Now. How to count directional angles.
Directional angles are calculated by the formula:
for the right angles:
αn = αn-1 +180 –βn
for left corners:
αn = αn-1 –βn +180
αn-1 - Directional angle of the previous side
βn - measured angle right along the way ( for lines 12, 23, 34, 45, 51 - Average angle - 1 part of work table), (for lines AB - in your outline on the picture)
So you count the directional angles:
Important! You can calculate the directional angle of line 12 in two ways:
1- from the directional angle of line B1 (pink color)
2- from the directional angle of line 51 (black color)
This is calculated in two ways for control.
Select control angles in red (they should be the same)
F ormulas for calculating the diffraction angles
αB1=αAB+180-∠ βmeasuret B right=
αB1=αAB+∠ βmeasuret B left - 180=
α12=αB1+180-∠ βmeasuret 1 right0 =
α23=α12+180-∠ βmeasuret 2 right =
α34=α23+180-∠ βmeasuret 3 right =
α45=α34+180-∠ βmeasuret 4 right =
α51=α45+180-∠ βmeasuret 5 right =
α12=α51+180-∠ βmeasuret 1 right =
αAB– look your variant
βmeasuret B right –take from outline, Example for 4 variant βmeasuret B right = 197° 19'
βmeasuret B left –take from outline, Example for 4 variant βmeasuret B left = 162° 41'
because imagine that you are going from point A to point B along the line AB and then you go from point B to point 1 along the line B1. If you stop at point B. Angle 197 19 will be on the right side, and angle 162 41 will be on the left side. Therefore, if the corners are right and left. They are calculated according to different formulas.
Variant 4/ outline
Next For 4 variant in formula α12=αB1+180-∠ βmeasuret 1 right0,
βmeasuret 1 right0 = 240° 49' (the angle of the right side between the lines B1 and 12 at point 1)
But in formula α12=α51+180-∠ βmeasuret 1 right
βmeasuret 1 right - this is the angle between lines 51 and 12 at point 1, this angle you calculated in table 1. Example for variant 4 - βmeasuret 1 right = 68°24'30''
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Table 1.
log of angles measured by theodolite / variant 4
| Point Numbers | КП(КЛ) RR(RL) FR(FL) | Отсчеты по горизонтальному кругу Result in horizontal ring Result in horizontal face | Угол в полуприеме Half Reception Angle | Средняя величина угла Average angle | Длина линии, D. м distance, m | Tilt angle a | Горизнтальное проложение , (d) Horisontal line Proection on the level ground | |
| Станции station | Наблюдаемые точки Points | |||||||
| 1 | 5 | КП | 215°48'0'' | 68°24'30'' | (1-2) | - | 101,100 | |
| 68°24'0'' | 101,1 |
| ||||||
| 2 | 147°24'0'' | 101,1 | ||||||
| 5 | КЛ | 159°41'0'' | 101,1 | |||||
| 68°25'0'' |
| |||||||
| 2 | 91°16'0'' | |||||||
| 2 | 1 | КП | 184°11'0'' | 69°56'30'' | (2-3) | - | 70,260 | |
| 69°57'0'' | 70,25 |
| ||||||
| 3 | 114°14'0'' | 70,27 | ||||||
| 1 | КЛ | 308°54'0'' | 70,26 | |||||
| 69°56'0'' |
| |||||||
| 3 | 238°58'0'' | |||||||
| 3 | 2 | КП | 264°56'0'' | 242°59'0'' | (3-4) | - | 63,610 | |
| 242°59'0'' | 63,6 |
| ||||||
| 4 | 21°57'0'' | 63,62 | ||||||
| 2 | КЛ | 123°16'0'' | 63,61 | |||||
| 242°59'0'' |
| |||||||
| 4 | 240°17'0'' | |||||||
| 4 | 3 | КП | 336°38'0'' | 56°17'0'' | (4-5) | - | 72,190 | |
| 56°17'0'' | 72,2 |
| ||||||
| 5 | 280°21'0'' | 72,18 | ||||||
| 3 | КЛ | 277°49'0'' | 72,19 | |||||
| 56°17'0'' |
| |||||||
| 5 | 221°32'0'' | |||||||
| 5 | 4 | КП | 117°47'0'' | 102°25'0'' | (5-1) | 3°35'0'' | 133,399 | |
| 102°25'0'' | 133,68 |
| ||||||
| 1 | 15°22'0'' | 133,64 | ||||||
| 4 | КЛ | 47°58'0'' | 133,66 | |||||
| 102°25'0'' | 133,68 | |||||||
| 1 | 305°33'0'' | |||||||
PART 3
Calculate the angular residual
1. rewrite Average angle (or measure angles), and Horizontal line Proection on the level ground from table 1 to table 2, as in the example.
