Gheorghe M. T. Radulescu

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Gheorghe M. T. Radulescu

  1. 1. GHEORGHE M.T.RADULESCUGENERAL TOPOGRAPHY TUTORIALS
  2. 2. FOREWORD The practical applications represent the basis for learning General Topography. Irecommend to all those who wish to be initiated and to improve themselves in this area,to use the three manuals (lecture notes, tutorials, problems) in parallel, by chapter, in thepresented order. I ensure them that, if they respect this suggestion, the results will be as expected. I want to thank my two colleagues from the department within the PolytechnicInstitute from Cluj-Napoca: Mrs. Viorica Balan and Mr. Gheorghe Bendea, for thesuggestions they have given me concerning the presentation of practical applications. The Author II
  3. 3. TABLE OF CONTENTSFOREWORD....................................................................................................................IITABLE OF CONTENTS................................................................................................III1. TOPOGRAPHIC ELEMENTS OF THE TERRAIN – MEASURING UNITSAND COMPUTATIONAL MEANS IN TOPOGRAPHY.............................................1 1.1. THE TOPOGRAPHIC ELEMENTS OF THE TERRAIN.......................................1 1.1.1. CLASSIFICATION............................................................................................1 1.2.2. DISTANCE MEASURING UNITS.....................................................................3 1.2. TOPOGRAPHIC SURFACES.................................................................................3 1.2.1. ACTUAL SURFACES AND HORIZONTAL SURFACES.................................3 1.2.2. SURFACE MEASURING UNITS......................................................................4 1.3. ANGULAR TOPOGRAPHIC ELEMENTS............................................................4 1.3.1. ANGLES MEASURED IN TOPOGRAPHY.......................................................4 1.3.2. THE ANGLE IN GEOMETRY AND TOPOGRAPHY.......................................5 1.3.3. ANGLE MEASURING UNITS...........................................................................5 1.3.4. REVISION OF TRIGONOMETRIC NOTIONS, THE TRIGONOMETRIC CIRCLE.......................................................................................................................7 1.3.5. ORIENTATIONS, THE RELATION BETWEEN COORDINATES AND ORIENTATIONS........................................................................................................11 1.4. PROBLEMS FOR TUTORIAL 1...........................................................................13 1.4.1. SOLVED PROBLEMS.....................................................................................13 1.5. EXAMPLE FOR SOLVING THE HOMEWORK (FOR (N) = 0)..........................152. STUDYING THE THEODOLITE.............................................................................19 2.1. THE GENERAL CONSTRUCTION SCHEMA OF A THEODOLITE...............19 2.1.1. GENERAL NOTIONS, CLASSIFICATIONS...................................................19 2.1.2. THE PRINCIPLE SCHEMA – AXES AND COMPONENT PARTS................20 2.1.3. THE DETAILED SCHEMA OF THE THEODOLITE (figure 2.4.)................22 2.1.4. THE DETAILED SCHEMA OF OTHER TYPES OF THEODOLITES...........24 2.2. WORKING PROCEDURE FOR THE THEODOLITE.........................................25 2.2.1. WORKING PRINCIPLES................................................................................25 2.2.2. VERIFYING THE DEVICE.............................................................................25 2.2.3. PLACING INTO THE STATION.....................................................................26 2.2.4. AIMING AND POINTING...............................................................................28 2.2.5. DEVICES FOR READING ANGULAR VALUES ON THE THEODOLITE...29 2.2. THE HOMEWORK OF THE TUTORIAL............................................................313. MEASURING ANGLES WITH THE THEODOLITE...........................................32 III
  4. 4. 3.1. THE NATURE OF TOPOGRAPHIC ANGLES....................................................32 3.2. ANGLE MEASURING METHODS......................................................................34 3.2.1. THE CASE OF MEASURING ONE ANGLE...................................................35 3.2.2. MEASURING MORE ANGLES FROM ONE THEODOLITE STATION........39 3.3. THE HOMEWORK OF THE TUTORIAL............................................................42 3.4. EXAMPLE FOR SOLVING THE HOMEWORK.................................................434. DIRECT AND INDIRECT DISTANCE MEASURING METHODS....................47 4.1. MEASURING DISTANCES DIRECTLY.............................................................47 4.1.1. INSTRUMENTS FOR THE DIRECT MEASUREMENT OF DISTANCES.....47 4.1.2. PREPAIRING THE TERRAIN FOR MEASUREMENTS................................48 4.1.3. CORRECTIONS APPLIED TO LENGTHS MEASURED DIRECTLY...........50 4.2. MEASURING DISTANCES INDIRECTLY.........................................................52 4.2.1. TACHEOMETRIC METHODS AND INSTRUMENTS...................................52 4.3. THE HOMEWORK OF THE TUTORIAL............................................................575. PLANIMETRIC TRAVERSE....................................................................................61 5.1. GENERAL PROBLEMS........................................................................................61 5.1.1. CLASSIFICATIONS........................................................................................61 5.1.2. CONDITIONS THAT CHARACTERIZE A TRAVERSE:................................64 5.2. DESIGNING THE TRAVERSE............................................................................65 5.2.1. MEASUREMENT PREPARING WORKS........................................................65 5.3. FIELD WORKS......................................................................................................65 5.3.1. PERFORMING MEASUREMENTS................................................................65 5.4. OFFICE WORKS...................................................................................................66 5.5. THE HOMEWORK OF THE TUTORIAL............................................................66 5.6. THE SOLUTION OF THE HOMEWORK FOR N = 0...........................................666. SURVEYING PLANIMETRIC DETAILS...............................................................78 6.1. SURVEYING DETAILS THROUGH RADIATION (POLAR COORDINATES) ........................................................................................................................................78 6.2. SURVEYING DETAILS THROUGH SQUARING (SQUARE COORDINATES) ........................................................................................................................................83 6.3. SURVEYING DETAILS THROUGH THE METHOD OF ALIGNMENTS.......84 6.4. THE HOMEWORK OF THE TUTORIAL............................................................857. PROBLEMS SOLVED ON MAPS AND PLANS.....................................................87 7.1. PROBLEMS CONCERNING USING MAPS AND PLANS................................88 7.1.1. SYMBOLS........................................................................................................88 7.1.2. THE GRATICULE OF MAPS AND PLANS....................................................88 7.1.3. THE SCALE OF MAPS AND PLANS.............................................................92 7.1.4. ORIENTING MAPS AND PLANS IN THE FIELD.........................................93 7.2. SOLVING SOME PLANIMETRY AND LELVELING PROBLEMS ON A TOPOGRAPHIC PLAN................................................................................................94 7.2.1. THE HOMEWORK OF THE TUTORIAL.......................................................94 7.2.2. SOLVING METHOD EXAMPLE....................................................................968. THE STUDY OF LEVELING INSTRUMENTS....................................................107 IV
  5. 5. 8.1. SIMPLE INSTRUMENTS – WITHOUT TELESCOPE.....................................107 8.8.1. THE LEVEL HOSE........................................................................................107 8.1.2. THE LEVELING LONG BOARD AND THE AIR-BUBBLE LEVEL............109 8.2. LEVELING INSTRUMENTS WITH TELESCOPE...........................................110 8.2.1. RIGID INSTRUMENTS FOR GEOMETRIC LEVELING.............................110 8.2.2. AUTOMATIC INSTRUMENTS FOR GEOMETRIC LEVELING.................114 8.2.3. ACCESSORIES FOR GEOMETRIC LEVELING DEVICES........................116 8.3. THE HOMEWORK OF THE TUTORIAL..........................................................1219. METHODS FOR MEASURING ALTITUDE DIFFERENCES...........................122 9.1. GENERAL PRINCIPLES....................................................................................122 9.1.1. THE PRINCIPLES OF GEOMETRIC LEVELING.......................................122 9.1.2. THE PRINCIPLES OF TRIGONOMETRIC AND TACHEOMETRIC LEVELING..............................................................................................................125 9.2. APPLICATIONS..................................................................................................12910. GEOMETRIC LEVELING TRAVERSE WITH RADIATIONS......................138 10.1 THE GENERAL CONDITIONS OF A GEOMETRIC LEVELING TRAVERSE ......................................................................................................................................138 10.2. RECOGNIZING AND PREPARING THE ROUTE OF THE TRAVERSE.....139 10.3. FIELD WORKS – MEASUREMENTS THAT ARE PERFORMED IN THE CASE OF SIMPLE MIDDLE GEOMETRIC LEVELING TRAVERSE, SUPPORTED AT THE ENDS............................................................................................................139 10.4. OFFICE OPERATIONS.....................................................................................14411. LEVELING OF PROFILES AND SURFACES...................................................148 11.1 LONGITUDINAL AND TRANSVERSAL LEVELING THROUGH PROFILES ......................................................................................................................................148 11.1.1 THE CONDITIONS OF THE ROUTE.........................................................148 11.1.2. RECOGNIZING THE TERRAIN AND PREPARING THE ROUTE...........149 11.1.3. PERFORMING FIELD MEASUREMENTS................................................149 11.1.4. OFFICE OPERATIONS..............................................................................150 11.2. SURFACE LEVELING......................................................................................152 11.2.1. SURFACE LEVELING THROUGH THE METHOD OF SMALL SQUARES .................................................................................................................................152 11.2.2. SURFACE LEVELING THROUGH THE METHOD OD LARGE SQUARES .................................................................................................................................154 11.3. USING THE DATA OBTAINED THROUGH SURFACE LEVELING..........155 11.4. THE HOMEWORK OF THE TUTORIAL........................................................157 V
  6. 6. 1. TOPOGRAPHIC ELEMENTS OF THE TERRAIN – MEASURING UNITS AND COMPUTATIONAL MEANS IN TOPOGRAPHY The content of the tutorial: The topographic surveys needed for drafting plansand maps consist in measuring the relation in which the topographic points that define asurface are, either using a control network (the planimetric problem), or using ahorizontal datum level (the leveling problem). Actually – the linear elements (horizontal and vertical distances) and the angularelements (horizontal and vertical angles) are measured in the field, formed bytopographic points and reference elements. The purpose of this work is to determine thetopographic elements of the terrain, the relation between them: what does measuringthem represent, which are the measuring units that are used and which are the auxiliarymeans used for computations.1.1. THE TOPOGRAPHIC ELEMENTS OF THE TERRAIN1.1.1. CLASSIFICATION a. The nature of topographic elements Consider two topographic points A and B in the field, materialized in some form(wood or metal stake, concrete boundary marks, etc.). In what concerns these points, we can identify the following topographicelements: - THE AB ALIGNMENT – which represents the intersection of the topographic surface of the terrain with a vertical plan that passes through the given points. In practice, the sinuous line is geometrized (approximated) by a right line, which represents the direction materialized in the field by the points A and B. - THE INCLINED DISTANCE LAB – represents the line segment limited by the points A and B on the direction mentioned above. 1
  7. 7. B L AB ΔZ AB Z ZB A ϕ DAB ZA Vertical datum Figure 1.1. The linear elements measured in the field - THE HORIZONTAL DISTANCE DAB – represents the projection of the slanted distance on the horizontal plan, having as value the horizontal segment between the verticals of the given points. - THE HEIGHTS ZA and ZB – of the points A and B – represent the value of the vertical segment between the vertical datum and that point. - THE ALTITUDE DIFFERENCE ∆ZAB – between the given points – represents the vertical distance measured between the horizontal plans that pass through these points ∆ZAB = ZB – ZA (1.3’). b. Relations between the topographic elements The relation in which are the elements presented above results from theexpression of the trigonometric functions of the angle ϕ – called slope angle (being theangle formed by the distances LAB and DAB). ∆ZAB sin ϕ = -------- (1.1) LAB DAB cos ϕ = ------- (1.2) LAB 2
  8. 8. ∆ZAB tg ϕ = --------- (1.3) DAB L²AB = D²AB + ∆Z²AB (1.4) Using these formulas we can determine the unknown elements based on theknown (measured) ones. Measuring the linear elements presented above consists in comparing their size,using a chosen etalon (measuring unit).1.2.2. DISTANCE MEASURING UNITS Most of the countries use the meter (m) as distance measuring unit. Determined in 1799 by the French DELAMBRE and initially considered as beingthe 40,000,000th part of the length of the terrestrial meridian, and by more recentcomputations, as the 40,000,000.42th part, now it is defined (since 1960) as being equal to1,650,763.73 wavelengths of the orange radiation produced by the KRYPTON 86 gas. The multiples of the meter are: 1 km = 10 hm = 100 dam = 1000 m, and thesubmultiples: 1 m = 10 dm = 100 cm = 1000 mm. The measuring units of the Anglo-Saxon system are given the appendix tables inGENERAL TOPOGRAPHY – lecture notes.1.2. TOPOGRAPHIC SURFACES1.2.1. ACTUAL SURFACES AND HORIZONTAL SURFACES The topographic surface (St) is the actual surface of the terrain, which, not havinga regular form, cannot be mathematically expresses, and therefore, it cannot bemathematically represented. For this reason a geometric schematization of the terrain is performed by choosingthe characteristic points. It should be mentioned that, since the constructions are 3
  9. 9. performed with horizontal foundations, the horizontal projection of the surface of theterrain (S in figure 1.2.) is represented on all topographic maps and plans. St S Figure 1.2. Surfaces in topography1.2.2. SURFACE MEASURING UNITS Derived from the metric system, the measuring unit for surfaces is the squaremeter (m2) with the following multiples and submultiples: 1 km² = 100 ha, 1 ha = 100 ari = 10,000 m²; 1 m² = 100 dm² = 10.000 cm² = 1,000,000 mm². In the appendix tables from GENERAL TOPOGRAPHY – lecture notes there arepresented the measuring units of the Anglo-Saxon system, too.1.3. ANGULAR TOPOGRAPHIC ELEMENTS1.3.1. ANGLES MEASURED IN TOPOGRAPHY In topography there are measured horizontal and vertical angles. In figure 1.3, the angle α is horizontal, being the angle formed by the horizontalprojections of two aiming lines. The vertical angles (ϕ) are formed by some direction with its horizontalprojection. 4
