Module – 2 Tacheometry 
                Tacheometry: 
                Basic principle, types of tacheometry, distance equation for horizontal and 
                inclined line of sight in fixed hair method, problems.                                     
                Tacheometry:  
                       Tacheometric        is     also     called 
                tachemetry or telemetry. It is the branch of 
                surveying  in  which  both  horizontal  and 
                vertical  distances  between  stations  are 
                determined from instrumental observations. 
                Tacheometric is also called tacheometry. The                      Theodolite 
                method  is  rapid  and  convenient.  Although 
                the  accuracy  of  tacheometry  does  not 
                compare favourable with that of chaining but 
                it is best adapted in obstacles such as steep 
                ground, deep ravines, stretches of water which make chaining difficult.   
                       The  instrument  employed  for  tacheometric  purpose,  is 
                generally  know  as  a  tacheometer  which  is  similar  to  theodolite                   Stadia hair 
                having diaphragm fitted with two additional horizontal wires, called 
                stadia hairs.   
                Situation where tacheometry can be used: 
                     When obstacles like river, broken ground, stretches of water,              Diaphragms 
                       tacheometer gives speed and accuracy to work. 
                     In rough country where measurement of horizontal and vertical distances are 
                       difficult inaccurate and slow. 
                     In locating contours and filling details in a topographic survey, this method is 
                       fast and best. 
                Purpose: 
                       The primary project of tacheometry is the preparation of contoured plans. It 
                is considered to be rapid and acurate in rough country and has thus been widely 
                used by engineers in location surveys for railways, canals, reservoirs, etc. Whenever 
                surveys of higher accuracy are carried out, tacheometer provide a good check on 
                distances measured with a tape or a chain. 
                                Difference between Theodolite and Tacheometer 
                                Theodolite                                    Tacheometer 
                It is used for measurement of horizontal  It is used for measurement of horizontal 
                and vertical angle.                             and vertical distances. 
                In  Theodolite  survey  distances  are  In             Tacheometer         survey      direct 
                measurement by chain or tape.                   measurement of distances are possible.  
                Suitable for plane and hilly area with less  Suitable  in  case  obstacles  like  never 
                obstacles.                                      broken ground.  
                More statices are required in Theodolite  Less           statices     are     required      in 
                survey                                          Tacheometer survey  
                 
                G. Ravindra Kumar, Associate Prof, CED, Govt Engg College, Chamarajanagar         Page 1 of 23 
                  Module – 2 Tacheometry 
                    Systems of tacheometric measurement: 
                  The  various  systems  of  tacheometric  measurements  may  be  classified  as 
                  under: 
                      1.  The stadia hair systems 
                      2.  The tangential systems 
                  The Stadia hair systems: the stadia hair system may further be divided into two:  
                      1.  Fixed hair method 2. Moveable hair method. 
                  Fixed hair method: In this, stadia hairs are kept at fixed interval and the intercept 
                  on the levelling staff (or stadia rod) varies, depending upon the horizontal distance 
                  between the instrument station and the staff. The intercept used in computation is 
                  deduced by subtracting the lower stadia reading from the upper stadia reading. 
                  When the staff intercept is more than the length of the staff, only half intercept is 
                  read, which is equal to the difference between central stadia hair reading and the 
                  lower/upper stadia hair reading. 
                           This method can be suitable employed even when horizontal sight are not 
                  possible. For inclined sights, reading may be taken by holding the staff either vertical 
                  or normal to the line of sight. This is most common method of tacheometry and the 
                  name stadia hair method generally refers to this method. 
                           Movable hair method: In this method, the intercept on the levelling staff is 
                  kept constant and the distance between the stadia hairs are variable. Targets on the 
                  staff at a known distance apart are fixed and the stadia hairs are adjusted such that 
                  the upper hairs bisects the upper target and the lower hair bisects the lower target, 
                  In  this  case  a  provision  is  made  for  the  measurement  of  the  variable  interval 
                  between the stadia hairs. For inclined sights, readings may be taken by holding the 
                  staff either vertical or normal to the line of sight as in the case of fixed hair method.  
                  2. The  tangential  method:    In  this  method,  the  stadia  hairs  are  not  used. 
                      Readings on a staff are taken against the horizontal cross hair. To measure the 
                      staff intercept, two pointing of the telescope are therefore, necessary. Full metre 
                      value readings are taken to avoid the decimal part and also for simplification of 
                      computations. This method is not adopted as two vertical angles are required to 
                      be measured for one single observation. 
                  Dist
                  anc                               v                                     u 
                  e               b                                 B’                                           A 
                  for         i                                O 
                  mul               c                                                                            C   S 
                  a                                                                  F 
                  for             a                       d         A’      f                                      B 
                  hori                                                        D 
                  zon
                  tal sight: 
                   
