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How to calculate distances and directions using the Geographic Coordinates

Before starting this study, it is necessary to remember the concepts of:
DLA (latitude difference) and DLO (longitude difference)
  • DLA - is the angular difference between two latitudes, being at most 180 degrees, since it is the difference between 90ºN and 90ºS.
  • DLO - is the smallest angular difference between two longitudes, and it can also be at most 180 degrees, since it is the difference between the length of any meridian and its anti-meridian (opposite to 180º).
To calculate the distance between two locations only by knowing the coordinates, we will also need to remember how to convert these DLA and DLO values ​​into distance.
Note: In order to calculate the direction between two locations it will be necessary to remember concepts of trigonometry, as we will see later.
Turning a DLA or DLO value into distance
To transform an angular value into distance, just remember their equivalences.
As you know:
1º = 60 NM
Thus it can be concluded that:
60 '= 60 NM \ 1' = 1 NM
Occurs that:
1 '= 60 "
Thus it can be concluded that:
60 "= 1 NM
That is:
1 "= 1/60 NM
By knowing these equivalences, it is easy to transform any DLA or DLO value into distances. Observe the following example:
Let's convert the value 23º 30 '36 "into distance.
Just isolate each value and convert individually, adding up the results:
23º x 60 = 1,380
30 'x 1 = 30
36 "÷ 60 = 0.6
1.380 + 30 + 0.6 = 1,410.6 NM x 1,852 = 2,612.4 Km
Obviously, this method is valid for small distances (smaller than 800 NM), because the correct one would be to take into account the terrestrial curvature.
However, this method works very well, as we will see below.
Calculating the distance between two geographical points
It may occur that at some point you can have the coordinates between two points, but you do not have the letter or some equipment in hand to calculate the distance between them.
When this happens, just use what is already known about geographic coordinates.
It has already been seen that a geographical coordinate uses the Cartesian system to indicate localities.
By doing a simple analysis, any coordinate can be represented in a system of axes of type "x" and "y".
Let's take as an example the geographical coordinates of the two headwaters of the SBMT track (Campo de Marte Airport, São Paulo):
SBMT: TRACK 12 (23 ° 30 '29.93 "S / 046 ° 38' 32.90" W)
SBMT: TRACK 30 (23 ° 30 '36.50 "S / 046 ° 37' 53.01" W)
Let us now calculate the length of the track using the two coordinates.
A small analysis is enough to realize that the length of the lane is defined by a line connecting the two points and that this line is nothing more than the hypotenuse of a right triangle defined by the latitude (DLA) and longitude differences (DLO ), Which are the cathets between these points.

See the diagram below:
As by the Pythagorean Theorem, the square of the hypotenuse is equal to the sum of the squares of the legs, we can consider that one of the legs is the DLA and the other the DLO, the hypotenuse being the length of the lane (or the distance between the two points) .
Thus, the formula will always be valid:
LENGTH 2 = DLA 2 + DLO 2
Let us then calculate the DLA and DLO:
           DLA = 23 ° 30 '36.50 "- 23 ° 30' 29.93" = 6.57 "
           DLO = 046 ° 38 '32.90 "- 046 ° 37' 53.01" = 39.89 "
Knowing the value of DLA and DLO, simply turn them into distance, dividing them by 60:
           DLA = 6.57 "÷ 60 = 0.1095 NM x 1.852 = 202.8 meters
           DLO = 39.89 "÷ 60 = 0.6648 NM x 1.852 = 1.231.2 meters
By placing the values in the formula:
           LENGTH 2 = 202.8 2 + 1,231.2 2 = root (41,127.84 + 1,515,853.44)
           LENGTH = 1,247.8 meters
To prove that the calculation is correct, let's use the Google Earth ruler tool:
Calculating the direction between two geographic points
So far, only a simple calculator has been used for calculations, requiring only the value of a square root.
We will now see that, although somewhat complex, there is the possibility of calculating the direction between two geographical points.
For this, it will be necessary to review basic concepts of trigonometry and the theory of triangles.
As the triangle that we are going to study is a triangle rectangle, we will have the following drawing:
By the theory of triangles, the internal sum of all angles is always equal to 180 °.
Like this,
α + β + 90 ° = 180 °
It is enough, therefore, to find α to find β or vice versa:
           α = 90 ° - β
           β = 90 ° - α
To calculate the value of the angles, it is necessary to remember the concepts of trigonometry.
The value of an angle in a right triangle can be calculated as follows:
Tangent at an angle is equal to the opposite leg on the adjacent
Sine from an angle is equal to the opposite leg on the hypotenuse
An angle cosine is equal to the adjacent side of the hypotenuse
Knowing this, on the basis of the angle α, we can deduce that:
           tan α = DLA ÷ DLO
           sin α = DLA ÷ distance
           cos α = DLO ÷ distance
Since the values of DLA and DLO are more easily found, we will then apply these values using the formula of the tangent of α:
Tang α = 202.8 ÷ 1,231.3 = 0.1647
Knowing the value of the tangent, simply calculate the inverse tangent.
That is, the arc tangent of this angle.
The result of this operation, which should be done using a calculator with this function or Excel - as we will see next - can be represented as follows:
arctan α = tan-1 α
This operation gives the value in radians, which must be converted to degrees.
A more advanced calculator makes this calculation quickly by simply clicking the "inverse" function and then the "degrees / radians" function.
In Excel, simply put the following formula:
           = degrees (atan (tanα))
           = degrees (atlan (DLA / DLO))
Applying this formula in Excel, we have:
α = degrees (atan (0.1647))
The result will be:
That is, rounding up to integers will be:
Thus, if:
α = 9 °, β = 90 ° - α  / β = 90 ° - 9 ° = 81 °
That is:
           α = 9º
           β = 81 °
It is important to note that these values are from the inside of the triangle, which will look like this:
Therefore, the True Course (RV) values of lanes 12 and 30 will be, respectively:
           RV TRACK 12 = 180º - 81º = 99º
           RV TRACK 30 = 270º + 9º = 279º
As the magnetic declination of the SBMT is 21ºW, the Magnetic Roads will be, respectively:
           RM TRACK 12 = 99 ° + 21 ° = 120 °
           RM TRACK 30 = 279 ° + 21 ° = 300 °
This proves that the calculations are correct, otherwise the clues would not be 12 and 30!
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