  10. 10. (V AB) (VBC ) A ZA ZB B CA Z ϕA ϕB HC ∆ α D CA DCB A0 B0 Figure 1.3. The angular elements measured in the field The vertical angle formed by a straight line, which represents the support of aslanted distance, between two points, with its horizontal projection, is called slope angle(figure 1.3. the angles ϕA and ϕB). Usually, the theodolites (topographic devices used for measuring angles) recordthe angle Z, called zenithal angle, and the vertical angles result by computation.1.3.2. THE ANGLE IN GEOMETRY AND TOPOGRAPHY The geometric notion of angle – as shape formed by two half-lines having thesame origin – is incomplete for topographic use, knowing the sign and the sense ofmeasuring the angle being necessary. Thus, the topographic angles are oriented angles, the first side of the angle and thesense of measuring being known. By measuring an angle one can understand comparingit with another angle, chosen as unit.1.3.3. ANGLE MEASURING UNITS In topography the new (centesimal) degrees are usually used as measuring units. 5
  11. 11. One centesimal degree (1g) represents the hundredth part of the right angle (D) orthe 400th part of the entire circle (C). D C 1g = ----- = ------ (1.5) 100 400 Submultiples: 1g = 100c (centesimal minutes); 1c = 100cc (centesimal seconds). Most of the measuring instruments in topography are graduated in centesimaldegrees. The advantage of this system consists in the simplicity of the operations, thedivision of degrees being done in the decimal system. E.g. 123g32c17cc = 123g.3217 Other measuring units: Sexagesimal degrees (10), which represent the 90th part of the right angle (D) orthe 360th part of the entire circle (C): D C 0 1 = ----- = ------ (1.6) 90 360 Submultiples: 10 = 60’ (sexa minutes); 1’ = 60” (sexa seconds). The radian (1 RAD) is the central angle corresponding to the arc of circle equal tothe circle radius. It is known that a circle has 2πRAD. For various computations is required to pass from one gradation system toanother, this transformation being performed through one of the equivalence relations: π C 100g = 900 = ------ RAD = 1D = ------ (1.7) 2 4 α0 αg a(RAD) ------- = ------ = ------------- (1.8) 1800 2000 π 1 RAD ≅ 63g66c20cc≅ 57017’5” (1.9) 10 = 1g.111… 1g = 54’ 1’ = 1c85cc.2 1c = 52”.4 (1.10) 1” = 3cc.09 1cc= 0”.34 6
  12. 12. For transforming in radians, with the use of formula (1.4), the followingcoefficients are obtained for the centesimal gradation: 200g ρg = -------- = 63.661977… π 200g x 100c ρ = -------------- = 6366.1977… c π 200g x 100c x 100cc ρcc = ------------------------- = 636619.77… π and for the sexa gradation: (1.11) 1800 ρ0 = -------- = 57.295779… π 1800 x 60’ ρ = ------------- = 3437.7467… ’ π 1800 x 60’x 60” ρ = -------------------- = 206264.80… ,, π Letting π = 3.14159265… It can be said that: 1g = 0.015708 RAD (1.12) 10 = 0.017453 RAD1.3.4. REVISION OF TRIGONOMETRIC NOTIONS, THE TRIGONOMETRIC CIRCLE a. The trigonometric circle and the topographic circle The computations performed in topography need a thorough knowledge oftrigonometric functions, of the trigonometric circle, which, in topography, is transformedin the topographic circle. We define the trigonometric circle as the circle having the center in a pointdenoted by 0, the radius equal to the unit, having the origin for measuring arcs in thepoint A and the measuring sense in left-handed direction (figure 1.4.a). 7
  13. 13. Y II -ctg B +ctg V I ctg P +cos N tg -cos R=1α +sin α +sin A -sin A’ 0 M -sin α -tg X α +cos - cos III IV B’ Figure 1.4.a. The trigonometric circle A V X II - tg A +tg I tg T M +cos N +ctg -cos θ’ =0 c α cos cos + + α θ” P B’ 0 θ” cos B cos - -ctg Y α - θ ”’ α +sin -sin III IV A’ Figure 1.4.b. The topographic circle In topography, the trigonometric circle is replaced by the topographic circle(figure 1.4.b), for the following reasons: 8
  14. 14. - The reference direction in the field, and therefore in topography, is the direction of the topographic North, which coincides with the ordinate axis (this is why this axis is denoted here by 0X); - The sense of measuring angles, in topography, is the right-handed direction. Consequently, it can be seen, comparing the two circles, that in the topographiccircle, the quadrants II and IV are switched, and the quadrants I and III stay in the sameposition as in the trigonometric circle. Therefore, the order of the quadrants is given by the sense of measuring angles.Since one of the characteristics of the trigonometric circle is that the origin and the senseof measuring arcs can be changed without changing the rules and formulas established onthe quadrants, in the two circles, the formulas and the signs of the trigonometric functionsare identical. Hence: defining the trigonometric functions and the variation oftrigonometric lines is equivalent in both circles (see tables 1.1 and 1.2). b. Reduction to the first quadrant, determining the values of trigonometric functions The trigonometric functions of some given angles θ, located in the quadrants II –IV, can be determined as functions of some corresponding angles in the first quadrant –α. The transformation formulas for passing to the first quadrant presented in table 1.3 arebuilt in the following way:Table 1. 1 – The corresponding sign and line of the trigonometric functions, in the four quadrants The function The trigonometric The sign in the quadrants line I II III IV sin MN + + - - cos OM + - - + tg AT + - + - ctg BV + - + - - The sign of the function for the four quadrants is the one specified in table 1.1; - For quadrant I, the functions have the significance that results from figure 1.4.a and 1.4.b and from table 1.1; 9
  15. 15. - For quadrant III, the function of the angle from this quadrant is equivalent to the function of the angle from the first quadrant, obtained by subtracting 200 g from the initial angle;Table 1. 2 – The variation of trigonometric lines The quadrant I II III IV θ The trigonometric line 0g 100g 200g 300g 400g sinθ 0 +1 0 -1 0 cosθ +1 0 -1 0 +1 g g tgθ +∞ 100 +ε 0 +∞ 300 +ε +1 g g 0 100 -ε -∞ 300 -ε -∞ g ctgθ +∞ 200 -ε +∞ 0 -∞ 200g+ε 0 - For quadrants II and IV, the function is equivalent to the cofunctions of the angles from the first quadrant (obtained by subtracting 100g and 300g, respectively, from the initial angle); These rules lead to establishing table 1.3, which is used in the following way:Table 1. 3 – The values of the trigonometric functions in the four quadrants Quadrant I II III IV θ = given angle θ* =α θ” =α + 100g θ” =α + 200g θIV =α + 300gα= reduced angle sinθ + sinα + cosα - sinα - cosα cosθ + cosα - sinα - cosα + sinα tgθ + tgα - ctgα + tgα - ctgα ctgθ + ctgα - tgα + ctgα - tgα - Having an angle θ, which can be found in one of the four quadrants and knowing the fact that there exist tables of natural values of trigonometric functions only for the angles situated in the first quadrant, it becomes necessary to transform the function of the angle θ into that corresponding to quadrant I. Depending on the quadrant in which is situated the angle θ, it can be expressed as: 10
  16. 16. θI = α (1.13) θII = α + 100g θIII = α + 200g θIV = α + 300g corresponding to the quadrants I, II, III, and IV. Extract the trigonometric function, from the mentioned table, from theintersection of the line corresponding to the initial function (of the angle θ) with thecolumn corresponding to the quadrant in which θ is found.1.3.5. ORIENTATIONS, THE RELATION BETWEEN COORDINATES AND ORIENTATIONS The orientation is the horizontal angle formed by some direction in the field, or onthe plan (map) with the direction of the topographic North, parallel to the 0X axis of thecoordinate system, and it is denoted by θ. We specify that the orientation is an oriented angle, measured in right-handeddirection, starting from the direction of the North, until the given direction isencountered. The Cartesian coordinates of a point A represent the distances from this point tothe rectangular axes of the chosen system, and are denoted by (XA, YA). As it has already been mentioned, the axis of ordinates in rectangular topographicsystems is denoted by 0x, and the axis of abscissas is denoted by 0y. In figure 1.5.a, b, c, d there are presented the four possible situations concerningthe relative position of points in the field. N The relations between orientations and coordinates result by expressing thetrigonometric functions of the angle θ, the computation of the unknown elements being N N B N B the known elements.possible depending on θAB ∆XAB X θAB A ∆YAB ∆XAB A ∆YAB A ∆XAB ∆XAB θAB θAB ∆YAB ∆YAB B A B Y 11 a) QUADRANT I b) QUADRANT II c) QUADRANT III d) QUADRANT IV Figure 1.5.
