102_2015_0_b - unisa

SRC2601/102/0/2015

Tutorial letter 102/0/2015 Surveying I (THEORY) SCR2601 Year module

Department of Civil and Chemical Engineering

IMPORTANT INFORMATION:

This tutorial letter contains important information about your module.

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These notes were drafted by JN Wiesner in January 2015 and is intended only as supplementary notes to

cover and clarify topics in the official syllabus omitted in the current study guide

As lecturer for the Theory and Practical of the subject Surveying 2, I would like to welcome all those who

have registered for this subject for 2015. I would like to extend my hand as far as possible to be always

available and being of help to you all. You must also know that I am just as demanding from you in the

sense that I expect just as mush from you as I expect from myself. I expect of you to become thinking

constructive persons. Think before you ask a question, be part of the solution not part of the problem. Think

about possible solutions and come with proposals rather than just throwing questions in the bush, hoping that

two or more solutions and answers will just pop out. I do not tolerate plain laziness, if it is clear from your

question or problem that you did not read up or understood the underlying foundation theory, I will rather

direct you to study or read up the relevant theory than giving the answer to you on a platter!

I set high standards for myself and expect nothing less from you! I will actively use MyUnisa discussion

forum and will soon open a forum for every topic of the syllabus for us to actively communicate.

I demand active participation from all of you so that we can learn from each other, but please do not use this

forum just as a private chat- or moaning room. Let us use it for any related questions, problems and even for

new discoveries and sharing survey related civil engineering practical experiences with each other.

Surveying is a service entity to the Civil Engineering field and we do not want to make surveyors from you

as civil Engineering Technicians! The intention is to make you active and able users of survey so that you

will be able to understand what the surveyor have to do to supply control or establish structures on the

ground so that we can effectively communicate and evaluate the service we are getting out of the marriage.

The full syllabus of Surveying 1 is part and parcel fully part of the survey 2 syllabus and therefore you

cannot leave it alone yet (pun - the job is not done until all the paperwork is finished!) Studying surveying is just like building a big jigsaw puzzle. You have to complete it as a whole to be able to see the bigger

picture!

In Surveying 1 you hopefully looked at each little piece of the puzzle, turning it around, up and down, left an

right and even inside out, to be able to see how it operates within the whole and in which places it may fit in

and could be applied, but eventually find the spot where it fits best and correctly tying all together.

In Surveying 1 you have mostly done calculations on the monkey see monkey do principle. You were not assessed on the absolute correctness and or accuracy of your answers because it mostly revolved around

mastering the formulae and applying it in a standard form of calculation.

THE STANDARD FORMS OF CALCULATION IS JUST AS IMPORTANT TO THE SURVEYING

INDUSTRY AS STANDARD BOOK KEEPING IS TO THE ACCOUNTING FRATERNITY.

Every other surveyor or user of survey data, such as civil engineers, must directly be able to identify

the calculated data and be able to assess the usability and correctness thereof.

Surveying 1 form the foundation blocks for the full picture and the extent of how good you have laid those

foundation blocks will eventually influence your ability to understand the applied applications that we will

be dealing with in Survey 2. If the foundation work is not fully grasped and understood some pieces may be

in the wrong places, meaning that you will have to adapt your picture, otherwise it will not all tie and fit in

and will most probably fall apart so that you will never see the full picture. The moral of the story is that you

have to be adaptable, open for new ideas and applications and most important avoid your own Resistance to Change (RC) factor!

NB. Please all note:

(i) That it is really a waste of time in modern days to still note down the Sin and cosine functions in calculations as set out in the new Surveying I study guide by HM Labuschagne! Stick to the standard forms of calculations as set out in the according to my knowledge the official Surveying I

study guide!! (Green book by JIP Bisschoff)

(ii) The terms CWA and ACWA as used in the survey 2 study guide is commonly used in Mine surveying but never in Engineering survey. In your survey 2 study guide it is even used wrongly

where it sometimes should have been named Circle Left (L) and Circle Right (R). Please ignore and or scratch out those wrong terms

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(iii) It is very important that you practice neat handwriting because we cannot afford any miss understanding in Civil Engineering! Any miss understanding may lead to injuries and or even deaths

to the public we serve in designing and constructing civil engineering structures. This serves as a very

urgent warning to you that in this course (and hopefully in all the others) you will not earn any

marks for unreadable writing (to me as lecturer and Assessor) of text and or figures. Figures must be

formed correctly so that everybody understands the same value, not making for example a 4 that

could be mistaken for a + or even a 1! Like this: 1;2;3;4;5;6;7;8;9;0 and nothing else! Do not try to write too small or too big, the ideal size is a 12 point size, same as the size of these notes!

(iv) Any observed and or calculated survey data must be usable in practice and that means that in survey 2 you will be assessed on the correctness and accuracy of observed data and the answers obtained in

the calculated data. You will be heavily penalised if you do not offer the calculations in the standard

forms of the calculations and where there are signs (+ or -) applicable it is worthless if not shown.

Just as a chicken hops around in any direction when its head is chopped off, we are like headless

chickens without signs to height differences, coordinates and delta Y or Delta X values!

(v) Sighs in front of these values does not say the value is positive or negative, but it indicates direction, telling us to move up or down, left or right, North or South and East or West.

(vi) Always calculate to one decimal place more that the given decimal places of observed or calculated data, but not more than three (3) decimal places. Carry on with one decimal place more than given

right up to the end of the calculation and only then round off to the same amount of decimal places

of the given data. The reason for not more than 3 decimal places is the fact that we very seldom work

to sub-millimetre accuracies in civil Engineering

(vii) Should you round off intermittently as you carry on with the calculation you will accumulate small discrepancies to such an extent that the accumulated result may be too big to fall within the

required degree of accuracy as prescribed and or required.

(viii) Any constants used in calculations and or transformation of linear distances and coordinates must however always be used with as many decimal places possible but never less than six (6) decimal

places.

(ix) We always round down for any value between 1 and 4 while up for any value between 6 and 9. If the last digit is however on a 5 it is as close to the lower value as it is to the higher value and we have to

make an informed calculated decision. The rounding off rule in survey stipulates that if the last

digit is a five (5) we always round to the nearest even number.

Take the following cases as an example:

145,345 will be rounded off to 145,34 to two decimal places.

134,275 will be rounded off to 134,28 to two decimal places.

In this way you will note that in case 1 we throw away the 5, while in case two we add the 5, so that

according to the principle of progression of errors, we will add as many fives as we will throw away

fives in a series of involved combined calculations and therefore accumulate very little errors!

(x) In rounding off you may not use your calculator to do the rounding off for you because it is pre-programmed to always round up, which will cause this accumulation of errors. Always set your

calculator to display one more decimal place than given. You probably know that, depending on the

type and manufacturer of the calculator, they can display from six to thirteen decimal places, but

you may only look at one more decimal place than the given data. If you start at the back of a value

to say six decimal places rounding off to eventually three decimal places it boils down to the same

thing as to accumulate errors. Lets look at the following example: In the value 1,254546 you may actually only look at the 4

th decimal place to round of to 3 decimal

places and it will be rounded off as 1,254.

If you however start at the back at the 6th

decimal place it will become 1,25455 rounded to 5 places.

It will then become 1,2546 to 4 decimal places which will then be rounded off as 1,255 to 3 places.

You will note that this caused this answer (1,255) to be 1 mm higher than the correct value (1,254).

Good luck and may you truly benefit from your studies in this subject

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POSSIBLE EXAMINATION QUESTIONS FROM SURVEYING I

Revise all Surveying I theory and practical. SA-Co-ordinate system Bearings and back bearings Azimuth, also known as The True Direction of a Line at a point Orientation Observed and oriented directions

Joins and Polar theory

Plans and Maps Plotting accuracy Relation between the purpose, the scale and the accuracy of a plan Requirements for the layout -, orientation of the grid- and compilation of a plan.

