Hello, this part of the course on measuring distances

is dedicated to trigonometric leveling.

Trigonometric leveling

allows, from measuring distances and angles,

to determine height differences.

The question that must be asked

in topographic measurements

is: how to define a distance unambiguously?

I take here a small example with a station P

an aiming point Q, a slant distance S' measured here

as well as an angle of height beta (β).

We can reduce this distance to the horizontal

here S horizontal (HZ) and calculate a height difference.

The question that we must ask:

at what point should we consider the distance

like a spherical distance?

And also the question is knowing

that the measurement is done at a certain altitude

in this case here Hp = 1000 meters

and that a distance measured at a certain altitude

is not necessarily the same

as a measured distance

at a lower or higher altitude.

So we must proceed with what we call reductions

and these are the elements that we will see

in this part of the lesson

with the reduction due to altitude

as well as the reduction in the projection plane

in order to have distances

that are compatible with the national coordinate system.

Flat Earth, round Earth

at what point do we take into account

the roundness of Earth

in the measurements of distances?

We take here an example between point P and point Q

with the chord which is the straight line between these two points

and the arc that follows Earth's terrestrial reference surface.

If I have a chord of 10 000 meters

my arc will be 10 000,001 meters

a difference of about 1 millimeter.

So, in this case, for our topographic measurements

the arc and chord will be undistinguishable.

On the contrary the arrow, which is the distance here in the center, between the arc and the chord

- I have here my arc and my chord -

in this case, the arrow is equivalent to about 2 meters.

To illustrate this principle

we must stop for a moment

at this figure, fundamental

to understand the problems

with reduction of distance.

Firstly, we have here our station P

and the point Q which has been measured.

At station P I have a vertical here with an altitude HP.

At station Q I have another vertical.

These two verticals are not necessarily parallel.

From P I can consider

the distance perpendicular to this vertical

namely the horizontal distance S

as well as the arc Sp here, which will connect P to Q'.

I can also consider Cp

as the chord that connects P to Q'.

I have thus three types of distances

to be considered in this figure.

How about for our topographic work?

We recall here our three concepts of distance:

namely the arc, chord

and the horizontal here at P.

So we have Sp, Cp and S.

Cp - Sp = -1mm for Sp = 10 km

and then S - Sp = 1mm

for Sp = 5 km.

So in topographic work

where we have some kilometers of surface

we can neglect this effect.

Here we come back to the effect of the roundness of Earth.

In this figure, we have first of all

the angle at the center of Earth which is equal here to 2 ɣ (Gamma).

We can consider the triangle OPQ'

and its relation to the radius of Earth R x 2 ɣ = distance S between P and Q'.

Gamma ɣ is expressed in radians.

Then we will consider the triangle PQ'Q''

in this case we can write that E = ɣ x S.

Finally, our value E, by combining these two equations,

E = S²/2R (radius of Earth).

For a distance of 10 km, if you do the calculations,

you will get a value of E = 8 meters.

After having considered these spherical models

we now get closer to

a model called flat Earth

that we will use for

most topographic works and construction sites.

We have in this figure

a station P at a certain altitude Hp.

We measured here an oblique distance S'

with a altitude angle Bêta (β)

we can reduce at the horizontal the distance S.

We also have to consider

the height here of the aiming point, namely Z

as well as the height of the instrument namely i.

We have thus all geometric informations

to calculate the height difference.

In the case of flat Earth

the trigonometric formulas are relatively simple.

We have here the Delta (Δ ) Δ h which is equal to S' times sine of the altitude angle β.

If we look at the figure we can calculate

the altitude HQ = Hq + Δ h which I already calculated

+ I (height of instrument) - Z (height of shown signal)

I start from P, I have the height of the instrument

+ Δh - Z which gives us Hq.

We will now consider

the different steps leading

to the reduction of distances

to finally have an unambiguous definition.

The first step is the reduction due to altitude.

We see in this figure

that the distance considered at altitude Q or at altitude P is not the same.

We must therefore bring it back a distance called S zero

to the sea level.

To calculate the reduction factor

we will simply take the proportional ratio

between the spherical distances and the altitudes.

I consider here Sp/S0 and I take the relation of the distance to the centre of the Earth

so r + altitude Hp and then r (radius of Earth) at altitude H=0.

I can write that Sp, so at altitude Hp

= r + Hp/r x S0.

This is significant

I will here take an example with

Hp = 500 meters and a distance S equal to 1000 meters

Sp - S0 = 8 centimeters, in this case.

For trigonometric leveling

the following problem involves

the question of measuring the altitude angle.

We have actually in this figure

at station P, measured an angle beta

with respect to a horizontal

Or, we are interested here

in the reference surface, namely the sphere

at altitude P, here the reference P Q'.

In addition, we have a refraction effect

which will impact the measured angle.

In topography we must, in fact,

consider the phenomena of atmospheric refraction

when measuring altitude angles.

We have here our angle Tau (τ) which is the angle of refraction

which has the effect to curve the aimed beam

It is accepted in practice

that τ = a factor K that multiplies the angle at the centre of the Earth

so the farther away the aiming point is

the more important is the effect of refraction.

Gamma (ɣ) is obviously expressed in radians

and with experience, K is a value that we fix to 0.13.