                  G. Ravindra Kumar, Associate Prof, CED, Govt Engg College, Chamarajanagar                       Page 2 of 23 
                Module – 2 Tacheometry 
                 
                 
                 
                 
                 
                 
                 
                Assuming  ‘O’  is  the  optical  centre  of  the  object  lens  of  the  external  focusing 
                telescope:  a, b, c represent the three horizontal hairs. A, B, C represent respective 
                points on the staff which appear cut by three hairs, ‘ab’ is the length of the image on 
                the staff intercept AB.  
                Let,   f = Focal length of the object lens 
                       i = Stadia hair interval ab 
                       S = staff intercept AB 
                       D = Horizontal distance from the axis of the theodolite to staff 
                       d = The distance between the optical centre of the object glass and the axis  
                             of the theodolite. 
                With the basic knowledge of optics, it is clear that the rays from A and B which pass 
                through the exterior principal focus of the objective F, travel parallel to the principal 
                axis after refraction at A’ and B’. 
                Proof: 
                       A’B’ = ab = stadia hair distance = i 
                From similar triangles ABF and A’B’F 
                         CF   OF  
                         AB A'B'         
                         CF  OF AB
                                 A'B'
                       AB = s,         OF = f,       A’B’ =i.        
                                CF  f s 
                                      i
                But D = CF + f + d =   f s  f  d 
                                           i
                    f
                                 
                D  i s   f  d  Tacheometric  Distance  equation formula      
                D  Ks C Tacheometric  Distance  equation formula      
                K and C are Tacheometric constants, K is multiplying constant and C = Additive 
                constant. 
                Determination of tacheometric constants (K and C) : 
                                                                                                        
                                                                                                        
                                                                                               S        
                                                           S                 S                  4
                                          S                 2                 3                         
                                           1
                                                                           0.924 
                G. Ravindra Kumar, Associate Prof, CED, Govt Engg College, Chamarajanagar    B    Page 3 of 23 
                  A                 25 m              50 m             75 m              100 m 
                          25 m               25 m              25 m            25 m 
               Module – 2 Tacheometry 
                
                
                
                
                
                
                   1.  Measure a line AB 100 meters long on a fairly level ground and fix pegs at 
                      25m intervals. 
                   2.  Set up the instrument at A and centre it over the ground point accurately. 
                   3.  Obtain the staff intercepts s1, s2, s3 and s4 by taking stadia reading on a staff 
                      held vertically at each peg, keeping telescope horizontal by setting the vertical 
                      circle verniers to read zero. 
                   4.  Substitute  the  different  values  of  D  and  s  in  the  tacheometric  distance 
                      formula, i.e. D= k s + C, to get four quadratic equation. 
                   5.  Solve  the  quadratic  equations  in  pairs    to  get  the  values  of  tacheometric 
                      constants. 
                   6.  Mean values are the required values of the constants. 
                                        f 
                                                    
                         D  K S  C      S   f  d    
                                        i 
                                                       
                     f = focal length of the object lens = distance between the object lens and the 
                              plane of cross hairs, 
                      d = distance between the object lens and the vertical axis of  the theodolite. 
                      i =  Stadia interval. 
                
               Problem: (A.M.I.E, 1972 summer) 
               Determine the constants of a tacheometer from the following taken with it:  
                Distance of staff from the                   Reading against stadia wires 
                tacheometer vertical axis              Lower wire                     Upper wire 
                           30m                           1.086m                         1.383m 
                           60m                           0.924m                         1.521m 
                                                            
               Solution: 
               Staff intercept (S1) = THR – BHR =  1.383 – 1.086 = 0.297m                THR = Top hair reading 
               Staff intercept (S2) = THR – BHR =  1.521 – 0.924 = 0.597m                MHR = Middle hair reading 
                                                                                         BHR = Bottom hair reading 
                                                                               1.521 
                
                                                        1.383                          S  
                                                                S                       2
                                                        1.086    1
                                                                               0.924 
                                     30 m 
                                                60 m 
                              D  = KS  + C ---- (1),       D  = KS  + C ----(2) 
                               1      1                      2      2
               G. Ravindra Kumar, Associate Prof, CED, Govt Engg College, Chamarajanagar       Page 4 of 23