Traverse Revision of Surveying I theory on traverse calculation and adjustment Reduction of the field book Reduction of the oriented directions in the fieldbook. Traverse Calculation Standard form Adjustment of the traverse by the Bowditch rule Acceptability of the calculated values according to the relevant regulations of the Survey Act

Tacheometry Revision of theory on the deduction of Tacheometric formulae and reduction of the fieldbook. Topographic surveying Rules for placing of traverse stations Spot heights Scales Spot height accuracies Contour intervals in relation to the scale of the plan

The Surveying telescope Lenses

Lens errors Chromatics aberration Spherical aberration

Focussing Parallax

The properties of a surveying telescope Definition of a surveying telescope. Resolving power Enlargement The clarity of the image The size of the field

The enlargement and resolving power of the telescope are two very important properties that have to be taken into account in the choice of purchase and use of a theodolite. Discuss!

Which one of the properties of a surveying telescope is the most important when triangulation observations are done? Motivate your answer in full and give the formulae to calculate this property. (5)

Levelling Reduction of the readings Rise and fall method Height of instrument method

Adjustment of the levelling line Applications Reasons for balancing of back and fore sights.

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State and discuss the important requirements that have to be complied with, when a very long line of pegs must be levelled, and a high degree of accuracy is required.

It is now time to invest in a good calculator. Know and understand how to correctly use your calculator,

otherwise there is no way that you will be able to do all the calculations in the expected time to finish an

assessment or examination paper.

Know how to use and address the available memories on your calculator, it can save a lot of time rather to re type the same values over and over.

Know how to set the number of decimal digits to be displayed Know how to use the programmed conversion keys from degrees to DMS; DMS to Deg; Deg

Radians; Rad. deg.; the pre-programmed keys for polars and joins, depending on the type of calculator [(P-R) (Polar Rectangular)]; [(R P) (Rectangular Polar)] or P; R.

Remember these functions are only available on the more expensive sophisticated calculators. It is pre-programmed keys and is not making it a programmable calculator, which is not permitted in some

subjects. These keys will save you a lot of calculation time!

An examination paper is mostly set out of 100 marks to be completed in 3 hours (180 minutes). Thus you have 1,8 minutes for each mark that a question counts and you could calculate how much time you

have available to spend on any question. Where a Join and a Polar both counting 6% in an examination

paper it should not take more than (6 x 1,8) = 10,8 minutes to do a Polar with its Join check as well as

the other way around a Join with its Polar check (correct and written down). You therefore have to

practice to do a Polar with its Join check as well as the other way around a Join with its Polar check

within eleven (11) minutes maximum.

This same principle of time calculation also applies to descriptive theory questions. Theory questions cannot always just be answered cryptically in a point wise fashion. Read the questions properly and do

what was asked. The marks allocated will specify how much time you have to spend on thinking,

constructing and writing the answer. If a question counts for instance 10 marks you have to spend

about 18 minutes otherwise you have not given enough scope to the answer!

Writing an examination paper is a skill you have to study, practice and apply: Define means to give a definition, if possible in your own words, showing your understanding of the

topic.

Describe means to give a, preferably tabularised, description of the topic in your own words showing insight and or understanding of the topic, with reasons for your statements.

Discuss means to identify and write notes with reasons for your statements on certain outstanding matters such as discrepancies, mistakes, errors, strong points, weak points or any other important

matter so that again it is clear that you understand and show insight on the topic.

Evaluate means to assess the calculated or observed data in terms of identified mistakes; errors, omitted data and accuracies achieved

In assignments you have the luxury that it can be answered open book, but this does not mean that you may just copy verbatim from my notes or any other book or source. Remember plagiarism is an

offence. I can and have read my own notes as well as probably most of the worthwhile textbooks on

surveying. I definitely do not want to read copies over and over again. I need to be able to assess your

own work to be able to see if you understood the work. It is no use to fool yourself thinking to get

good marks by copying but not understanding anything! It serves no use either to just put some

wording in another word- or fact order, it is still copying. I also ran each suspect quotation through a

copy detection program and if any commonalities are found it will identify it. So if you get a marked

assignment back with the remark B/S you must know it is worthless, probably copied work.

Remember that the internet is a very patient source of information. Anyone may post their own interpretation and or views on any topic, but if it is not reviewed by a trusted source it can be totally

miss-leading and wrong. So be very careful to believe and or relate to any info that you may find on

the internet or chat-rooms.

The real test of time comes when you eventually have to sit for an examination confronted with the same type of question on the same topic and you cannot answer it, when it really matters!

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SUGGESTED WORK PROGRAM To be successful in this subject, I honestly believe that a person should spend at least 5 notional hours per week

reading, exercising and studying the Surveying subject material. Where, when and how you fit it in is your choice!

Open Distance Learning (ODL) demands a lot of personal discipline from students and therefore you have to come

into a study rhythm as quickly as possible. I trust that you have already experienced this in Survey 1.

To be able to perform well in the practical you have to be fully prepared, knowing all the subject theory before

attending the practical. We are planning for 10practical weeks of 24 students each at the Florida campus, starting on

the 11th August 2014. Please plan your availability for practical as soon as possible and e-mail me your full particulars

and preferred week through, with any specific valid reasons why you can only attend a certain week /s, as quickly as

possible. I will schedule you in order of first come first serve, but take note that you will only get one chance if

you miss it you will have to come back next year. My e-mail address is [email protected]

I accept that everybody will start at the latest studying this subject in the first week of February 2015, which gives us

27 weeks to study the full syllabus before the first practical week starts. In full time studies they work on 15 weeks per

semester times 6 hours contact time per week = 15 x 6 = 90 hours contact time! If you spend 5 notional hour per week

times 27 week = 27 x 5 = 135 notional hours. Anything less will not be enough to master this subject! I therefore

suggest that you follow this suggested work plan so that everybody are able to post your questions, comments and

discussions at the same time on the discussion forum. Naturally there will be some late starters, but you will have to

accept that you will have to play catch up to get on par with the rest. The discussions on the forum will still be

available to you and you will have to consult the forum first before asking the same questions. I will only entertain

new valid questions and problems if you only join in late.

WEEK 1

02 08 Feb. WEEK 2

09 15 Feb. WEEK 3

16 22 Feb. WEEK 4

23 01 March. WEEK 5

02 08 March WEEK 6

09-15 March

REVISION

Lo. system; Join

& polar theory

and calculations

REVISION

levelling, long.-

& X- sections,

area & volumes

Writing theory

model answers

and Calculation

Exercises

Revision

Theodolite

Checks

Adjustments

Traverse

orientation &

Calculation

Bowditch adj.

Start working on

Assignment 1

WEEK 7

16-22 March

WEEK 8

23 29 March WEEK 9

30 05 April WEEK 10

0612 April WEEK 11

13 19 April WEEK 12

20 26 April

Writing theory

model answers

and Calculations

Reconnaissance Triangulation

Intersection

Continue working

on Assignment 1

Observation

& fieldbook

Reduction

Solving the

triangle using the

Sin Method

Writing theory

model answers

and Calculations

Compl. & submit

Assignment 1

WEEK 13

2703 May WEEK 14

0410 May WEEK 15

1117 May WEEK 16

1824 May WEEK 17

2531 May WEEK 18

0107 June

Writing theory

model answers

and Calculations

Start working on

Assignment 2

Resection

q-Point

Continue working

on Assignment 2

Writing theory

model answers

and Calculations

Compl. & submit

Assignment 2

Horizontal curve

Basic Geometry

Start working on

Assignment 3

Writing theory

model answers

and Calculations

Horizontal curve

Apply Geometry

Continue working

on Assignment 3

WEEK 19

0814 June WEEK 20

1521 June WEEK 21

2228 June WEEK 22

2905 July WEEK 23

0612 July WEEK 24

1319 July

Horizontal curve

Calculate

Geometric data

Horizontal curve

Setting out

Continue working

on Assignment 3

Horizontal curve

Exercises

Compl. & submit

Assignment 3

Writing theory

model answers

and Calculations

REVISION

Levelling

Reverse &

reciprocal leveling

WEEK 25

2026 July WEEK 26

2702 August WEEK 27

0309 August WEEK 28 - 39

10 Aug04 Oct WEEK 40 to

END EXAM.

Trigonometric

leveling

(LEVEL)

LEVEL Map projections SA Map series

(6 to -8) Practical

weeks

Revision and

exercises

You will note that by following this programme you will be ready and fully prepared for the practical

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ADDITIONAL NOTES NOT COVERED IN THE CURRENT SURVEYING I & II STUDY GUIDE

1. JOIN AND POLAR THEORY

In the measurement and use of Directions in calculations different instruments can be used to

determine either angles or directly directions between points on the earths surface. It is not intended in this chapter to discuss the different instruments, but rather to investigate in which manners

directions are determined and used in calculations.

