How Far Away is the Horizon

In this experiment we want to calculate the distance (h2) between the ground and the line of sight (d) which extends from the observer's eyes to the horizon boundary that can be seen. Thus, since the observer is at a height (h) above sea level, and at a given distance (L) from an obstacle (can be a tree, a hill, a mountain), one can calculate whether this obstacle (height h2) can block the view of the far horizon.

To do this, simply provide the height of the eyes of the observer (h) and the distance on the surface of the earth to the point where you want to know (L). Therefore L is the arc length or distance on the earth's surface, formed between the perpendicular lines of the observer and the obstacle. This information is useful for knowing the height of an obstacle, far away from the observer, which may limit the view of the observer when looking directly at the horizon.

In short, if the obstacle is greater than the height h2, it will not be possible to see beyond this, assuming the horizon is at the same height, relative to sea level, as the base of the observation site.

 

According the this  website the distance (d) from the horizon is given by:

By doing some other trigonometric relationships, we find that::

 

 

Where h2 is the altitude from a object located L2 far from the observer. L2 is related to the ground distance or the arc length.

Distance from the observer to the obstacle:

After a little algebra:

Where,

 

h (km)

d (km)

phi

phi1

phi2

h2 (L2=5.05 km)

0.001

3.57

0.032117253

-0.013341299

0.045458552

0.000172552

0.002

5.05

0.045420652

-3.79002E-05

0.045458552

1.39282E-09

0.003

6.18

0.055628707

0.010170155

0.045458552

0.000100272

0.005

7.98

0.071816343

0.02635779

0.045458552

0.000673506

0.01

11.28

0.101563613

0.056105061

0.045458552

0.003051598

0.02

15.96

0.143632545

0.098173992

0.045458552

0.00934365

0.03

19.54

0.175913108

0.130454555

0.045458552

0.016498429

0.05

25.23

0.227102548

0.181643996

0.045458552

0.031986525

0.08

31.91

0.287263962

0.241805409

0.045458552

0.056683726

0.1

35.68

0.321170452

0.2757119

0.045458552

0.073695057

0.2

50.46

0.454200636

0.408742084

0.045458552

0.161968689

0.5

79.78

0.71814016

0.672681607

0.045458552

0.438699559

0.7

94.40

0.849703772

0.80424522

0.045458552

0.627098525

1

112.83

1.015570316

0.970111763

0.045458552

0.912469945

2

159.57

1.436139318

1.390680765

0.045458552

1.875360058

5

252.34

2.270289956

2.224831404

0.045458552

4.801647674

10

356.93

3.20962521

3.164166658

0.045458552

9.718384189

12

391.03

3.515508655

3.470050103

0.045458552

11.69119391

15

437.24

3.929687778

3.884229226

0.045458552

14.65430593

How Far is the Horizon

Click the button to calculate the distance from the observer to the horizon.

Observer's Height (h) in meters[m]:

Obstacle distance (L2) in kilometers [km]:

Calculated distance (d) in kilometers[km]:

Calculated height (h2) in kilometers[m]:

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Trilateration (under development)

Click the button to calculate the observer coordinates:

Satellite 1:

Coordinate x: Coordinate y: Distance P :

Satellite 2:

Coordinate x: Coordinate y: Distance P:

Satellite 3:

Coordinate x: Coordinate y: Distance P:

Solution:

Coordinate x: Coordinate y: