## Triangulation Theory

Triangulation can be defined as process of determining the position of a point on the ground by measuring angles to it from known point at either end of a fixed baseline rather than measuring distance to the point directly.Triangulation can also refer to the accurate surveying of systems of very large triangles, called triangulation network. Triangulation can be used to find the coordinates and sometimes distance from the shore to the ship. The observer at *A* measures the angle

*α* between the shore and the ship, and the observer at *B* does likewise for *β* . If the length *l* or the coordinates of *A* and *B* are known, then the law of sines can be applied to find the coordinates of the ship at *C* and the distance *d*.

- The coordinates and distance to a point can be found by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other known reference points.

- Alternatively, the distance RC can be calculated by using the law of sines to calculate the lengths of the sides of the triangle.

- The distance AB is known, so we can write the lengths of the other two sides as RC can now be calculated using either the sine of the angle α, or the sine of the angle β:

- We know that γ = 180 − α − β, since the sum of the three angles in any triangle is known to be 180 degrees; and since sin(
*θ*) = sin(180 - *θ*), we can therefore write sin(γ)=sin(α+β), to give the final formula.

This formula can be shown to be equivalent to the result from the previous calculation by using the trigonometric identity sin(α + β) = sin α cos β + cos α sin β.

## Tolerance

To ensure that the data taken are accurate, some limitation has been introduced:

- E1 ≤ 1
^{o}20"

- E2 ≤ 40"

E3 ≤ 40"

## Equal Shift Adjustment

- Is intended to obtain the correct value which is individually satisfying the conditions of angles and sides and distribute them equally to the angles (before proceeds to the sides and coordinate computations).

- Satisfy the figure equation, side equation, station equation (if only).

- Using angle equation and side equation to know the required condition for brace quadrilateral triangulation.

Angle equation

n_{1} = m – n +1

where,

n = the number of station

m = the number of sides

The brace quadrilateral triangulation has 4 stations and 6 sides. Therefore;

n_{1} = m – n +1

n_{1} = 6 – 4 +1

n_{1} = 3

The brace quadrilateral triangulation has 3 angle conditions.

3 angle conditions of the brace quadrilateral triangulation

- Angle 1 + angle 2 + angle 3 + angle 4 + …….+ angle 8 = 360
^{o}.

- (angle 5 + angle 6) = (angle 1 + angle 2)

- (angle 7 + angle 8) = (angle 3 + angle 4)

Side equation

n_{2} = m – 2n +3

where,

n_{ }= the number of station

m = the number of sides

n_{2} = 6 – 2(4) +3

n_{2} = 1

Therefore, the brace quadrilateral triangulation has 1 side conditions.

1 side conditions of the brace quadrilateral triangulation