Different instruments that can be used to observe directions and angles are the Prismatic compass,

the Tache, Theodolite, Electronic Theodolite, Total Station and GPS

1.1 Definitions

1.1.1 Geographic meridian

The true or geographic meridian through a point is the line where the earth is cut by a plane that

cuts through the north and south poles as well as the point in question.

1.1.2 The magnetic direction

The magnetic direction of a line at a point is the angle measured in a horizontal plane, from the

northerly direction of the magnetic meridian, through the point, clockwise to the vertical plane,

which contains the line.

1.1.3 Magnetic declination

The magnetic declination is the angle between the directions geographic - and magnetic north. The

magnetic declination is measured East or West of True North.

Magnetic declination increases for a number of years to the one side of true north, then decreases

again towards true north and carries on increasing to a point on the other side. The maximum values

either side are not constant or even always the same because of different factors working in on the

turning of the earth round its own axis.

Magnetic declination is always indicated clearly on the side of Topo Cadastral maps with the year it was drawn up, the direction of and average movement over a number of years up to the date of the

map.

1.2 Introduction to the SOUTH AFRICAN COORDINATE SYSTEM

1.2.1 Angles

B

A Angle BAC

C

Figure 3.1

An angle is the difference between two directions out of the same point.

There are different types of directions such as:

Magnetic (compass) directions

True directions

Oriented directions

Oriented observed directions

Observed directions

Un-oriented observed directions

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The use of angles makes up a very large part of survey calculations and it is therefore very important

that one takes note of the following points to ensure that angles are calculated correctly:

(i) The two lines must both pull out or both must push in on the point or

(ii) The two directions must both be of the same type

B (Un-Oriented direction xAB) E (Oriented direction xDE)

A OR D

C (Un-Oriented direction xAC) F (Oriented direction xDF)

(iii) Two lines form an outer and an inner angle. Make sure that you know which one you are calculating or need to calculate. Inner angle

Outer Angle

(iv) According to Azimuth the directions will always increase in value as we move clockwise

+Y

90 K

135

Figure 3.2

The full definition for an angle will therefore be as follows:

An angle is the difference between two directions out of the same point and is obtained by

subtracting the smaller direction from the larger, where the larger direction is always clockwise

from the smaller.

1.2.2 Directions and Back Directions

Any line between two points will have two directions, namely the forward direction and the back

direction. In a line AC in the sketch below the forward true direction is xAC and the true backward

direction xCA. The forward direction and the back direction differs by 180

+Y

90

C

250

A 70 P

Figure 3.3

L +X

0

A

xAL = True direction AL = 45 xAK = True direction AK = 135

The inner Angle KAL = xAK Xal

Angle KAL = 90

45

L

+X

0

xAC = True forward direction AC = 250 Angle CAP = 270 - 250

Angle CAP = 20

Angle APC = 90

Angle ACP = 180 (90 + 20)

Angle ACP = 70 = xCA

xCA = True Back direction CA = 70

xCA = xAC 180

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1.2.3 Construction of a grid and plotting of coordinated points.

The earth is a sphere and for mapping purposes the sphere must be projected on a flat horizontal plane.

Geographically the earth can be divided into North - South lines of longitude and East West lines of latitude. The lines of longitude will not be straight lines when projected in a flat horizontal plane, but the

lines of longitude will be straight. These projected lines will form a graticule as shown in Figure 3.4.

GRATICULE Figure 3.4

In surveying we however need straight lines to be able to measure and or plot directions and distances easy

and correctly. We use Y- and X ordinates, running North South and East West respectively, perpendicular to each other. These projected lines will therefore form a grid as shown in Figure 3.5:

Y Y Y Y

X X

X X

Y Y Y Y

GRID Figure 3.5

Any point plotted on this grid can thus be identified in terms of an Y-ordinate and an X-ordinate, called the

co-ordinates of that point.

In South Africa we always write the Y-ordinate first and the X-ordinate last. (Note the difference to what

you are used to in mathematics)

P - 86 468,357 Y + 1 367 254,137 X

A coordinate without a sign means nothing. The signs do not say the value is positive or negative, it

indicates the quadrant in which the point lies and for plotting purposes tells you in which direction to move,

using the gridline values!

Latitude

Longitude

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1.2.4 True direction (AZIMUTH) of a line at a point in South Africa

Azimuth or true direction of a line at a point is the angle measured in a horizontal plane from the

southerly direction of the meridian through the point, clockwise to the vertical plane, which

contains the line.

The definition will explained by the following sketch in Figure 3.6.

+Y

90

Meridian through A

B Southerly direction of the Meridian through A

0

South

Figure 3.6

1.2.4.1 Implications of the difference between surveying and mathematics direction systems

It is important that notice must be taken here between the mathematical and surveying directions and

how it will influence calculations, using trigonometry.

A circle is 360 and is divided into four rectangular quadrants of 90 each. In mathematics the zero direction of the system is West

o Directions increase anticlockwise

In surveying, as seen from the definition of true direction, the zero direction of the system is South o Directions increase clockwise.

The differences can best be demonstrated by means of a sketch.

In Figure 3.7a the surveying system will be shown and in Figure 3.7b the mathematical system.

180 90

2nd

Quadrant 3rd

Quadrant 2nd

Quadrant 1st Quadrant

+ - - - + - + +

Y X Y X X Y X Y

SIN TAN SIN ALL

90 270 180 0

1st Quadrant 4

th Quadrant 3

rd Quadrant 4

th Quadrant

+ + - + - - - +

Y X Y X X Y X Y

ALL COS TAN COS

0 270 South

Figure 3.7a Figure 3.7b

Note:

Apart from directions increasing clockwise and anti clockwise it is only the first and third quadrants that exchanged positions.

+X

0

A

45

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The sign convention in the four quadrants are the same on both systems

Sin, Cos, Tan and their inverses are still positive and negative in the same quadrants in both.

All calculations using trigonometric functions stays exactly as you are used to.

It is in effect then only the directions, quadrants and Y & X ordinates that change.

1.2.5 Orientation

a) Orientation of a plan is the turning of the plan in a horizontal plane so that corresponding lines

on the map and on the ground are parallel.

b) Orientation of a theodolite is the adjustment of the horizontal circle so that the true direction will

be read off the circle when the instrument is pointed to a point. It also implies that if the

instrument is pointed to true South the reading on the horizontal circle will be zero (00 00 00)

An important rule when orienting is to always orient on the longest, best visible, available ray and to check

the orientation on at least two other rays. In calculations a mean orientation can be obtained by giving more

weight to a long line than to shorter rays. Will be discussed in detail in later sections!

1.2.6 The South African Coordinate System

The spheroid cannot be projected onto a flat horizontal plane without some distortion. There are three

manners in which the projection could be done namely; directly to a flat surface, or using a cylinder, or a

cone. These projections are then known as flat, cylindrical and conical projections. Under each type of

projection there are a number of projections each emphasizing or avoiding different aspects to ensure least

distortion in the projection process. Map projection is however a topic or subject on its own and we will not

dwell deeper into it here.

The important factors that must however be considered in the choice of a projection is:

o The form or shape of the earth or part thereof to be transformed o The size of the land mass or figure o Measurement of true directions and distances (true to scale)

The South African Coordinate System is based on the Gauss Conform projection, which is a cylindrical

projection where the cylinder is placed in a North-South position around the sphere, touching it around the

equator.

Figure 3.8

If the earth can be presented by an orange cut it in two halves and the fruit taken out, so that we only have

the two hollow halves of peel. If you should press the one halve hollow peel flat on a sheet of paper it will

form a circle. The circle will have a number of deeper or shallower tears at different places in the peel. The

smaller piece of peel pressed down will have less and more shallow tear marks as shown in figures 9a and

9b. The amount and depth of the tears is the distortion that takes place in the process of the projection.

Equator

Latitude

Longitude

North Pole

South Pole

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Figure 3.9b

Figure 3.9a

It is important that in any projection used in surveying the distortion must be kept to a minimum and is

therefore apparent that the smaller part of the earth projected at a time the less the distortion will be.