  17. 17. Hence we have: ∆YAB sinθAB = -------- (1.14) DAB ∆XAB cosθAB = -------- (1.15) DAB ∆YAB YB - YA tgθAB = ------- = ----------- => θAB from tables (1.16) ∆XAB XB - XA ∆XAB ctgθAB = -------- from tables (1.17) ∆YAB Knowing the orientation of the direction formed by two points A and B, θAB, thedistance between the points DAB and the coordinates XA and YA of one of the points, wecan compute from the relations (1.14) and (1.15) the relative coordinates ∆XAB and ∆YABof the second point (B), with respect to the known one (A). Thus: ∆XAB = DAB · cosθAB (1.15’) ∆YAB = DAB · sinθAB (1.14’) Since: ∆XAB = XB - XA (1.18) ∆YAB = YB - YA (1.19) 12
  18. 18. It will result that: XB = XA + ∆XAB (1.18’) YB = YA + ∆YAB (1.19’) From the relations (1.16) and (1.17) we can determine the orientation θAB,depending on the known coordinates of two points. Returning to figure 1.5.a, b, c, and d it can be seen that the sign of the relativecoordinates ∆XAB and ∆YAB indicates the position of the orientation in the topographiccircle. Therefore: in quadrant I + ∆X, + ∆Y, θI = α in quadrant II + ∆X, + ∆Y, θII = α + 100g (1.20) in quadrant III - ∆X, - ∆Y, θIII = α + 200g in quadrant IV - ∆X, + ∆Y, θIV = α + 300g α being the angle reduced to the first quadrant.1.4. PROBLEMS FOR TUTORIAL 11.4.1. SOLVED PROBLEMS Problem #1: The following data concerning the topographic points A and B wascollected as result of field measurements: LAB = 147.32 m; Z = 97g 31c; Also, the height of the point A is known: ZA = 300.53 m + n (mm); Determine: DAB, ∆ZAB, ZB. Remark: (n) represents the number of the student from the half-group. Problem #2: The following data is known concerning two points A and B: LAB = 121.56 m – n (m); 13
  19. 19. ∆ZAB = 2.454 m. Compute DAB and ϕ. Problem #3: Transform the following angles, from the given basis into therequired one: a) From centesimal degrees into sexagesimal degrees: 32g 43c36cc + nc 121g 52c37cc + ncc 237g 82c58cc + ng 321g 52c84cc - nc b) From sexagesimal degrees into centesimal degrees: 52°36’28” – n’ 131°52’42” + n” 236°58’36” – n” 321°31’43” + n’ Problem #4: Express the functions of the following angles through trigonometricfunctions of the angles from the first quadrant: 121g 36c42cc + ng 237g 52c38cc - nc 346g 82c56cc + nc 98°52’36” - n’ 231°36’48” + n” 303°21’52” + n’ Problem #5: Determine the angles θ corresponding to the following values: ∆XAB = 148.05 m + n (m); ∆YAB = - 136.21 m - n (m); ∆XAB = - 121.37 m + n (m); 14
  20. 20. ∆YAB = - 111.66 m + n (m); in centesimal and sexagesimal degrees. Problem #6: The coordinates of two points A and B are known: XA = 1321.52 m + n (m); YA = 3436.48 m; XB = 1464.49 m; YB = 3542.64 – n (m); Compute DAB and θAB. Problem #7: The coordinates of a point A, the distance to the point B and theorientation of the direction formed by the two points are known. Compute the coordinatesof the point B. XA = 1336.92 m ; YA = 2438.84 m; DAB = 184.52 m + n (m); θAB = 236g 51c36cc.1.5. EXAMPLE FOR SOLVING THE HOMEWORK (for (n) = 0) Problem #1: See figure 1.1, the relations (1.2), (1.1), (1.3), (1.3’). DAB = LAB cosϕ It can be seen that ϕ = 100g 00c00cc – Z = 100g 00c00cc – 97g 31c00cc = 2g 69c, Hence DAB = 147.32 m x cos2g 69c = 147.32 x 0.999107 = 147.19 m. ∆ZAB = LAB sinϕ = 147.32 m x sin2g 69c = 147.32 m x 0.042242 = 6.223; ZB = ZA + ∆ZAB = 300.53 m + 6.223 = 306.753 m. Problem #2: DAB = √L²AB - ∆Z²AB 15
  21. 21. Hence DAB = √1477.8017 = 121.54 m ∆ZAB 2.454 m sinϕ = ------- = ------------- = 0.020187562 => ϕ = 1g 28c00cc LAB 121.56 mProblem #3:a) 32g 43c36cc = see the appendix table 2, part one 32g = 28048’ 43c = 23’13”.2 36cc = 11”.66 32g 43c36cc = 29011’24”.86 ≅ 29011’26”b) 52036’28” = see the appendix table 2, part two 520 = 57g 77c77cc.00 36’ = 66c66cc.70 28” = 86cc.40 52036’28” = 58g 45c30cc.90 ≅ 58g 45c31cc.The other transformations will be solved similarly.Problem #4: sin 121g 36c42cc = cos 21g 36c42cc cos 121g 36c42cc = sin 21g 36c42cc tg 121g 36c42cc = - ctg 21g 36c42cc ctg 121g 36c42cc = - tg 21g 36c42cc sin 237g 52c38cc = - sin 37g 52c38cc cos 237g 52c38cc = - cos 37g 52c38cc tg 237g 52c38cc = tg 37g 52c38cc ctg 237g 52c38cc = ctg 37g 52c38cc 16
  22. 22. sin 346g 82c52cc = - cos 46g 82c52cc cos 346g 82c52cc = sin 46g 82c52cc tg 346g 82c52cc = - ctg 46g 82c52cc ctg 346g 82c52cc = - tg 46g 82c52cc For the angles expressed in sexagesimal gradation, we shall proceed in a similarmanner. Problem #5: DAB = √∆X2AB + ∆Y2AB = √(XB - XA) 2 + (YB - YA) 2 = = √(1464.49 – 1321.52)2 + (3542.64 – 3436.48)2 =√ 142.972+ 106.972 = 178.07 m; ∆YAB 106.16 tgθAB = ---------- = ----------- = 0.742533 ∆XAB 142.97 θAB = arctg 0.742533 = 40g 66c12cc Problem #6: ∆XAB = DAB cosθAB = 184.52 cos236g 51c36cc = 184.52 · (-0.939978) = -154.99 m; ∆YAB = DAB sinθAB = 184.52 sin 236g 51c36cc = 184.52 · (-0.542621) = -100.12 m; XB = XA - ∆XAB = 1336.92 – 154.99 = 1181.93 m; YB = YA - ∆YAB = 2438.84 – 100.12 = 2338.72 m. Problem #7: ∆YAB - 136.21 tgθAB = ------- = ----------- = - 0.920027 ∆XAB 148.05 - ∆YAB HavingtgθAB =---------- we are in quadrant II. ∆XAB Hence, we are looking for the angle β, for which ctgβ = 0.920027. We have for 0.920001 – ctg52g 65c10cc 17
  23. 23. 0.920027 – 0.920001 = 27 units 1cc …………………… 2.91 units Xcc …………………… 27 27 · 1cc Xcc.= ----------- ≅ 9cc 2.91 Hence ctg (52g65c10cc - 9cc) = 0.920027 => β = 52g65c01cc; => tgθAB = β + 100g = 152g65c01cc; The other computations are performed similarly. Proposed problems: Problem #1’: The following data is known concerning two topographic points Aand B: LAB = 136,54 m – n (m); ϕ = 2g51c32cc - nc; Compute DAB and ∆ZAB. Problem #2’: The following data was determined concerning two points C and D,as result of field measurements: LAB = 243.76 m + n (m); ∆ZAB = 12.345 m; Compute DAB and ϕ (the slope angle of the terrain). Problem #3’: Transform the following angles in the required basis: a) From centesimal degrees into sexagesimal degrees: 64g31c12cc + nc; 356g17c24cc - ncc; b) From sexagesimal degrees into centesimal degrees: 18