EXAMPLE
Table 2/1
Coordinate calculation
| № точки | Углы | Горизнтальное проложение , (d) Horisontal line Proection on the level ground | Приращения | Координаты | |||||||
| Измеренные углы / Average angle/ measured angles (βmeasuret) | Исправленные углы/ corrected angles (βcorrected) | Дирекционные/ directional angles (α) | Вычисленные | Исправленные | X | Y | |||||
| ∆X | ∆Y | ∆X | ∆Y | ||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
| 1 | 68°24'30'' | ||||||||||
| 101,100
| |||||||||||
| 2 | 69°56'30'' | ||||||||||
| 70,260 | |||||||||||
| 3 | 242°59'0'' | ||||||||||
| 63,610 | |||||||||||
| 4 | 56°17'0'' | ||||||||||
| 72,190 | |||||||||||
| 5 | 102°25'0'' | ||||||||||
| 133,399 | |||||||||||
| 1 | 68°24'30'' | ||||||||||
| 440,559 | |||||||||||
2. Calculate the angular residual (correction):
Our territory is the pentagon (5 corners). The sum of the angles of a pentagon is 540. We need to compare the sum of the angles (which we measured) corresponds to 540 or not. Any measurements have an error, so we will calculate the error (the difference between the ideal amount is 540 and the amount that we received). If the difference between them is acceptable, then we will distribute the error between 5 angles with the opposite sign (in order to compensate for the error). This error - there is an angular residual or correction.
Sum of measured angles:
∑βmeasuret=68°24'30'' +69°56'30'' + 242°59'0'' + 56°17'0'' + 68°24'30'' = 540°2'0''
Theoretical sum (ideal):
∑βteoretical =180˚(n-2)= 180˚(5-2)540˚00'
n - the number of angles of the closed theodolite traverse
Find the difference between them:
fβ=∑βmeasuret-∑βteoretical=0°2'0''
Permissible residual:
fβдоп=± 
=± 
= 2,2'
If you get fβ ≤ fβдоп , you measured correctly. So you can consider the angular residual.
Ϭβ=-fβ / n =2' / 5 = 0°0'24''
the angular residual is written over the measured angles. If the disagreement is not divided without remainder by n (we have 5 corners), then you should write large numbers over corners with smaller sides.
We write corrections over the measured angles (highlighted in yellow in the Table 2/2).
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3. We find the corrected angles:
The corrected angle is equal to the measured angle plus correction.