Gauss calculated that if he uses a cylindrical projection where the cylinder is placed in a North-South

position around the sphere touching it around the equator, projecting only a strip of 2 longitude wide at a time the distortion will be a minimum and he could derive correction formulae for the distance and direction

distortions in his Gauss Conform projection. A belt of 2 strips will then look as follows:

Lo 17 Lo 15 Lo 13 Lo 11

18 16 14 12 10

Boundary Meridians

Central Meridians

Figure 3.10

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Figure 3.11 shows an enlarged single strip of 2 longitude width projected as a plane rectangular coordinate system. The corrections for a line AB, with (A) close to the central meridian and (B) on the boundary

meridian will be indicated in a simplified way.

Lo 19 Central Meridian (CM)

20 18 Equator

+Y 0,0 Y 0,0 X Y

A Plane Distance

Spheroidal Direction (T)

Plane Direction (t)

BM Meridian through A

0 Boundary Meridian (BM)

Spheroidal Distance

Correction for (t T) B B

+ X

Correction for scale enlargement Figure 3.11

In the Gauss Conform projection the size of the land mass or figure, the measurement of true directions and

distances are the priority.

o Strips of 2longitude width are projected separately as a Plane Rectangular Coordinate system.

o The uneven degree of longitude is the central meridian of each 2 wide strip.

o Each projected strip will be named after the degree of longitude of the central meridian for the strip, thus LO 13, Lo 15 etc. In Figure 3.11 the Plane Rectangular Coordinate system is called Lo 19.

o The Southerly direction of the meridian through any point is the zero direction of the projection.

o Directions increase clockwise from South 0 to 360 South again

o From any point the true direction, to any other point, can fall in any one of the four quadrants of 90

around the point. The true direction to any other point can be from 0 to 360.

o Coordinates in use are Y and X

o The intersection of the central meridian (CM), for each strip, and the equator is the origin of the Plane

Rectangular Coordinate system. Thus 0,0 Y 0,0 X

o The Yordinates increase POSITIVE to the left of the (CM) and increase NEGATIVE to the right.

o The Xordinates increase POSITIVE to the South of the equator and increase NEGATIVE to the North. Because South Africa falls in total on the Southern side of the equator our X-ordinates will always have a

POSITIVE sign on the South African Lo Coordinate System.

o The central meridian will be projected as a straight line. Directions and distances are true on the CM.

o True directions and distances true to scale can be measured on the central meridian.

o The boundary meridians (BM), both sides of the central meridian, will be projected as curved lines converging at the poles and will always be the even numbered longitude.

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o The BMs will however be drawn as straight lines, therefore the scale will enlarge as you move to the left and right of the central meridian, and the system makes provision for scale enlargement.

o To make things easier for surveyors in practice there is a 15 minute overlap on both sides of every Boundary Meridian. Within the total overlap area of 30 minutes all the trigonometrical beacons

coordinates were transformed by the then Department of Surveys and Mapping, so that the coordinates

are available on both the systems.

o Lines of direction between any two points will be projected as curved lines and the curvature will increase as you approach the boundaries. The varying small differences between the curved line and the

tangent to it, at different positions in the projection, can be corrected by the correction for (t T)

o The (t T) is mostly a very small angular correction and is only applied to the highest order of accuracy work in practice. The (t T) correction increases the further we move away from the CM towards the BM and it is therefore important to investigate when the Y values get bigger.

o The South African Lo Coordinate System is only applicable up to plus minus the 79 Southern Latitude because the meridian convergence to the pole after that is so irregular and at a very fast rate that the

formulae for Scale enlargement and (t T) are not applicable anymore.

o From the 79 to 90 Southern Latitude Polar Coordinates are used.

o A Polar Coordinate in the Southern hemisphere is the angle East or West of the true or geographic

meridian (the line between the South and North Poles) (between 0 and 90) and the horizontal distance from the South Pole, as can be seen in Figure 3.12.

North

P

Horizontal Distance = 152,3m 25 West of True North

South

Figure 3.12

The Polar Coordinate of P = 25 West of True North and 152,3 meters

The term Polar originated from the term Polar Coordinate and is therefore described further in 3.3.1

At the beginning of survey in South Africa a Base Line was measured very accurately in Port Elizabeth. The

two end points of the baseline were fixed by means of astronomic observations and that is where our system

of trigonometric beacons all over South Africa originated from. A series of triangles was extended to trig

beacons placed on high outstanding points all over the country. These trig beacons were always used by all

surveyors as the known points to start any survey operation anywhere in the country.

South Africa is covered by more or less five Lo coordinate systems, i.e. 5 by 2 wide longitude strips. Cape Town in the West falls on Lo 31 up to Durban in the East at Lo 31.

Coordinate lists of all trig beacons in South Africa within the relevant strip and the overlap areas they fall in,

specifying the X, Y and Z ordinates are available from the Surveyor Generals (SG) offices and was generally

known as Lo Coordinates.

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Trig beacon

Figure 3.13

In modern days with the introduction of the Global Positioning System (GPS) the extensive use of trig

beacons for orientation and Control became less important, but in order to tie any survey in on the existing

control (Coordinates) the GPS receiver has to be set up at a trig point to initialize the instrument.

With satellite observations with the GPS equipment the world changed to WGS 84 Coordinates.

In the process of change we originally changed to a system where the Geoid for the WGS system referred to

PE as it was with the original Baseline. This proved to be problematic because very large distortions were

experienced as we move further away from PE, especially in the North West Province.

In 1994 the Geoids origin and dimensions was recalculated, with all the new knowledge about the worlds dimensions. As a central location for South Africa the observatory at Hartebeeshoek near Hartebeestpoort

dam was chosen, to facilitate the least amount of distortion in the coordinates all over South Africa.

The coordinate system now in use is therefore known as the HARTEBEESHOEK 94 SYSTEM and it is

very important that one must make very sure which system coordinates are given and or required.

The form of the world before 1994 and now

1.3 Calculations

From the field observations it is necessary to be able to orientate observed directions, calculate

coordinates from observed Polars and calculate the true direction and horizontal distance between

two coordinated points (Joins).

We will concentrate on the theory and mathematics to be able to Orientate, calculate the true

direction and horizontal distance between coordinated points (Joins) and calculate coordinates from

Polars. As a further basic application we will also look at the necessary calculations to set out points

in the horizontal plane and the solving of simple triangles.

Geoid

Ellipsoid before 1994

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In Figure 3.14 below, coordinate measurements are shown.

90 + Y

+XA

A

+XB

S X

B P

Y +YA

+YB + X

0 Figure 3.14

The line AB lies in the first quadrant and the true direction xAB = degrees From triangle APB in Figure 3.14, all the basic formulae necessary in polar and join calculations can

be derived using the trigonometric functions Sin, Cos, Tan and Cot plus basic adding and subtraction.

Table 3.1

It is always important in all calculations to visualize the sketch and remember the trigonometric functions!

180

Y X Y X

+ - - -

2nd

3rd

90 Sin Tan 270

+Y Y X Y X -Y

+ + - +

1st 4

th

All Cos

0 + X

Given:

Coordinates of A = +YA +XA Coordinates of B = +YB +XB Horizontal Distance line AB = S

True direction of line AB = xAB = +YA = Y-ordinate of A

+XA = X-ordinate of A

+YB = Y-ordinate of B

+XB = X-ordinate of B

Y = Increase / decrease in Y-ordinates X = Increase / decrease in X-ordinates S = Horizontal Distance line AB

= True direction of line AB Figure 3.14 represents a case where points

A and B lies in the first quadrant (+, +)

Y = [+YB (+YA)]

X = [+XB (+XA)]

Y = S Sin xAB = S Sin

X = S Cos xAB = S Cos

S = Sin

Y

S = Cos

X

X

Y

= Tan

Y

X

= Cot

First quadrant Third quadrant

Sin (0 + ) = + Sin Sin (180 + ) = - Sin

Cos (0 + ) = + Cos Cos (180 + ) = - Cos

Tan (0 + ) = + Tan Tan (180 + ) = + Tan Note quadrants 1 & 3 use function for the function

Second quadrant Fourth quadrant

Sin (90 + ) = + Cos Sin (270 + ) = - Cos

Cos (90 + ) = - Sin Cos (270 + ) = + Sin

Tan (90 + ) = - Cot Tan (270 + ) = - Cot Note quadrants 2 & 4 use co-function for the function

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Although the sophisticated calculators used nowadays are preprogrammed to directly use the trigonometric

function correctly in any quadrant, you still have to know all the basic mathematics to be able to write your

own programmes for the calculator. At the same time if you do not want to become a slave of the calculator,

accepting any answer as if it is always correct, you always have to be a step ahead to be able to question the

answers.