  24. 24. 126°31’15” + ng; 223°17’38” – nc.2. STUDYING THE THEODOLITE The content of the tutorial: In different topographic works there appearsfrequently the need to measure the angular elements of the terrain – horizontal andvertical angles – the theodolites being the optical instruments that serve this purpose. We shall study some types of these devices that are more frequently used in ourcountry.2.1. THE GENERAL CONSTRUCTION SCHEMA OF A THEODOLITE2.1.1. GENERAL NOTIONS, CLASSIFICATIONS So, the theodolites are optical devices that allow measuring angles with anapproximation of minutes (or seconds), being used in topographic measurements. The tacheometer-theodolites (the tacheometers) are optical devices that can beused to measure optically (hence, indirectly) both angles and distances, by the use ofstadia hairs traced on the reticule of the devices. B V(V’) L SB Clinome ter L AB DSB ϕB A B’ DAB α S ϕA D SA Be aring circle A’ Horiz ontal projection plan V(V’) S: station point; DiJ : the horizontal distance; VV: the vertical axis ofthedevice; A, B: aimed points; ϕA, ϕB : vertical angles; V’V’: the verticalofthestation point; LiJ : the slanted distance; α: horizontal angle. 19 Figure 2.1. Measuring angles with theodolites
  25. 25. Types of theodolites: - Classical type, with metallic graduated circles, equipped with decentralized reading devices; - Modern type, with crystal graduated circles and centralized reading devices; Measuring angles is performed with these devices placed in geodetic ortopographic station points, obtaining (figure 2.1): - Horizontal angles αi (the plan dihedral angle formed by the station point as angle apex and the directions that unite it with other topographic points); - Vertical angles (formed by some direction with its horizontal projection).2.1.2. THE PRINCIPLE SCHEMA – AXES AND COMPONENT PARTS Theoretical axes: - MAIN VERTICAL (VV) AXIS – The device can be rotated around this axis (rotation r1). During measurements, the VV axis coincides with the vertical of the station point (V’V’); - SECONDARY HORIZONTAL (HH) AXIS – The complex telescope- clinometer is rotated around this axis, on the vertical plan (rotation r2); - AIMING AXIS OF THE TELESCOPE (Γ0) – Materializes the aiming line of the topographic points. These three axes meet in one point (C v), called the aiming center, in the followingway: HH ⊥ VV, Γ0 ⊥ HH (figure 2.2). The main parts of the device are the following (figure 2.3): THE GRADUATED HORIZONTAL CIRCLE (or the bearing circle) ((1) infigure 2.3) is a metallic disc (in the case of classical theodolites) or a crystal disc (in thecase of modern theodolites), having a diameter of φ 70 ÷250 mm, and being graduated onthe entire circumference with centesimal (sexagesimal) degrees, increasing towards right-handed direction, being used for determining horizontal angles. 20
  26. 26. THE GRADUATED VERTICAL CIRCLE (or the clinometer), having the samecharacteristics as the previous one, being used for determining vertical angles ((2) infigure 2.3). 0 V Γ2 H H Cv Γ Γ1 V Figure 2.2. The main axis and rotations of the theo-tacheometer THE ALIDADE ((3) in figure 2.3) is a plate in the interior of the bearing circle,which bears two diametrically opposed reading indexes (i1 and i2). With the use of a distaff ((a) and (b)), the alidade supports the complexclinometer-telescope. THE TELESCOPE ((4) in figure 2.3) is the optical device with the use of whichaiming the topographic points is possible. THE BASE ((5) in figure 2.3) consists of a metallic support, equipped with threefoot screws (C) and a threaded inlet (d), which allows fixing the device on the trivet, bythe use of a screw. 21
  27. 27. 0 4 H H 2 Cv r a b i2 i1 1 5 c d c Figure 2.3. The main parts of the theodolite Operating the foot screws, the device can be brought to horizontal. One difference between measuring horizontal angles and vertical angles is thatwhile in the first case, the bearing circle is fixed and the reading indexes are mobile (withthe alidade and the telescope), in the case of vertical angles, the clinometer is mobile(with the telescope) and the indexes are fixed.2.1.3. THE DETAILED SCHEMA OF THE THEODOLITE (figure 2.4.) 1. THE TELESCOPE (1’: LENS, 1”: OCULAR, 1”’: APPROXIMATE AIMING DEVICE, 1IV: IMAGE FOCUSING – CLARIFYING MUFF, 1V: RETICULE CLARIFYING MUFF, 1VI: RETICULE). 2. VERTICAL GRADUATED CIRCLE – CLINOMETER (2’: INSCRIPTION THAT INDICATES THE POSITION OF THE DEVICE). 3. HORIZONTAL GRADUATED CIRCLE – BEARING CIRCLE (the area in which it can be found is indicated). 22
  28. 28. 4. THE ALIDADE (4’: DISTAFFS FOR SUPPORTING THE TELESCOPE WITH CLINOMETER). 5. THE BASE (5’: FOOT SCREWS, 5”: THE TENSION PLATE OF THE BASE). Parts that ensure the correct position of the device for measurements: 6. THE AIR-BUBBLE LEVEL FROM THE HORIZONTAL CIRCLE (6’: ADJUSTING SCREW OF THE LEVEL). 7. THE SPHERICAL LEVEL (FOR APPROXIMATE HORIZONTAL SETTING). 8. THE AIR-BUBBLE LEVEL FROM THE VERTICAL CIRCLE (8’: THE REFLECTING MIRROR, 8”: MICROMETRIC SCREW). Parts that ensure reading angular values: 9. MICROSCOPE (9’: THE OCULAR OF THE MICROSCOPE, 9”: READING CLARIFYING MUFF, 9”’: RELFECTING MIRROR). Parts that ensure the motion of the device, the motion of the telescope withclinometer around the horizontal axis: 10. LOCKING SCREW (CLAMP) (10’: SLOW MOTION SCREW). Parts that ensure the motion of the device around the vertical axis: 11. LOCKING SCREW (CLAMP) (11’: SLOW MOTION SCREW). Parts that ensure the motion for recording angular values: 12. THE CLAMP FOR FASTENING THE BEARING CIRCLE TO THE ALIDADE AND FOR LOCKING THE RECORDING MOTION. Parts that ensure fastening the device to the base: 13. CLAMPING SCREW. TRIVET (t) PLATE (t’) FEET (t”) SCREW FOR FASTENING THE BASE TO THE TRIVET (t”’) PLUMB-BOB WIRE THAT ENSURES CENTERING THE DEVICE ON THE STATION POINT (tIV). 23
  29. 29. 2 1 8’ 1’ 8 10 2’ 4’ 1IV IV 10’ 1 8” 6 1’’’ 9, 9” 1IV 1” 11 6’ 1 V 9’’’ 11’ 4 9’ 7 13 3,4 12 5 t,t’ 5’ 5” t” t’’’ tIV Figure 2.4. The detailed schema of the tacheometer-theodolite THEO 030 Carl Zeiss Jena (Germany) – position I2.1.4. THE DETAILED SCHEMA OF OTHER TYPES OF THEODOLITES In the volume GENERAL TOPOGRAPHY – problems and practical applicationsII, we present the detailed schema for other types of theodolites, frequently used onconstruction sites. 24