βmeasuret 1 + Ϭβ = βcorrected 1
βmeasuret 2 + Ϭβ = βcorrected 2
βmeasuret 3 + Ϭβ = βcorrected 3
βmeasuret 4 + Ϭβ = βcorrected 4
βmeasuret 5 + Ϭβ = βcorrected 5
Table 2/2
Coordinate calculation
| № точки | Углы | Горизнтальное проложение , (d) Horisontal line Proection on the level ground | Приращения | Координаты | |||||||
| Измеренные углы / Average angle/ measured angles (βmeasuret) | Исправленные углы/ corrected angles (βcorrected) | Дирекционные/ directional angles (α) | Вычисленные | Исправленные | X | Y | |||||
| ∆X | ∆Y | ∆X | ∆Y | ||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
| 1 | 68°24'30'' -24 | 68°24'6'' | |||||||||
| 101,100 | |||||||||||
| 2 | 69°56'30'' -24 | 69°56'6'' | |||||||||
| 70,260 | |||||||||||
| 3 | 242°59'0'' -24 | 242°58'36'' | |||||||||
| 63,610 | |||||||||||
| 4 | 56°17'0'' -24 | 56°16'36'' | |||||||||
| 72,190 | |||||||||||
| 5 | 102°25'0'' -24 | 102°24'36'' | |||||||||
| 133,399 | |||||||||||
| 1 | 68°24'30'' -24 | 68°24'6'' | |||||||||
| 440,559 | |||||||||||
4. Correct directional angles:
4.1. The same that in part 2:
αB1=αAB+180-∠ βmeasuret B right =
αB1=αAB+∠ βmeasuret B left - 180=
α12=αB1+180-∠ βmeasuret 1 right0 =
βmeasuret B right –take from outline, Example for 4 variant βmeasuret B right = 197° 19'
βmeasuret B left –take from outline, Example for 4 variant βmeasuret B left = 162° 41'
αAB– look your variant
4.2. Correct angles in part 2:
α23=α12+180-∠ βcorrected 2 right =
α34=α23+180-∠ βcorrected 3 right =
α45=α34+180-∠ βcorrected 4 right =
α51=α45+180-∠ βcorrected 5 right =
α12=α51+180-∠ βcorrected 1 right =
4.3. Example correct direction angles for variant 4.
αAB=110°20'0''
αB1=αAB+180-∠ βmeasuret B right = 110°20'0'' +180 - 197° 19' = 92°34'0''
αB1=αAB+∠ βmeasuret B left - 180= 110°20'0'' +162° 41' - 180= 92°34'0''
α12=αB1+180-∠ βmeasuret 1 right0 = 92°34'0'' +180 – 240° 49' = 31°45'0''
α23=α12+180-∠ βcorrected 2 right = 31°45'0''+180 – 69°56'6'' = 141°48'54''
α34=α23+180-∠ βcorrected 3 right =141°48'54'' + 180 - 242°58'36'' = 78°50'18''
α45=α34+180-∠ βcorrected 4 right =78°50'18'' +180 - 56°16'36'' = 202°33'42''
α51=α45+180-∠ βcorrected 5 right = 202°33'42'' + 180 - 102°24'36'' = 280°9'6''
α12=α51+180-∠ βcorrected 1 right =280°9'6'' + 180 - 68°24'6'' = 31°45'0''
31°45'0'' = 31°45'0'' - directional angles are correct
write directional angles in table
Table 2/3
Coordinate calculation
| № точки | Углы | Горизнтальное проложение , (d) Horisontal line Proection on the level ground | Приращения | Координаты | |||||||
| Измеренные углы / Average angle/ measured angles (βmeasuret) | Исправленные углы/ corrected angles (βcorrected) | Дирекционные/ directional angles (α) | Вычисленные | Исправленные | X | Y | |||||
| ∆X | ∆Y | ∆X | ∆Y | ||||||||
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | |
| 1 | 68°24'30'' -24 | 68°24'6'' | |||||||||
| 31°45'0'' | 101,100 | ||||||||||
| 2 | 69°56'30'' -24 | 69°56'6'' | |||||||||
| 141°48'54'' | 70,260 | ||||||||||
| 3 | 242°59'0'' -24 | 242°58'36'' | |||||||||
| 78°50'18'' | 63,610 | ||||||||||
| 4 | 56°17'0'' -24 | 56°16'36'' | |||||||||
| 202°33'42'' | 72,190 | ||||||||||
| 5 | 102°25'0'' -24 | 102°24'36'' | |||||||||
| 280°9'6'' | 133,399 | ||||||||||
| 1 | 68°24'30'' -24 | 68°24'6'' | |||||||||
| 31°45'0'' | 440,559 | ||||||||||
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