Latitude and Departure in Surveying

In surveying, when we measure a line, we often need to determine its projection onto the North-South and East-West axes. These projections are called Latitude and Departure, respectively.

• Latitude (L): The north-south component of a line. It is considered positive if the line bears towards the North and negative if it bears towards the South.
• Departure (D): The east-west component of a line. It is considered positive if the line bears towards the East and negative if it bears towards the West.

Understanding Quadrants and Reduced Bearing

The Earth is divided into four quadrants based on the cardinal directions: North-East (NE), South-East (SE), South-West (SW), and North-West (NW).

The reduced bearing (θ) of a line is the acute angle that the line makes with the nearest North or South meridian. It is measured from either North or South towards East or West, depending on the quadrant.

The question specifies that the line lies in the South-West (SW) quadrant. In the SW quadrant, the reduced bearing θ is measured from the South meridian towards the West.

Calculating Latitude and Departure for a Line in the South-West Quadrant

For a line of length 'l' with a reduced bearing θ measured from the South towards the West (as is standard for the SW quadrant), the Latitude and Departure can be calculated using trigonometry.

Consider a right-angled triangle formed by the line 'l' as the hypotenuse, the South meridian as one leg, and the West line as the other leg. The angle between the line 'l' and the South meridian is θ.

• The side adjacent to the angle θ (along the South meridian) represents the magnitude of the Latitude. Its length is $l \cos(\theta)$.
• The side opposite to the angle θ (along the West direction) represents the magnitude of the Departure. Its length is $l \sin(\theta)$.

Now, we apply the sign convention based on the South-West quadrant:

• Since the line is in the South direction, the Latitude is negative. So, $L = -l \cos(\theta)$.
• Since the line is in the West direction, the Departure is negative. So, $D = -l \sin(\theta)$.

Therefore, for a line with length 'l' and reduced bearing θ in the South-West quadrant, the latitude 'L' and departure 'D' are computed by the formulas:

$L = - l \cos(\theta)$

$D = - l \sin(\theta)$

These formulas correctly account for the direction (South and West) by assigning negative signs to both Latitude and Departure.