  30. 30. 2.2. WORKING PROCEDURE FOR THE THEODOLITE2.2.1. WORKING PRINCIPLES It should be specified that the theodolite, being a very sensitive optical device,should be kept in certain conditions, being carefully and gently handled, but not beforethoroughly knowing the purpose of each part. The device should be kept away from: hits, shocks, moisture, high temperaturedifferences, high pressures, dust, vibrations, etc. After operating the screws (clamps) 10 and 11, only small amplitude motion ofthe device will be performed, with the use of the screws 10’ and 11’.2.2.2. VERIFYING THE DEVICE In order to ensure the precision of measurements, the theodolites must satisfy aseries of required conditions. The errors during angle measurement, due to the device, can be classified into thefollowing categories: ERRORS DUE TO: 1) Construction imperfections of the device; 2) The degradation, wrong-going, destruction of some component parts; 3) The alteration of component parts. The errors from the first category can be significantly improved by choosing someadequate working methods. When the device is found in a state characteristic to thesecond category, the operator will turn to shops specialized in repairing optical devices.The errors from the third category can be identified and reduced by adjustment. All problems presented above are discussed in detail in the lecture and we do notthink it is necessary to treat them here. It should be specified that the device should be periodically verified-rectified, andannually, the devices are presented to specialized shops for verification. 25
  31. 31. 2.2.3. PLACING INTO THE STATION It is performed through some operations that have the purpose to ensure thecoincidence of the vertical axis of the device (VV) with the vertical of the station point(V’V’). The working steps are the following: 1) Centering the device, on the vertical of the station point, is performed in the following way: - Place the trivet above the station point, watching the plumb-bob wire to be as close to the point as possible, the plate of the trivet to be horizontal, in the same time ensuring the stability of the trivet, by successively pushing the foot on the shoes of the trivet; - Remove the device from the casing, place it on the plate of the trivet and fasten it temporarily with the screw of the trivet; - Moving the device on the plate (or, if this is not possible, adjusting the height of the feet of the trivet), the plumb-bob wire is brought to the vertical of the station point. V Instrument Plate n Clamping screw Plumb- bob wire Trivet Shoe V’ A V’ Figure 2.5. Centering 26
  32. 32. 2) The horizontal setting of the device is ensured in two steps (figure 2.6). 2 2 1 I 1 II Figure 2.6. Horizontal setting Therefore, the air-bubble level is brought to be parallel, with the use of two footscrews, which are operated in separate directions (1 or 2), so that the bubble of the levelis brought between the benchmarks. Then, we rotate the device such that the level tobecome normal to the previous position, and operating the third foot screw, the air-bubbleis brought again between the benchmarks of the level. If the level is adjusted, after these operations the device is set horizontally, that is,its horizontal axis should be parallel to the horizontal of the location. This can be checkedbringing the level in different positions, the bubble remaining between the benchmarks.Otherwise, the level does not operate adequately, therefore its verification andrectification is necessary, after which the horizontal setting operation should be doneagain. We mention that in the case of devices that are equipped with spherical level, anapproximate horizontal setting is possible to be previously performed, bringing thebubble of this level in the benchmark circle. 27
  33. 33. After the device was horizontally set, the centering operation is performed again,maybe optically, and then the horizontal setting is verified again, and so on, until the twooperations have satisfactory results. In this moment, the device is ready for measurements, its vertical axis (VV) beingidentical with the vertical of the station point (V’V’). Centering can be performed with the use of the centering stick or optically, withthe so-called “optical-plumb wire” (which is an optical device, inserted in the base ofmedium and high precision devices).2.2.4. AIMING AND POINTING Their purpose of to bring the image of the point (of the aimed signal) in the centerof the reticule and they are performed in the following way: a) Clarify the cross-hairs (figure 2.7.a), which can be done aiming a bright background with the telescope and operating the clarifying muff of the reticule (1V figure 2.4); b) Aim approximately the signal (figure 2.7.b), overlapping the device 1”’ (figure 2.4) on the free image of the signal; c) Lock the motion of the device, with the screws (clamps) 10 and 11 (figure 2.4); d) Clarify (focusing) the image of the aimed signal, operating the muff 1 IV (figure 2.4); the image of the aimed signal will appear in the center of the reticule; e) Point the signal operating the screw 10’ (obtaining the image (2)) and then 11’ (obtaining the image (3) of the pointed signal). Then, read and record the angular values. Handling conveniently the reflecting mirror 9”’ (figure 2.4), the image ofgradations is lit up. The muff 9” (figure 2.4) is used to clarify these gradations, and thenthe reading is performed. 28
  34. 34. Operating muff IV a. Clarifying cross-hairs Aimed signal Aimers Telescope b. Approximate aiming with various devices 10’ 11’ 1 2 c. Pointing 3 Figure 2.7. Aiming and pointing2.2.5. DEVICES FOR READING ANGULAR VALUES ON THE THEODOLITE We shall discuss only those devices that the medium precision topographicinstruments that are frequently used by the construction engineer are equipped with. a. The microscope with lines has a fix hair carved on its reticule, whichoverlaps on the image of the graduated circles, ensuring the reading. The precision of the devices is equal to the smallest division: 1g 100c P = ------- = ---------- = 10c 10 div 10 div The reading consists of an integer part (PI) and an approximate part (PII). On the horizontal circle (Hz) we have: PI = 274g30c 29
  35. 35. PII = 3c CHz = 274g33c On the vertical circle (V) we have: PI = 302g50c PII = 7c CV = 302g57c V 302 303 READINGS V = 302g54c HZ = 274 33c g 274 Hz 275 Figure 2.8. The microscope with lines b. The scale microscope has a graduated scale carved on its reticule, equal invalue to the apparent value of a division on the graduated circle. The scale is divided into 100 divisions (minutes) grouped by ten. The precision of the device (and the smallest division) is: 1g 100c P = --------- = ----------- = 1c 100 div 100 div The readings are (figure 2.9.a and b): On the horizontal circle (Hz): PI = 372G67C PII = 50CC CHZ = 372G67C50CC On the vertical circle (V): 30
