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Physics LibreTexts

56.6: Vincenty’s Formulæ- Inverse Problem

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In the inverse problem, we're given two points on the Earth's surface (ϕ1,L1) and (ϕ2,L2) and want to calculate the distance s between them, as well as the direction from one to the other. We'll use the constants defining the Earth's ellipsoid as before:

  • Earth ellipsoid semi-major axis (i.e. equatorial radius): a=6378137.0 meters.
  • Earth flattening factor f=1/298.257223563. This is defined as the difference between semi-major and semi-minor axes, divided by the semi-major axis: f=(ab)/a.

In performing the following calculations, work with all angles in radians. We begin by calculating

U1=tan1[(1f)tanϕ1]U2=tan1[(1f)tanϕ2]L=L2L1b=(1f)a

Now set an initial value λ=L. Then iterate on Eqs. (53.28) through (53.35) until λ converges:

sinσ=(cosU2sinλ)2+(cosU1sinU2sinU1cosU2cosλ)2cosσ=sinU1sinU2+cosU1cosU2cosλσ=arctansinσcosσsinα=cosU1cosU2sinλsinσcos2α=1sin2αcos2(2σm)=cosσ2sinU1sinU2cos2αC=f16cos2α[4+f(43cos2α)]λ=L+(1C)fsinα{σ+Csinσ[cos(2σm)+Ccosσ(1+2cos2(2σm))]}

When λ has converged to the desired degree of accuracy, continue calculating:

u2=(cos2α)(a2b2b2)A=1+u216384{4096+u2[768+u2(320175u2)]}B=u21024{256+u2[128+u2(7447u2)]}Δσ=Bsinσ{cos(2σm)+14B[cosσ(1+2cos2(2σm))16Bcos(2σm)(3+4sin2σ)(3+4cos2(2σm))]}s=bA(σΔσ)α1=arctan(cosU2sinλcosU1sinU2sinU1cosU2cosλ)α2=arctan(cosU1sinλsinU1cosU2+cosU1sinU2cosλ)

Then s is the distance between the two points.

Example 56.6.1

Find the distance between the sounding rocket in Chesapeake Hall at Prince George's Community College ( 385316.87N,764923.14W ) and the top (apex) of the Great Pyramid of Giza in Egypt (295845.03N,310803.69E).

Solution

The given parameters are the coordinates ϕ1=38.888019,L1=76.823094,ϕ2= 29.979175,L2=+31.134358. Employing Vincenty’s formulæ (inverse method), we find:

U1=38.794230U2=29.895958L=339.15856744b=6356752.3 meters λ=108.139490u2=0.00393162979A=1.00098218405082B=9.809796134747123×104Δσ=0.054160886s=9351378.858 meters α1=55.910048α2=131.801775

So the distance s=9351.378858 km (5280 miles, 3576 feet, 10 inches), in the direction 55.910048 (10.91 south of northeast).


56.6: Vincenty’s Formulæ- Inverse Problem is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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