Looking at the sign convention in the first quadrant:

90 + Y

+XA

A

+XB

S X

B P

Y +YA

+YB + X

0 Figure 3.15

In Figure 3.15:

AP = [+XB (+XA)] is Positive because XB & XA are both positive and XB bigger positive than XA

PB = [+YB (+YA)] is Positive because YB & YA are both positive and YB bigger positive than YA

Sin = (YB-YA)/S = +/+ = Positive Sin is Positive in the 1st Quadrant.

Cos = (XB-XA)/S = +/+ = Positive Cos is Positive in the 1st Quadrant.

Tan = (YB-YA)/ (XB-XA) = +/+ = Positive Tan is Positive in the 1st Quadrant.

The True direction is (0 + )

Looking at the sign convention in the second quadrant:

+YB 180

Y - X

B P

S X

-XB

A

+YA -XA

90 + Y Figure 3.16

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In Figure 3.16:

AP = [-XB (-XA)] is Negative because XB & XA are both Negative and XB bigger Negative than XA PB = [+YB (+YA)] is Positive because YB & YA are both positive and YB bigger positive than YA Sin = (YB-YA)/S = +/+ = Positive Sin is Positive in the 2

nd Quadrant.

Cos = (XB-XA)/S = -/+ = Negative Cos is Negative in the 2nd

Quadrant.

Tan = (YB-YA)/ (XB-XA) = +/- = Negative Tan is Negative in the 2nd

Quadrant.


Looking at the sign convention in the third quadrant:

180 -YB

- X Y

P B

X S

Figure 3.17

-XB

A

-XA -YA - Y

270 In Figure 3.17:

AP = [-XB (-XA)] is Negative because XB & XA are both Negative and XB bigger Negative than XA PB = [-YB (-YA)] is Negative because YB & YA are both Negative and YB bigger Negative than YA Sin = (YB-YA)/S = -/+ = Negative Sin is Negative in the 3

rd Quadrant.

Cos = (XB-XA)/S = -/+ = Negative Cos is Negative in the 3rd

Quadrant.

Tan = (YB-YA)/ (XB-XA) = -/- = Positive Tan is Positive in the 3rd

Quadrant.


Looking at the sign convention in the fourth quadrant:

90

- Y

+XA -YA Y

A

+XB

S X

P B

+ X +YB

360 Figure 3.18

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In Figure 3.18:

AP = [+XB (+XA)] is Positive because XB & XA are both positive and XB bigger positive than XA

PB = [-YB (-YA)] is Negative because YB & YA are both Negative and YB bigger Negative than YA

Sin = (YB-YA)/S = -/+ = Negative Sin is Negative in the 4th

Quadrant.

Cos = (XB-XA)/S = +/+ = Positive Cos is Positive in the 4th

Quadrant.

Tan = (YB-YA)/ (XB-XA) = -/+ = Negative Tan is Negative in the 4th

Quadrant.


NB: Did you notice that according to the definition of AZIMUTH the directions increase clockwise and are

always written in the different quadrants as (0+); (90+); (180+) and (270+).

In survey we never use (180) or 360) although it is mathematically correct.

1.3.1 The Polar

A Polar is the determination of the true direction and horizontal distance of a line between two points in

order to calculate coordinates of unknown points.

The Polar problem and calculation

Coordinates of A = +YA +XA True direction of line AB = xAB = Horizontal Distance line AB = S

Calculate the Coordinates of B?

Equations and in table 3.1 can be re-written to make YB and XB the subject of the equations.

3.3.1.1 Standard tabulated form of calculation of the Polar

A +YA +XA

True direction of line AB = xAB = Y = S Sin xAB X = S Cos xAB

Horizontal Distance line AB = S B : YB = +YA + S Sin xAB) XB = +XA + S Cos xAB)

Note: Directions are always observed and or given in dd mm ss and must be decimalized before it can be used in calculations! On the other hand all angles calculated on the calculator will be displayed as a

decimal degree and must be transformed to dd mm ss before the answer can be noted down.

1.3.1.2 Worked example of a Polar calculation in standard form

A +7 987,344 + 89 657,433

xAB = 125 34 25 = 125,573611 + 3 715,385 - 2 657,367

S = 4 567,897 B +11 702,729 + 87 000,066

Y = [+YB (+YA)]

YB = (+YA) + (Y)

Equation Y = S Sin xAB = S Sin

Then: YB = (+YA) + (S Sin xAB)

X = [+XB (+XA)]

XB = (+XA) + (X)

Equation X = S Cos xAB = S Cos

Then: XB = (+XA) + (S Cos xAB)

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1.3.1.3 Further single polar exercises

A + 87 987,344 + 1 489 657,433

a) xAC = 205 43 52 and AC = 3 546,789m

b) xAD = 25 27 29 and AD = 7 563,479m

c) xAE = 287 54 37 and AE = 6 427,865m

1.3.1.4 Checking the Polar calculation

It is a known fact that in surveying one must always check your work, preferably by means of an

independent check, so that your results will be free from mistakes and errors. There are two checks that

could be done on the polar calculation namely the Auxiliary angle check for the polar and using a Join.

The Auxiliary Angle Check for the polar uses a different direction in the check and is therefore an

independent check, while the Join must give the same direction and distance as a check. The problem that

can make the Join check not so reliable is the fact that Y and X must be calculated from the polar calculated coordinates to be able to calculate the direction and distance as a check. Persons with lots of

confidence and a strong belief that they do not make mistakes often use the same deltas as calculated in the

polar (without re-calculating them). If you are one of those you are bluffing yourself, because you are not

checking the calculation in full.

Because The Auxiliary Angle Check for the polar is not commonly used in modern days, mostly because the

Join is much easier and quicker to calculate, we will not discuss it any further!

1.3.2 The Join

A Join is the calculation to determine the true direction and horizontal distance of a line between two

coordinated points.

From the definition it will be noted that the join is just the reverse of a polar. It must also be noted that the

computer will give back any angle as a decimal degree and to be able to use the direction in observations

with a theodolite it must now inversely be transformed to dd mmss 90

+ Y

+XA

A

+XB

S X=(XB-XA)

B P

Y=(YB-YA) +YA

+YB + X

0 It is important to note that:

could be calculated from either of the two equations X

YTan

1 and

Y

XCot

1

It is convenient to choose to use only Tan because it saves time in the number of calculator keys to be pressed in comparison with Cot.

Given:

Coordinates of A = +YA +XA Coordinates of B = +YB +XA

Asked:

Calculate the true direction xAB and

the horizontal distance AB = S

Solution:

X

YTan

and

Y

XCot

X

YTan

1 and

Y

XCot

1

S = 22 XY

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If the signs of Y and X are +,+ or -,- use X

YTan

1 (first and third quadrants)

If the signs of Y and X are +,- or -,+ use Y

XTan

1 (second and fourth quadrants)

in this solution is not the direction yet, because in a rectangular triangle it will always be an angle smaller than 90. Look at the signs and add the quadrant factor

Join formulae

1.3.2.1 Standard tabulated form of calculation of the Join

A + YA + XA xAB =

X

YTan

1 if +,+ or -,- (plus 0 or 180)

or xAB =Y

XTan

1 if +,- or -,+ (plus 90 or 270)

B YB XB AB = S = 22 XY

Y=(YB-YA) X=(XB-XA)

1.3.2.2 Worked example of a Polar calculation in standard form

A +7 987,344 + 89 657,433 35,573611 + 90 = 125 34 25

B +11 702,729 + 87 000,066 AB = 4 567,897

+ 3 715,385 - 2 657,367

Note:

Y is always the lower Y- minus the top Y- value X is always the lower X- minus the top X- value The signs of Y and X are + , - which means the true direction will lie in the 2nd quadrant Therefore we have to divide /X/ by /Y/ with Tan-1 and add 90 as the quadrant factor. Always calculate to one decimal place more than the given values, but not more than 3 places!