  36. 36. PI = 267G14C PII = 50CC CHZ = 267G14C50CC 1 267 2 V 267 V = 26714c50 cc g 266 0 1 2 3 4 5 6 7 8 9 10 6 372 7 Hz = 372 67 40cc g c b- reading details 0 1 2 3 4 5 6 7 8 9 10 373 Hz 372 a - readings V = 267g14c 50cc Hz = 372g67c 40cc Figure 2.9. The scale microscope2.2. THE HOMEWORK OF THE TUTORIAL Problem #1: Draft the detailed schema of a theodolite, explaining the purpose ofeach component part and the conditions that the theoretical axes must satisfy. Problem #2: Set up a station, parsing all the steps from centering until reading theangular values, explaining the purpose of each operation, the method and the parts thatare used for it. Problem #3: Draft the schemas for the following readings on the microscope withlines (V = 321g32c + nc; Hz = 268g52c + nc) and on the scale microscope (V = 321g32c20cc+ ncncc; Hz = 268g52c 60cc + ncncc). Remark: Problems 1 and 3 represent the topic of the paper, and problem 2 will beperformed practically. 31
  37. 37. 3. MEASURING ANGLES WITH THE THEODOLITE The content of the tutorial: In the previous tutorial we have studied thetheodolite and its usage, therefore, we shall see in the sequel how this device is used intopographic measurements.3.1. THE NATURE OF TOPOGRAPHIC ANGLES We call theodolite station placing it on the vertical of a topographic point (calledstation point) and performing some topographic measurements from this position. In what follows we shall present the angles that can be measured from such apoint: a) HORIZONTAL AIMS (c): formed as horizontal angles between the origin on the bearing circle and the vertical aiming plan towards a given point (figure 3.1.a), being the reading performed on the bearing circle. b) HORIZONTAL DIRECTIONS (ω): in the case when the horizontal graduated circle (the bearing circle) is placed with the origin towards one of the given points, then they are measured on the direction of the second point, reading the angle formed by the station as angle apex, with the directions towards the given points (figure 3.1.b). The horizontal angles (α) are formed between the vertical plans for aiming two given points. It results in value the difference of the aims performed to the two points. c) ZENITHAL ANGLES (Z): are the angles read usually on the vertical graduated circle, being formed by the vertical of the location (of the station) with the aimed direction (figure 3.1.e). d) VERTICAL ANGLES (V): are formed by some direction with its horizontal projection (figure 3.1.c). e) SLOPE ANGLES (ϕ): of the terrain are vertical angles, obtained by aiming a given point, at the height i of the device, in the station; they represent the 32
  38. 38. angle formed by the direction determined by the station and the aimed point, with the horizontal of the station point. In figure 3.1.d. it can be seen that in this case we measure an angle ϕ’ equal to the slope angle ϕ of the terrain, which are angles with parallel sides. In the general case, there are measured the angles from the categories a and e, theother ones being computed based on the measured values. 1 α 100g 1 ω 100 g 2 C 0g : Origin on the 0g bearing circle; S S: Station point; S 200g 1: Aim ed point; 300 g 200 g ω : Horizontal 1, 2: Aimed points; direction 300g a) α: Horizontal angle b) Aimed direction Aimed direction s UV Horizontal s= ϕ’ Device i AP ϕ horizon i i Station point ϕ horizontal i: The height of the instrument in the station; ϕ: Angle measured with the theodolite; s: The aim ing height; ϕ’ : Slope angle of the terrain; UV:Vertical angle i ≠ s ϕ’ = ϕ . c) d) V ZI i ϕ’ i ZII ϕ S V Z : zenithal angle read in position I I Z : zenithal angle readin position II II ϕ’I + ϕ’ II 100 g - ZI+ ZII - 300 g e) ϕ= ϕ’ = ------- = --------------------------- 2 2 Figure 3.1. Topographic angles 33
  39. 39. We mention that a zenithal angle is the angle of type Z I, the angle ZII beingassimilated as angle measured in the second position of the telescope (the clinometer onthe right of the telescope).3.2. ANGLE MEASURING METHODS As result of the operations of placing the theodolite in the station, presented in theprevious tutorial, the device is ready for measurements, satisfying the followingconditions: - The main (VV) and secondary (HH) axes of the device become vertical, and horizontal, respectively; - The (VV) axis coincides with the vertical (V’V’) of the station point; - The device placed on the trivet is stable, in completely fixed position, little sensitive to the touches during handling; - As result of the verification and rectification procedures of the device, the correctness and the precision needed for using it, the integrity and sound functioning of each component part have been ensured. The topographic angles can be measured through various methods, chosendepending on the following factors: - The precision needed and the purpose of the measurement; - The existing equipment; - The number of points and measured angles; - The distance to the aimed points; - The condition of the atmosphere, and in general, the conditions in which the measurement is performed; - The existence or the lack of vegetation that could prevent or make difficult to perform some aims; - The available time, etc. 34
  40. 40. 3.2.1. THE CASE OF MEASURING ONE ANGLE a. Angle computed as difference of readings We mention that for all angle-measuring procedures listed here, we assume that the device is set up for measurement, in the theodolite station. In this case, we have the following steps (figure 3.2): 1) Aiming and pointing the first point (1) in the position I of the telescope (the clinometer on the left of the telescope); 2) Recording the reading on the bearing circle (CI1) and maybe on the clinometer (ZI1); 3) Rotating the device in right handed direction and aiming the second point in position I; 4) Recording the readings (CI2, ZI2); 5) Turning the device in position II and aiming – pointing the second point; 6) Recording the readings (CII2, ZII2); 7) Rotating the device in left handed direction and aiming the first point in position II; 8) Recording the readings (CII1, ZII1); 9) Verifying the readings performed and recorded in table (3.1), as it follows: - CI1 with CII1, and CI2 with CII2, respectively, must differ with 200g to which the permissible error specified for each type of device is added; - ZI1 with ZII1, and ZI2 with ZII2, respectively should sum up to 400g plus the error specified above. Only after these checkings have been performed we can leave the station. Table 3.1Station Aimed Readings on the Mean Horizontal Readings on the Vertical point point bearing circle directions angle clinometer angle Horizontal directions Position Position Position Position I II I II 1 2 3 4 5 6 7 8 9 1 CI1 CII1 C1M ZI1 ZII1 V1 S1 α= 2 CI2 CII2 C2M ZI2 ZII2 V2 35
