# Earth Geodesics - Rhumb Line vs Great Circle

1. Jul 21, 2014

### 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

2. Jul 30, 2014

### Greg Bernhardt

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

3. Jul 31, 2014

### Matterwave

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

4. Jul 31, 2014

### SteamKing

Staff Emeritus
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'.

5. Jul 31, 2014

### 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 .

6. Aug 1, 2014

### 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?

7. Aug 6, 2014

### 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.