Earth Geodesics - Rhumb Line vs Great Circle

[GreenLRan]

I have an object (A) at some altitude above the Earth ellipsoid, and a point (B) on the surface of the Earth.

Since you're not confined to the surface of the Earth as you travel from A (at altitude), to B, I'm getting confused.

If I were to create a (Cartesian) vector pointing from object A in the air, to point B on the ground,
would a rhumb line, or a great circle be a more accurate representation of the vector if I were to put the discretized values of the vector in terms LLA (latitude, longitude, and altitude)?

Maybe a better way to ask is: What is the LLA projection of the Cartesian vector pointing from A to B? And would that projection be better represented as a rhumb line, a great circle, or some other calculation?

Thanks

 

[Greg Bernhardt]

I'm sorry you are not finding help at the moment. Is there any additional information you can share with us?

 

[Matterwave]

I have an object (A) at some altitude above the Earth ellipsoid, and a point (B) on the surface of the Earth.

Since you're not confined to the surface of the Earth as you travel from A (at altitude), to B, I'm getting confused.

If I were to create a (Cartesian) vector pointing from object A in the air, to point B on the ground,
would a rhumb line, or a great circle be a more accurate representation of the vector if I were to put the discretized values of the vector in terms LLA (latitude, longitude, and altitude)?

Maybe a better way to ask is: What is the LLA projection of the Cartesian vector pointing from A to B? And would that projection be better represented as a rhumb line, a great circle, or some other calculation?

Thanks

By "rhumb line" do you mean "plumb line"? I've never heard this term.

 

[SteamKing]

By "rhumb line" do you mean "plumb line"? I've never heard this term.

A 'rhumb line' is a term of art used by navigators to describe a path which cuts all meridians of longitude at the same angle.

http://en.wikipedia.org/wiki/Rhumb_line

A curve which does this is also known as a 'loxodrome'.

 

[Matterwave]

In that case, I have to ask are we allowed to move through the Earth? If so...the vector should just be directed along the straight line connecting point A and B. I don't see why not .

 

[lpetrich]

For navigation across the Earth, treating it as a sphere is a good approximation. The Earth's flattening is about 1/300, and an airliner typically travels at a relative altitude of 1/600. Mt. Everest is about 1/720 above sea level, the Mariana Trench is about 1/580 below sea level, and the Everest-Mariana difference is about 1/320.

As to geodesic vs. rhumb line, only some rhumb lines are geodesics: the equatorial and polar ones. That's because geodesics are great circles, and all of them have variable bearing except for the equatorial and polar ones.

Interested in the math?

 

[GreenLRan]

I believe the projection ends up being a great circle, this makes sense to me now. If you're following a rhumb line, you're constantly keeping a fixed azimuth with respect to North (a longitude line), in reality that works the vector off target. I verified by converting from LLA coordinates to ECEF, then making discrete points (X,Y,Z) along a line traveling from A to B. Once I converted the ECEF line back to LLA, it did indeed produce the same results you would get from a great circle.

Thanks for the help guys.