1.8. Example 1.26: Horizontal Distance and Elevation Calculation

The vertical angles to vanes fixed at 1 m and 3 m above the foot of the staff held vertically at station Q were 30°20' and 60°40', respectively from instrument station P. If the elevation of the instrument axis at station P is 101.520 m, calculate (i) the horizontal distance between P and Q, and (ii) the elevation of the staff station Q.

In this example, we need to calculate the horizontal distance and elevation using given vertical angles and heights of two vanes. The angles of depression to the vanes (1 m and 3 m above the ground) indicate how steeply the sight line is directed from the instrument to the vanes. First, we will convert these angles into tangents to find the necessary horizontal distance and vertical changes using trigonometric relationships.

Consider standing on a hill and spotting two objects (like a car and a tree) positioned at different heights below you. The angle to the top of the tree will be different than that of the car, and by measuring these angles and your distance from the base of the hill, you can find out how far away the tree or car is. Similarly, the example calculates these distances using angles.

D = S / [tan θ₁ – tan θ₂] = 1 / [tan 60°40' - tan 30°20'] = 34.13 m

In this step, 'S' is the difference in height of the vanes (3m - 1m = 1m). We use the tangent function, which relates the opposite side (height difference) to the adjacent side (horizontal distance) in a right triangle. The formula essentially finds how far away 'Q' is from 'P' given the angle of depression. By plugging in the angles into the tangent function and resolving, we can determine the horizontal distance.

If you have a slope and want to find out how far horizontally a person standing at the top is when looking down at someone directly below, using your angle measurements can help calculate this distance.

V = S tan θ₂ / [tan θ₁ – tan θ₂] = 2 x tan 30°20' / [tan 60°40' - tan 30°20'] = 1.99 m

This step calculates the vertical distance 'V' to the second vane using a similar trigonometric relationship. The height difference over the horizontal distance provides the information needed to find vertical displacement. The calculation considers the angle of depression to the lower vane (1.9995 m or about 2 m of vertical drop) and how that relates back to the first horizontal point. This helps us understand where the elevation stands compared to our initial starting position.

Imagine creating a ramp; finding how high the ramp should rise at different points can be thought of similarly. You take the angles to ensure you can get from one point to another while considering the vertical change.

Elevation of staff station Q = RL of HI + V – h = 101.520 + 1.99 – 1.0 = 102.510 m

In this final calculation, we determine the elevation of the vane at station Q. The reduced level (RL) represents the height of instrument axis at station P plus the vertical height change (V) we just calculated, minus the height of the V vane at 1 m. This gives us the final elevation of the point we are interested in.

Picture measuring a wall height from above on a balcony. If you know the height of the balcony plus how high the point below is relative to your view (up or down), you can sum this with the known wall height to adjust to find the final height of what you observe from above.

--