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Math Help - Vincenty's Formulae

  1. #1
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    Vincenty's Formulae



    This is the first step to Vincenty's formulae. I am trying to get an explanation of what sigma is representing in this case. Sigma being what is on the left hand side of the equation. What exactly is sigma supposed to represent and how is the formula on the right hand side doing?

    Thanks in advance...
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  2. #2
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    Re: Vincenty's Formulae

    Hey sepoto.

    What is the context of the equation? Is it from a science (physics/chemistry/etc)? Is it from engineering?

    To answer your question, we need the appropriate context because without it we have nothing to relate the equation, the individual variables, and any meaning to it be it physical, geometrical, or otherwise.
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    Re: Vincenty's Formulae

    I suppose if I had to group it into a category that Vincenty's would be an engineering question. It is used to take an distance between two latitude and longitude points on a flattened sphere. I took trigonometry but what's on the right hand side of the equation is not yet making sense to me.
    Last edited by sepoto; September 19th 2013 at 06:34 PM.
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  4. #4
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    Re: Vincenty's Formulae

    the argument sigma is definned by Vincenty as the arc length between points on the auxiliary sphere

    T. Vincenty
    DIRECT AND INVERSE SOLUTIONS OF
    GEODESICS 0N THE ELLIPSOID WLTH APPLICATION
    OF NESTED EQUATIONS

    Survey Revie\\ XXJl. 176, April 1975

    This problem is in spherical/ellipsoid trigonometry.
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  5. #5
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    Re: Vincenty's Formulae

    Without knowing the contents of the paper, I would start off by deriving the arc-length formula between two points on the sphere by using standard differential geometry.

    Basically you have to get derivatives of arc-length with respect to the other parameters. To do this, you use the appropriate directional derivative in the direction of the line segment from point a to point b. By evaluating the directional derivative and integrating over the right limits you can calculate the arc-length.

    This is the best way I can think of in deriving the formula after which you can use trigonometry to derive the RHS expression. This is of course without knowing the contents of the paper and its context.

    Note that sines and cosines can also represent forms of projections when you are dealing with spheres or ellipsoids because of the periodic nature of the geometry (they wrap around as opposed to that of flat or hyperbolic spaces).

    Given the title, I'm guessing there are other tricks that the author is using but again I can't really comment on that.
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