Now look back at the worked example of the polar in 3.3.1.2 and you will see that this Join confirmed that

the coordinates calculated for B was in fact correct, because we have calculated a direction and distance that

corresponds with the originally given direction and distance of the line AB!

Therefore we will use the Polar Calculation to check a Join calculation and inversely use a Join calculation

to check a Polar calculation.

X

YTan

1 if +,+ or -,-

Y

XTan

1 if +,- or -,+

S = 22 XY

+ + DIRECTION = ( + 0)

+ - DIRECTION = ( + 90)

- - DIRECTION = ( + 180)

- + DIRECTION = ( + 270)

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The combined standard form for the Join and its Polar check will be as follows:

JOIN POLAR CHECK

A +7 987,344 + 89 657,433

A +7 987,344 + 89 657,433 125 34 25 + 3 715,385 - 2 657,367

B +11 702,729 + 87 000,066 4 567,897 B +11 702,729 + 87 000,066

+ 3 715,385 - 2 657,367

The combined standard form for the Polar and its Join check will be as follows:

POLAR JOIN CHECK

A +7 987,344 + 89 657,433

125 34 25 + 3 715,385 - 2 657,367 A +7 987,344 + 89 657,433 125 34 25

4 567,897 B +11 702,729 + 87 000,066 B +11 702,729 + 87 000,066 4 567,897

+ 3 715,385 - 2 657,367

Without the tick marks () it will be assumed that you have only re-written the values and did not really check.

1.4 Further Join exercises

1.4.1 Calculate and check the Joins AC, AD, AE, CD, CE and DE.

A + 7 987,344 + 89 657,433

C + 12 323,427 + 92 479,185

D + 3 174,361 + 83 316,386

E + 4 534,725 + 85 547,937

1.4.2 Combined problems with calculations on joins and polars with orientation; setting out of points and

solving triangles

1.4.2.1 The coordinates of a two points P and Q is as follows:

P 8 417,42 + 5 672,56 Q - 2 941,75 + 2 815,92

a) Calculate the Join PQ (8)

b) If the oriented direction xPR = 131 16 50 and the reduced horizontal distance PR = 847,95m,

calculate checked coordinates for R (8)

1.4.2.2(a) Explain the following terms briefly

i) An Angle (2) ii) Observed angles of direction (2) iii) Oriented directions (2)

(b) Calculate the Join from the coordinates of two points M and N (8) M 9 723,41 + 7 164,98 N - 12 098,67 + 11 546,72

(c) Use the true direction calculated in (b) to orientate the following observed directions at M in tabulated form as given below: (2)

Observed directions Orientation Correction Oriented Directions

@M

N 297 14 03

P 46 27 37

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(d) The oriented direction from N to a point Q is 123 50 10 and the distance NQ = 347,67 meters. Calculate the coordinates for Q? (8)

(e) Would you accept the coordinates of Q for further calculations? If not why not? (3) (f) What further fieldwork and or calculations would you consider necessary before these

calculated coordinates of Q may be accepted as correct? (4)

1.4.2.3 Coordinates of A and B on Lo are:

Constants: - 80 000,00 + 1 600 000,00

A 7 462,35 + 15 819,59 B - 12 317,69 + 15 306,71

Observed un-oriented directions:

@A @B

B 253 48 02 A 93 58 12

P 01 44 16 P 66 49 03

Distances: AP = 3 155,97 BP = 6 579,38

(a) Explain what is meant by the coordinates are on Lo (3) (b) Calculate the distance and direction AB (8)

(c) Use the true direction calculated in (b) to orientate the observed directions at A and B in tabulated form. (4)

(d) Calculate fully checked coordinates for P? (8)

1.4.2.4 Coordinates of A and B on Lo are:

Constants: - 90 000,00 + 2 100 000,00

A 6 415,96 + 9 724,07 B - 9 172,93 + 7 324,81

Observed directions @A Horizontal Distance AC = 542,47

B 158 58 14

C 311 23 54

Calculate the coordinates of C? (10)

1.4.2.5 Points A, B and C are inter-visible. The oriented direction AC = xAC = 97 10 40 and the horizontal distance AC = 1 893,52 meters.

Coordinates of A and B on Lo are:

Constants: - 70 000,00 + 1 470 000,00

A 8 272,13 + 7 311,20 B - 21 438,17 + 12 194,10

(a) Calculate the direction and distance AB (8)

(b) Calculate coordinates for C (8)

(c) Would you accept the coordinates of Q for further calculations or setting out? If not why not? (3)

1.4.2.6 From the given data below:

(a) Calculate the Join EF (8)

(b) Orientate the observations @ F to obtain the oriented direction FP (2)

(c) Calculate the two polars FP and EP as a check on each other (8)

(d) Calculate the area of the triangle PFE from coordinates (4)

Coordinates Distances Oriented directions

E 8 441,37 + 28 885,70 EP = 4 740,08 xEP = 210 12 35 F - 10 108,35 + 20 086,39 FP = 4 757,48

Observations @ F

E 200 44 40

P 181 20 10

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1.4.2.7(a) Calculate the Join from the coordinates of two points M and N (8)

M 16 169,33 + 3 497,67 N - 14 998,70 + 4 668,30

(b) The point P lies directly East of N so that NP = 1 170,63 meters.

Without the use of your calculator or any calculations, write down the answers to the

following questions with reasons for each?

(i) The coordinates of P? (3)

(ii) The distance MP? (3)

(iii) The angle MPN? (3)

(iv) The direction PM? (3)

1.4.2.8 Point C lies on the western side of line AB. The Coordinates of A and B are:

A + 9 643,84 + 18 217,19 Angle BAC = 71 23 47

B + 16 675,74 + 9 640,92

(a) Calculate the direction and distance AB (8)

(b) Calculate the oriented direction of the line AC? (4)

(c) Describe fully how you would manipulate a Tacheometer which has been setup at A so that oriented

directions can be read off with it? (8)

1.4.3(a) The oriented direction from point A to point B is 150 41 10 and the distance AB after it has been

corrected for Slope and Temperature is 1 095,46 meters.

The Coordinates of A are: A - 25 265,64 + 24 945,67

Calculate coordinates for B? (8)

(b) Coordinates of C and D are: C - 23 456,91 + 23 986,53

D - 25 869,25 + 27 514,63

The following observations were taken with a Theodolite and a steel tape.

@ C

D 340 49 25

B 285 21 50

Measured distance CB = 1 274,26 at a slope of + 03 15 and a temperature of 29 C The tape is standard at 16 C and its coefficient of expansion is 0,000012 / C Calculate coordinates for B again. If there are no mistakes in the observations and or

calculations the coordinates should be the same as those calculated in (a) above. (20)

1.4.4 Point C lies on the western side of line AB. The Coordinates of A and B are:

A - 1 230,93 + 1 524,82

B - 7 459,08 + 9 459,32

Angle BAC = 60 14 10 and D is a point on line AB, perpendicular from C onto AB

Distance CD = 746,93

(a) Calculate coordinates for C (20)

(b) Calculate the area of the triangle ACD (5)

1.4.5 Coordinates of A and C are:

A - 1 426,84 + 9 495,23

C - 2 113,07 + 9 368,49

Measured distances:

Line Measured

distances

Temperature

Measured C Temperature

Standard C Coefficient of

expansion /C Slope

AB 416,99 29 15 0,000011 + 03 20 BC 393,19 29 15 0,000011 - 04 51

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Observed Directions

@A @C @B

C 209 32 10 A 79 32 30 A 80 14 40

B 180 14 40 B 110 51 50 C 320 51 30

C 209 32 10 A 79 32 10 A 80 14 50

(a) Calculate the Join AC (8)

(b) Calculate the horizontal distances AB and BC (10)

(c) Calculate the oriented directions xAB and xCB (Use all three setups observations) (8)

(d) Calculate Coordinates of B that are fully checked! (8)

2. ADDITIONAL NOTES NOT COVERED IN THE CURRENT SURVEYING II STUDY GUIDE

2.1 TRIANGULATION

2.1.1 Reconnaissance

The value of proper reconniasance is underestimated to a large extent. Reconniasance forms the basis for

planning and excecution of measurements in order to comply with the reqiurements of the task. Therefore it is

important that in surveys of a larger extent, such as triangulation networks, more time and attention should be

applied to reconniasance

Reconniasance can be devided into four phases:

Determine the purpose of the survey.

