Calculating Distance on a Spherical Earth Using Trigonometry

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In summary, the speaker is the captain of their school's academic team and one of their main topics this year is spherical trigonometry. They are struggling to find the shortest distance between two points on Earth given their longitude and latitude coordinates. They mention being able to convert the coordinates to degrees but not knowing what to do from there. The expert suggests using the dot product and trigonometric relationships to calculate the distance, as well as mentioning the concept of a great circle or geodesic being the shortest distance between two points on a sphere. They also provide resources for further information.
  • #1
thharrimw
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I am the captain of my schools academic team. One of the main topics this year is Spherical trig and I can't find out how to find the shortest distance between two points on Earth given the longitude and latitude of both points. I can easily convert the points from Degree- minute- second format to degree format but I don't know what to do from there. for example what is the distance between 53 09 02N ; 001 50 40W and 52 12 17N ; 000 08 26E?
 
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  • #2
thharrimw said:
I am the captain of my schools academic team. One of the main topics this year is Spherical trig and I can't find out how to find the shortest distance between two points on Earth given the longitude and latitude of both points. I can easily convert the points from Degree- minute- second format to degree format but I don't know what to do from there. for example what is the distance between 53 09 02N ; 001 50 40W and 52 12 17N ; 000 08 26E?

I'm not sure what level of math you're in. If you can make a triangle between the center of the Earth and the two points, then you, can use trigonometric relationships to get the distance. Have you learned about the dot product yet? Or are you trying to do this based on basic geometry? Anyway, once you know the angle to getting the distance is basic trig if you go through, the earth. If you want the arc length then you [tex]S=r \theta[/tex] where [tex]\theta[/tex] is in radians.
 
  • #3
I'm in calculus 2 but the academic bowl is algebra, basic geometry and Spherical trig. i know how to find the dot product between two vectors from the little that i have done with vector algebra but i needed to find out the length by moveing on the Earth's serface. how could i do that given the infromation stated above??
 
  • #4
A great circle is also called a Geodesic. A Geodesic is the shortest distance between two points on a sphere. The airlines call these Great Circles and they like to fly on them to save fuel. Click on this link for more information.

http://www.black-holes.org/relativity5.html [Broken]
 
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  • #5
If [itex]\theta[/itex] is the difference between the latitudes and [itex]\phi[/itex] is the difference between the latitudes, then you have a spherical right triangle with legs of angular length [itex]\theta[/itex] and [itex]\phi[/itex]. The spherical version of the Pythagorean theorem is [itex]cos(\mu)= cos(\phi)cos(\theta)[/itex] where [itex]\mu[/itex] is the angular distance between the points. The actual distance between them is [itex]\mu[/itex] times the radius of the earth.


Check
http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node5.html [Broken]
for more information.
 
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  • #6
HallsofIvy said:
If [itex]\theta[/itex] is the difference between the latitudes and [itex]\phi[/itex] is the difference between the latitudes, then you have a spherical right triangle with legs of angular length [itex]\theta[/itex] and [itex]\phi[/itex]. The spherical version of the Pythagorean theorem is [itex]cos(\mu)= cos(\phi)cos(\theta)[/itex] where [itex]\mu[/itex] is the angular distance between the points. The actual distance between them is [itex]\mu[/itex] times the radius of the earth.

so all that i would have to do is take 6 378.1km* ArcCos(Cos([itex]\phi[/itex])Cos([itex]\theta[/itex]) and that would give me the distance of my great circle?
 

1. What is the definition of distance of a great circle?

The distance of a great circle is the shortest distance between two points on a sphere. It is measured along the surface of the sphere, rather than through its interior.

2. How is the distance of a great circle calculated?

The distance of a great circle can be calculated using the Haversine formula, which takes into account the radius of the sphere and the latitudes and longitudes of the two points. Other methods, such as the Vincenty formula, can also be used for more accurate calculations.

3. What is the difference between distance of a great circle and distance of a small circle?

The distance of a great circle is the shortest distance between two points on a sphere, while the distance of a small circle is the shortest distance between two points on a plane that intersects the sphere. This means that the distance of a great circle will always be shorter than the distance of a small circle.

4. How is the distance of a great circle used in navigation?

The distance of a great circle is used in navigation to determine the shortest route between two points on a sphere. This is especially important for long-distance travel, such as air or sea travel, where the curvature of the Earth must be taken into account to determine the most efficient route.

5. Can the distance of a great circle be used on any type of sphere?

Yes, the distance of a great circle can be used on any type of sphere, as long as the radius of the sphere is known. It is a fundamental concept in spherical geometry and is applicable to any spherical object, including planets, stars, and even bubbles.

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