  41. 41. The bearing circle of The origin on the the theodolite bearing circle V(V’) 1” g Direction 1 V(V’) ≡0The bearing 0g 100g Direction 1 1”circle of the S’ 1” 300g 0gtheodolite 2” 1’’’ 300g 200g Direction 2 S’ 2”’ 200 2” g 1 100g 2” 1 1’ 1’ 2 α α S 2’ S 2 Horizontal projection Horizontal a) plan V(V’) projection plan V(V’) 2’ b)VV:The vertical of the device VV = V’V’(measuring condition) α = Direction 2V’V’:The vertical of the point SS: Station point S’1’’’ The horizontal formed by the S’2’’’ center of the bearing circle and theS’: The center of the bearing circle1,2 - Measured points signals1’,2’- The projection of the points on an imaginary horizontal plan1”,2”-Aims on the signals placed in these pointsIt can be seen: ∠1’’’ ’2’’’ = ∠ 1”S’2” = ∠ 1S2 =α = Direction 2 – Direction 1 S 4” V(V’) 1” d) Position II 4 Position I 1 i = the height of the device in 3” the station; 4” V3 1’ s = the aiming S i 2” 3’’’ height = 11”= 22” =….. α If i = s => V =ϕ 2 ϕ3 δ i = s => V ϕ ≠ 2’ 3’ V(V’) Figure 3.2. Angle measuring methods 36
  42. 42. 1 100g 2 CI2 C I1 0g 200g The origin on the bearing S1 C II1 CII2 circle Bearing circle 300g Figure 3.3. Data processing is performed in the following way: (CI2) + (CII2)² 1 I g - Column (5): C M = (C ) + ------------------ 1 2 Therefore, the degrees from position I are recorded and is computed the mean ofthe minutes from the two positions. - C2M is computed similarly; - Column (6): C2M – C1M, with the remark that if C 2M < C1M then 400 g are added to the first one (α= C2M + 400 g – C1M); 100g – ZI1+ ZII1 - 300g 1 - Column (9): V = -----------------------------; 2 - V2 is computed similarly. In table 3.2 we present another method for processing the measured values, bycomputing the mean of the horizontal angles (the vertical ones being computed as in thefirst table). Measuring the angles in the two positions of the telescope has the followingconsequences: - Removing (or reducing) the instrumental errors; - Increasing the measuring precision; 37
  43. 43. - The mutual control of the values measured in the two positions.Table 3.2 Readings on Horizontal angles Readings on the Vertical angle Station point Aimed point the bearing clinometer V circle (horizontal directions) Position Position αI αII α Position I Position I II II 100g-ZI1+ZII1-300g I II I II 1 C1 C Z Z V1=------------------- 1 αI + αII 1 1 2 S1 I C 2-C I 1 II C 2-C II 1 α= -------- 100g-ZI2+ZII2-300g 2 CI2 CII2 2 ZI2 ZII2 V2=------------------- 2 b. Angle measured through the method of “zeros in coincidence” (with zero origin on the bearing circle, towards the first point) Between the operation of setting up the device in the station and the firstoperation of measurement presented in the previous case, we insert the operation ofbringing the reading index of the alidade (in position I) in coincidence with zero, on thebearing circle. This can be obtained in the following way: - Rotate the device around the VV axis, watching the reading microscope on the horizontal circle; - When the gradation 0g appears in the microscope, lock the motion by the use of clamp 11 (figure 2.4); - Operating the slow motion screw in horizontal plan (11’), bring the zero gradation in coincidence with the reading index (the line of the microscope or 0 on the scale); - Lock the recording motion by the use of clamp 12 (figure 3.4), the bearing circle being locked now by the alidade; - Aim and point the first point (step I); - Unlock clamp 12; - Parse the steps 2-8 presented previously; 38
  44. 44. - The verification (9) in this case is that C II1 = 200g ± e, where e = the permissible error for the device used for measuring angles. In this case: αI = CI2; αII = CII2 - 200g αI + αII And α = ---------- 2 Or: C1M = 0g ± e/2; (CI2)c + (CII2)g 2 1 g C M = (C 2) + ------------------- 2 And: α = C2M - C1M The table to be used can be chosen in this case, too, between tables 3.1 or 3.2.3.2.2. MEASURING MORE ANGLES FROM ONE THEODOLITE STATION THE METHOD OF THE HORIZON TOUR Let S be the station point and 1, 2, 3, and 4 the points the have to be aimed fromthis station (figure 3.4). 4 The origin on the 1 The origin of aims bearing circle 0 g 10 0g 30 0g 200g 2 Position I Position II 3 Figure 3.4. Measuring by horizon tour 39
  45. 45. In this figure it can be seen that usually the ORIGIN OF AIMS does not overlapon the ORIGIN OF THE BEARING CIRCLE. But we know from the previous case that this can be done by applying themeasuring method with “zeros in coincidence”. The horizon tour in position I is performed in right handed direction, on the route1-2-3-4-1, and in position II it is performed in left handed direction, on the route 1-4-3-2-1. After the device is set up in the station, the following operations are performed: - Aiming and pointing the signal from point 1 (maybe with “zeros in coincidence”); - Recording the values from the bearing circle (CIi1) and from the clinometer (ZI1); - Unlock the alidade and the telescope with the vertical circle (the bearing circle stays fixed) and aim the point 2; - Record the values (CI2), (ZI2); - The values (CI3, ZI3), (CI4, ZI4) are obtained similarly; - Aim again the point 1, obtaining (CIf1). 1) The measured data are inserted in the columns 3 and 4 (readings on the bearing circle) and 11 and 12 (readings on the clinometer); 2) The computation for the mean of the directions: (CIi1)c + (CIIi1)c i I g C M1 = (C ) + -------------------- i1 2 3) The computation of the error (e) is: e = CfM1 – CiM1 4) The computation of the corrections: - The total correction: Ct = - e Ct - The unitary correction: Cu = ---- n (n = the number of measured points); 40

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