Determine the required accuracy of the survey, as prescribed in the contract spesifications or the purpose of the task

Decide on the instruments and method/s that will be used.

Decide where and when checks must be carried out in order to ensure that the work will comply to the required accuracy. Identify problem areas where mistakes and/ or errors are most likely to occur and plan

how independent and the most efficient checks can be carried out that will show up mistakes and errors

The purpose of the survey determines the scale and the scale determines the accuracy of the measurements.

The accuracy prescribes which methods and instrumentation will be the most effective under the specific

circumstances, and eventually determines the cost and time spent on a project.

Reconniasance is not limited to triangulation work. Reconnaissance must be done before any survey task can

be done. Existing topographical maps and or arial photos must be used where possible. On the one hand this will ease your work and save time, but on the other hand this prevents advance knowledge about strategic

surveys, that could lead to unfounded objections and property speculation.

Because reconniasance forms the basis of all further measurements, the basic principles of surveying must be

applied at all times, ie:

Work from the whole to the part

Economy of accuracy;

Independent checks.

2.1.1.1 TRIANGULATION RECONNIASANCE

Through reconniasance the best suitable positions for control, given the circumstances, are determined by

taking the following important points into account:

Collect all information relevant to the survey such as specifications; existing topographic maps; 1:50 000 topo-cadastral maps; co-ordinate lists; arial photographs etc.

Mark the proposed control points on the maps. Visit the terrain and determine whether the chosen positions are suitable and feasable.

26 SRC2601/102/0/2015



Avoid angles smaller than 30 or any vacant area larger than 120 where no control can be observed, to ensure a stable network..

Chosen control points must, as far as possible, be inter-visible to further strengthen the network stability.

Place control points at the centres of gravity of the figures formed by the known points.

Avoid grazing rays in order to eliminate the detrimental effect of refraction as far as possible.

In resection be aware of the danger circle.

2.1.2 Triangulation sketch

The triangulation sketch is a well constructed planning tool to be able to see the relative relationship

between the positions of knowns and unknowns as well as the lengths of the rays.

2.1.2.1 DRAWING UP THE TRIANGULATIN SKETCH, PLOTTING AND AREA

The following coordinates of the beacons of a property is known

CO-ORDINATES Lo 17

Constants: - 0,00Y + 1 200 000,00X

E - 8 441,37 + 11 885,70

F - 11 499,76 + 9 321,84

G - 10 108,35 + 86,39

H - 25 765,88 + 11 987,65

(a) Plot the points first on an A4 sheet, using a scale that will optimally fit the sheet

(b) Calculate the area of the closed figure EFGHE by coordinates using the given data.

Give your answer in hectare and show all calculations. (22)

You have to read the questions properly and do as asked

Answer:

(a) To be able to use an A4 sheet optimally you have to know the dimensions of an A4 sheet, which can be turned

either in the PORTRAIT or LANDSCAPE positions.

297mm

210mm 297mm

210mm Step 1: write down the minimum and maximum Y and X values, with the constants added, rounded to the nearest

1000m and determine the maximum Y and X movements, which will have to fit on the paper size.

Y X

MAX - 26 000,00 + 1 212 000,00

MIN - 8 000,00 + 1 200 000,00

18 000,00 12 000,00 Step 2: The Y movement is bigger than the X and therefore the paper will have to be turned landscape!

297mm 210 Step 3: determine the preliminary scale

297 mm = 18000 x 1000mm 210mm = 12000 x 1000mm

1 mm = 1mm =

1: 60 600 1: 57 100 Step 4: choose the nearest available smallest natural scale of the two. Thus 1:75 000

Step 5: determine the Grid Interval (GI) and Grid Value (GV)

In drawing the grid interval (GI) must be one tenth of the scale, thus 75 000 / 10 = 7500 m

The grid values (GV) must be continuous multiples of one tenth of the scale = 7500 m

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Step 6: Now choose new min and max values rounded to values exactly dividable by 7500 Y X MAX - 30 000,00 + 1 215 000,00

MIN - 7 500,00 + 1 200 000,00 22 500,00 15 000,00

297 mm = 22500 x 1000 mm 210mm = 15 000 x 1000 mm

1 mm = 1mm = 15 000 000/210

1: 75 750 1: 71 429

Thus a scale of 1: 75 000 is the smallest available scale of the two that will fit.

If not yet fitting you will have to repeat as much iteration as is necessary to find the largest possible scale

Thus a scale of 1: 75 000 will fit the A4 sheet maximally (optimally) placed landscape

Step 7: Now draw the gridlines according to your chosen maximum and minimum Y and X values for a

scale of 1: 75 000. Write in the grid values on the appropriate grid lines with the four corners fully

specified including the constants

Step 8: Using the grid line values and the coordinates of the points plot every point on the plan

(Triangulation sketch)

100mm

+1 200 000 X

G

100mm

TN + 7 500X F E H NB + 1 215 000X

Scale: 1: 75 000

This is the central meridian of the 17 Lo strip

Note this is a reduced scale illustration only to show how your A4 sheet will look like

Step 9: Draw in the True north ray to be able to orientate the plan and write in the scale

(b) Calculate the area of the closed figure EFGHE by coordinates using the given data

NB: Note down the coordinates starting at any point clockwise and ending on the same point, eg. CO-ORDINATES Lo 17

Constants: - 0,00Y + 1 200 000,00X

E - 8 441,37 + 11 885,70

G - 10 108,35 + 86,39

H - 25 765,88 + 11 987,65

F - 11 499,76 + 9 321,84

E - 8 441,37 + 11 885,70

NOW USE THE FORMULAE IN YOUR SURVEYING I STUDY GUIDE TO CALCULATE THE AREA

OF THE FIGURE EGHFE FROM COORDINATES

- 0,0

0Y

- 22 5

00,0

Y

28 SRC2601/102/0/2015



2.1.3 The Triangulation Fieldbook

With triangulation the angles between stations are observed as accurate as possible. A few important rules

must be kept in mind during the observation procedure to ensure the best possible observed directions.

The reference object (RO), the beacon on which the observing is started, is very important because it must be observed repeatedly to show up and eliminate observational reading and instrument errors.

Choose the best visible clear identifiable beacon as the (RO). This implies that it will not necessarily always be the absolute longest ray.

Normally we start with the instrument in the circle left (L) position sighted on the RO. The chosen beacons are then observed one after the other in strict clockwise order up to the RO again to close the

circle and in so doing will show up instrument and observational errors to be eliminated

The L observations are booked from top to bottom below each other in the fieldbook.

Take care to move the instrument slowly and at a constant tempo in the horizontal plane in observing the points one after the other.

Take care not to oversight. With this term we mean that you may not move past a point having to turn the instrument any amount back anti-clockwise to eventually observe the beacon.

The Theodolite normally has a constant forward movement of the horizontal circle, together with the turning of the instrument, called circle slip.

With the movements of circle right and left in opposite directions the circle slip must naturally always have opposite signs, if not there were probably oversight present

The purpose with the observation procedure is to ensure that during observation of L and R the circle slip of only one rotation of the circle, from RO to RO, has built up.

If backward movement is necessary, circle slip of more than one rotation will be build up and it cannot be eliminated by taking the means in the reduction process.

For high degree of accuracy work the set of observations will have to be rejected when you have over sighted and the set will have to be re-observed in total right from the start.

Observations only in L are not acceptable. A full set L and R must be observed.

After the completion reading, L on the RO was taken the instrument is transited and again pointed to the RO. The R reading on the RO must differ by 180 with the L reading.

The same beacons are then observed in strict anti-clockwise order, one after the other up to and including the RO to close the circle.

Resultantly the R observations will be booked from the bottom to the top, while one at the same time must look out whether the readings show more or less the same difference as on the RO.

