Thread: Vincenty's first calculation.

1. Vincenty's first calculation.

I have studied now spherical trigonometry although I do not consider myself at all an expert yet. I am wondering is the first calculation the proof of some theorem? I am wondering how Vincenty arrived at this conclusion. Is it arc length that is being measured? Unfortunately I am inquiring on the details of this equation and how it functions. I am very curious about this function and it does not make sense to me yet.

The greek letter in Tod Hunters book for geodetic calculations looks a little different but I believe it's still sigma with Lambda being a distance in this case. Although it isn't really clear to me how lambda is supposed to fluctuate in Vincenty's calculations. It says in wikipedia to iterate from Lambda but it doesn't say how the next iteration should look like.

2. Re: Vincenty's first calculation.

Hey sepoto.

What are the limits for sigma (since it is on a spherical geometry, it must be bounded)?

Remember that because you have a spherical geometry (with different curvature to that of a flat or hyperbolic geometry), that the limits will be bounded and that the sine of this distance will correspond to the circular arc on your sphere that corresponds with the distance away from a point.

A good way to visualize this is to think of a graph of a sine wave in two dimensions that only depend on the distance away from the origin. You will basically get a graph that shows ripples spreading out-ward from the origin (like when you cast a stone in a pond).

It would help immensely if you told us what your limits were and if they were "normalized" (say in the region of 0 to pi or something similar) and what this equation is being used for.

3. Re: Vincenty's first calculation.

In this case the center of the earth is O and the radius of the earth is 6 378 137 m. Vincenty takes the spherical model further by adding flattening to the sphere. So I believe that the left part of the equation I am calling sigma is an arc distance from what I am told however is it a leg of a spherical triangle? flattening in WGS84 is 1/298.257223563. I'm not sure yet but flat I am studying the equations because I want to find out why things fail at antipodal points and also how the equations react across hemispheres. I'm sure I will be referencing Spherical Trig a lot. Thank you for your response.

Actually in correct use of Vincentys for meteres

 WGS-84 a = 6 378 137 m (±2 m) b ≈ 6 356 752.314245 m f ≈ 1 / 298.257223563

Vincenty's formulae - Wikipedia, the free encyclopedia

4. Re: Vincenty's first calculation.

With regards to anti-podal points, once your metric goes past the distance of an anti-podal point then your metric acts differently.

Basically what happens is that the metric actually "decreases" since there exists a path (many paths in fact) where the metric is smaller because of the nature of the geometry (being spherical).

If you want to account for distances in some direction on the sphere that are greater than those of anti-podal points then you have to compensate for this and realize that the metric itself doesn't represent the actual distance, but just that of the shortest path between points (if it is indeed a proper metric).

If you are dealing with spherical triangles (i.e. multiple arcs) then this is going to compound the problem.

5. Re: Vincenty's first calculation.

Alright I see what you are saying about greater than antipodal distances but I have to see it for myself. I am a computer programmer that uses Vincenty's equation in my code and right now I don't have a full grasp on its details which I find to be problematic. I'm hoping someone might explain perhaps by spherical diagram what the first equation is actually doing. Right now I am trying to put it together and I know that sin of theta provides the y and cos of theta provides the x of the circle in trigonometry. That's about as far as I dare to go right now. I see that the sin of lambda is being taken and the sin and cos of the lat lng but what these actually denote at this time is still mysterious to me. I'm hoping someone might explain that.

I'm also curious at this point why both parenthesized parts of the equation are being squared and also the purpose of the square root.