/09 AMRC 2012 1-19

Figure 1.5 Bearing of Line AB is S 70 12' E Deflected angle at B is 28 52' Angle from South to Line BC is 99 04' Bearing of Line BC is 180 00' 99 04' = N 80 56' E Again, it must be noted that the bearing of the line depends on the direction in which the line is being viewed, which is a vital consideration in any survey calculations. The bearing for line BC in the above example is N 80 56' E when considered from B to C, or if your survey instrument is set up on B and sighting to C. However, if the line is being viewed from C to B, or if the survey instrument was set up on C and sighting to B, the bearing of the line would be 180 different, namely S 80 56' W. Additional examples in Figure 1.6 indicate the procedure used when the deflected angle changes quadrants. 1-20 AMRC 2012 WPC #27362 07/09

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Another method which may be used to calculate bearings in the Quadrant Method is as follows: 1. Convert the quadrant bearing to an azimuth bearing, i.e., S 70 12' E = 180 00' 70 12' = 109 48' Azimuth 2. Use Azimuth Rule, that is, add angles deflected right and subtract angles deflected left, i.e., 109 48' 28 52' = 80 56' 3. Convert azimuth bearing to quadrant bearing, i.e., 80 56' azimuth = N 80 56' E Quadrant If the azimuth bearing was 232 10', the quadrant bearing would be: 232 10' 180 00' = S 52 10' W. Angles between 0 and 90 in NE quadrant. Angles between 90 and 180 in SE quadrant. Angles between 180 and 270 in SW quadrant. Angles between 270 and 360 in NW quadrant. 1-22 AMRC 2012 WPC #27362 07/09

SECTION 1.7 Basic Elements of Trigonometry In order to convert slope distances to horizontal distances, or to calculate coordinates from angles, bearings, and distances, it is necessary to know a few basic elements of trigonometry. Trigonometry is defined as the study of the properties of triangles. Very often in the study of trigonometry and algebra, Greek symbols, instead of our alphabetic symbols, are used to denote various angles or quantities. Some of the more common symbols and names which you may encounter are as follows: " Alpha $ Beta ( Gamma ) Delta 2 Theta 8 Lambda : Mu B Pi N Phi T Omega The Right Triangle " is the angle formed by the two lines AB and AC $ is the angle formed by the two lines BA and BC ( is the angle formed by the two lines CB and CA; it is also a right angle or an angle of 90E a b c is the length of the line B to C is the length of the line A to C is the length of the line A to B The sum of the angles " + $ + ( = 180E WPC #27362 07/09 AMRC 2012 1-23

but ( = 90E therefore " + $ = 90E or " = 90E! $ and $ = 90E! " a is known as the side opposite to the angle " b is known as the side adjacent to the angle " c is known as the hypotenuse; it is the side opposite the right angle We can also see that: a is known as the side adjacent to the angle $ b is known as the side opposite to the angle $ c is known as the hypotenuse; it is still the side opposite the right angle. There is a definite relationship between the side and angles of a right angle triangle. First, let us review Pythagoras theory of solving right angle triangles. Remember the rule stated that in a right angle triangle the square of the side across from the right angle (hypotenuse) is equal to the sum of the squares of the other two sides: a 2 + b 2 = c 2. Then if a = 1 b = 3 (1) 2 + 3 2 = c 2 (1) 2 + 3 = c 2 4 = c 2 2 = c 1-24 AMRC 2012 WPC #27362 07/09

No matter how much we increase the length of a and c, we note that the angle remains the same. There is a definite relationship between the sides of the right triangle and the angles. If a increases 100 times, then c must also increase 100 times and b must also increase 100 times. Mathematicians long ago gave names to identify the following relationships. side opposite Sine is the ratio: hypotenuse It is usually abbreviated as sin side adjacent Cosine is the ratio: hypotenuse It is usually abbreviated as cos side opposite Tangent is the ratio: side adjacent It is usually abbreviated as tan hypotenuse Cosecant is the ratio: side opposite It is usually abbreviated as csc hypotenuse Secant is the ratio: side adjacent It is usually abbreviated as sec side adjacent Cotangent is the ratio: side opposite It is usually abbreviated as cot WPC #27362 07/09 AMRC 2012 1-25

Applying these ratios the following formulas were developed: a sin c c csc a cos b c sec c b tan a b cot b a and sin b c csc c b a cos c c sec a tan b a cot a b Reciprocals related to ratios: a csc sin 1 csc cos 1 sin csc 1 cos sec cot 1 tan 1 tan cot Tables of figures have been calculated for the relationship for the sides of a right triangle for each specific angle. These are known as natural functions of angles. Although hand calculators are used for computations by most people today, tables are still kept in many survey offices. If you again consider the right triangle: from the formula " = a b then sin " = 1 2 or sin " = 0.5. Now refer to tables or use a calculator and you will see that: sin 30E = 0.50000 therefore, in this case " = 30E 1-26 AMRC 2012 WPC #27362 07/09

What is the length of c if a = 25 and α = 30? sin a c a c sin c a csc 25 c 50 0.5 or c c (251) (2.0) 50 What is the length of b if a = 25 and α = 30? a tan b a 1 tan b or b (a) but b a cotan tan tan 25 b b (25)(1.73205) 0.57735 b 43.3 b 43/ 3 WPC #27362 07/09 AMRC 2012 1-27

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SECTION 1.8 Self-Test Provided below is a self-test for this module. Perform the test and compare your answers with those given in your student guide. If you have difficulty with the self-test, review the course materials and make notes in order to be able to successfully complete similar questions upon examination. 1. Describe the difference between plane and geodetic surveys. 2. List the five types of surveys which may be directly related to land subdivision and development projects. Define each briefly. 3. Add: A. 98E 18' B. 13E 37' C. 36E 36' D. 74E 19' 57" E. 3E 59' 08" 73E 33' 25E 58' 7E 59' 23E 45' 45" 57E 23' 32" 16E 03' 24E 13' 18E 19' 18E 15' 12" 12E 51' 49" 10E 10' 25E 01' 5E 03' 31" 108E 15' 25" 4. Subtract: A. 88E 54' B. 24E 12' C. 67E 28' D. 90E 00' 00" E. 56E 22' 17" 73E 31' 8E 56' 53E 37' 75E 54' 29" 13E 43' 21" 5. Multiply: A. 23E18' 47" by 4 B. 16E19' by 6 C. 63E03' 39" by 3 Compute C using degrees and decimals. 6. Divide: A. 143E41' 52" by 4 B. 139E48' by 6 C. 155E07' 39" by 3 Compute C using degrees and decimals. WPC #27362 07/09 AMRC 2012 1-29

7. What are the azimuth bearings of the lines A to B; B to C; C to D in the following diagram? 8. Compute the bearings by the Quadrant Method for the lines A to B; B to C; C to D; D to E shown in the following diagram. 1-30 AMRC 2012 WPC #27362 07/09

9. Compute the bearings by the Quadrant Method for the lines B to C; C to D; D to E; E to F; F to G; G to H; H to I; I to A shown in the following diagram. Note the bearing A to B is N 13E12" W. WPC #27362 07/09 AMRC 2012 1-31

10. A. A survey crew has measured the following distance on a slope and the slope angle as well. Determine the horizontal distance as well as the elevation difference between the top and the bottom points of the slope. B. Repeat the above exercise for a slope distance of 19.10 metres and an angle of 23E45'. 11. What check of the calculated distances could you use? 1-32 AMRC 2012 WPC #27362 07/09