Spherical Coordinates: Distance Between 2 Points

[jsbxd9]

Hello,

I am looking into finding geodesic distances for an ellipsoid. I will designate two points then find the distance between them. This will be my geodesic distance. I have put together a schematic (attached) for reference. Ultimately I need to know the distance D as shown on the attachment. I figured if I found β I could use the law of cosines to figure the distance D. I have also been reading a book that states D = √(ρ₁² + ρ₂² - 2ρ₁ρ₂ (cosθ₁cosθ₂ + cos (ø₁ - ø₂) sinθ₁ sinθ₂)) I cannot figure out the way this equation was derived so I am not sure if it will work.

Please let me know if you have a solution.

Thank you,
- Jonathan

View attachment Spherical Coordinate System.pdf

 

[jvicens]

Hello Jonathan,

Thank you for your inquiry. Geodesic distances for an ellipsoid can be calculated using the Vincenty formula, which takes into account the shape and size of the ellipsoid. The formula is as follows:

D = tan^-1(√((cosθ2sin(λ2-λ1))^2 + (cosθ1sinθ2-sinθ1cosθ2cos(λ2-λ1))^2) / (sinθ1sinθ2+cosθ1cosθ2cos(λ2-λ1)))

In this formula, θ1 and θ2 represent the latitudes of the two points, λ1 and λ2 represent the longitudes, and D represents the geodesic distance between the two points. This formula is derived from the geodesic equation, which takes into account the curvature of the ellipsoid.

I hope this helps with your calculations. If you have any further questions, please don't hesitate to reach out.