The more sets of observations that are read the better and more reliable the mean observed directions will be that are eventually obtained from the reduced fieldbook, for use in the direction sheet for

orientation.

For accurate work at least three sets must be observed.

Every protractor in different instruments have a internal calibration (manufacturing) or systematic error and the observation procedure must make provision that this error can also be eliminated. Every set of

observations must therefore not be observed at the same spot on the protractor. By reading on different

positions on the circle the calibration errors is distributed over the full circle in order to be eliminated.

Decide in advance during the planning phase (reconnaissance) how many sets will be observed for the project. Divide 360 by the Chosen number of sets to determine by estimate how much the protractor

must be turned forward after each set. If for example three sets are going to be observed the protractor

must be advanced 120 between sets.

Make sure that the nonius / vernier is also adjusted so that the minutes and seconds of the individual readings will also change. It is very important to eliminate the possible remembering factor where what

is remembered from the previous set is noted instead of what should actually be read off.

29 SRC2601/102/0/2015



2.1.4 Reduction of the Fieldbook

There are two methods to reduce the fieldbook and it is important that students should be able to

do anyone of the two as specified in an assessment or examination

(a) Reduction by taking the mean of circle left and right readings to eliminate instrument and observational errors as far as possible! The differences between circle left and circle right

directions enables us to evaluate the set as a whole for acceptability or not

(b) By booking the observed directions with a line spacing in between every two observed directions it is possible to determine the angles between every two rays. The means of the angles are then

determined to eliminate instrument and observational errors as far as possible! The differences or

comparison between circle left and circle right angles gives a better check and enables us to

evaluate the set as a whole for acceptability or not

The following two worked examples of the same two sets of observations in both methods will show

how both are done and the difference between the two.

2.1.4.1 Method 1: Reduction of the triangulation fieldbook by taking the mean of circle left and right

readings

In determining the difference between circle left and right readings we first check that they differ by

180 and then note down only the minutes and seconds that they do not differ by exactly 180 00 00.

In high order accuracy observations the differences in every set between circle left and right reading should be very close to the same, but very important that the differences should right through be

constantly bigger or smaller to the same side. I will use a (+) if the circle right reading is bigger than

circle left and a (-) the other way round.

Because collimation error is normally the biggest of any errors we commonly speak about the difference as an indication of the collimation error of the instrument used. It is however common

knowledge that the collimation error of the instrument is a constant for that instrument at that

specific moment in time. The signs and magnitude of the differences must therefore be the same

and if not it is a direct indication of poor observations and or may be due to bad weather conditions.

Very short rays in comparison to long rays may also cause the signs to change.

L SET I Diff.

R Mean I Mean II + * +239 49 59

Mean Adj. Final adjusted

observed

directions

@ Skop

Cap 277 24 04 + 11 97 24 15 10 10 10 +00 277 24 10

JHB 357 58 22 + 07 177 58 29 26 25 26 +00 357 58 26

Laagte 23 27 10 - 03 203 27 07 08 03 06 +01 23 27 07

Aero 29 20 39 + 05 209 20 44 42 47 44 +01 29 20 45

Iscor 31 30 54 + 07 211 31 01 58 59 58 +02 31 31 00

RO 277 24 12 - 06 97 24 06 09 06 08 +02 277 24 10

Diff = +08 Avg = +3,5 Diff = - 11 Diff 01 Diff 04 Diff 02

SET II

L Diff. R

Cap 37 34 00 + 22 217 34 22 11 *We have to add a swing of 239 49 59 to the 1st

reading set II to make it exactly the same as the

mean Set I on the first reading (277 24 10)

Rewrite mean II plus the swing calculated

above next to set I. In applying the swing (*)

between mean I and mean II you have to check

dd mm ss for each ray

JHB 118 08 17 + 17 298 08 34 26

Laagte 143 36 51 + 25 323 37 16 37 04

Aero 149 30 42 + 11 329 30 53 48

Iscor 151 40 49 + 23 331 41 12 41 00

RO 37 34 05 + 04 217 34 09 07

Diff = +05 Avg = +17 Diff = -13 Diff 04

30 SRC2601/102/0/2015



Evaluation of the observed two sets

A simplified method of evaluation of the observations is to decide on the acceptable standard deviation

according to the required accuracy. Let us for example say that the acceptable standard deviation is 03

seconds required for this task.

A standard deviation of 03 seconds means we are allowed a difference spread of 6 seconds, between +3 and -3.

Now look at the comparison between the average difference for SET I (+3,5) and every individual difference.

o +11 differs from +3,5 by 7,5 lying outside the acceptable 6 seconds o +07 differs from +3,5 by 0,5 lying within the acceptable 6 seconds o -03 differs from +3,5 by 6,5 lying outside the acceptable 6 seconds o +05 differs from +3,5 by 1,5 lying within the acceptable 6 seconds o +07 differs from +3,5 by 0,5 lying within the acceptable 6 seconds o -06 differs from +3,5 by 9,5 lying outside the acceptable 6 seconds

Now look at the comparison between the average difference for SET II (+17) and every individual difference.

o +22 differs from +17 by 05 lying within the acceptable 6 seconds o +17 differs from +17 by 00 lying within the acceptable 6 seconds o +25 differs from +17 by 08 lying outside the acceptable 6 seconds o +11 differs from +17 by 06 lying within the acceptable 6 seconds o +23 differs from +17 by 06 lying within the acceptable 6 seconds o +04 differs from +17 by 13 lying outside the acceptable 6 seconds

Evaluation decisions on the two sets observed

In set I the signs of the differences vary + and and therefore it shows possible poor observations by the observer and in practice you will directly in the field reject this set, redoing it directly!

In Set I the magnitude of the differences are not the nearly the same varying from +11 to -03 giving a spread of 14 seconds. Again reason to reject the full set!

In Set I comparing the differences with the standard deviation it is obvious that 3 of the rays do not fall within the acceptable 6 seconds. Again reason to reject the full set!

In SET II the signs of the differences are still positive (+) but the magnitude of the differences are suddenly much bigger (Average +17). This in itself does not make the set un-acceptable because there may be other reasons for it to differ by a bigger margin. It could have been that Set II was

observed on another day, by another observer and even using another instrument!

It is therefore very important that these facts be noted in the remark columns and page headings with proper date indications, otherwise it is impossible to do proper evaluation

In SET II the signs of the differences are all positive (+) making it a much more acceptable set of observations

In Set II comparing the differences with the standard deviation it is obvious that 2 of the rays do not fall within the acceptable 6 seconds. Giving reason to reject the full set!

In Set II the magnitude of the differences are not very bad except the last on the RO which is suddenly very small (+04), varying from +25 to +04 giving a spread of 21 seconds. Again reason to

reject the full set!

You will notice that the differences between the RO and starting reading in circle right and left, in both SET I and SET II do have opposite signs (Set I = +08; -11) and (Set II = +05; -13). It shows

that there was no obvious oversight in the observation process but it is worrying that the magnitude

of the circle slip between the circle right and left observations is differing too much. Again reason

to reject the full set!

31 SRC2601/102/0/2015



As a whole it is obvious that both sets should be rejected in practice and re-observed. As an example we will

however continue to do the rest of the reduction.

Please take note that if you were asked to comment or discuss any two sets of observations this same modus operandi will have to be followed to find points to write comments on or discuss.

2.1.4.2 Method 2: Reduction of the triangulation fieldbook by taking the mean of circle left and right angles

L SET I Diff.

R Mean I Mean II Mean Adjust Final adjusted observed directions

@ Skop

Cap 277 24 04 97 24 15 280 30 50

80 34 18 - 04 80 34 14 16 14 15 + 00 80 34 15

JHB 357 58 22 177 58 29 01 05 05

25 28 48 - 10 25 28 38 43 38 40 + 00 25 28 40

Laagte 23 27 10 203 27 07 26 33 45

05 53 29 + 08 05 53 37 33 44 38 + 00 05 53 38

Aero 29 20 39 209 20 44 32 27 23

02 10 15 +02 02 10 17 16 13 14 + 00 02 10 14

Iscor 31 30 54 211 31 01 34 37 37

245 53 18 - 13 245 53 05 12 53 06 09 + 04 245 53 13

RO 277 24 12 97 24 06

102_2015_0_b - unisa

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