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I'm using a tool I've found online (https://planetcalc.com/713/), which allows calculating the distance between two points on loxodrome using a formula that considers the WGS 84 ellipsoid model. In particular, I want to know the distance only along a meridian, thus the longitude is constant to zero. I set then the starting latitude point to 0°0'00'' and I adjust the end latitude point to specific values which allow returning specific distance values, i.e. multiples of 500 metres. The goal is to extract the latitude values for multiples of a given distance and export them to an Excel spreadsheet.

The formula used to calculate the distance on loxodrome is as follows:

$$S = a * sec \alpha \Biggl[\left(1-\frac14e^2\right)\Delta\phi - \frac38e^2(sin2\phi_{2}-sin2\phi_{1})\Biggl]$$

Where:

  • a: semi-major axis of the ellipsoid
  • α: course angle (which is zero, considering the longitude Δλ = 0)
  • e: eccentricity of the ellipsoid
  • ϕ1: starting latitude point
  • ϕ2: end latitude point
  • Δϕ: difference between ϕ2 and ϕ1

I thought I could automate this process by reversing this formula and isolating ϕ2, but I have no idea how to do it.

I've tried to use the SOLVER.XLAM add-in on Excel by forcing the distance (S) to exact values and find the latitude which returns that distance, but it is not as precise as I'd need.

NorthSon
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  • There will not be a closed form solution, but you can use some form of approximation or numerical method – Henry Nov 15 '21 at 20:19
  • @Henry, This is due to Δϕ which depends on the value I want to isolate? – NorthSon Nov 15 '21 at 20:43
  • The problem for a closed form is the combination of the $\Delta \phi$ and the $(\sin2\phi_{2}-\sin2\phi_{1})$ term – Henry Nov 16 '21 at 09:11
  • Thank you for the explanation! Alternatively, what would an approximation of it look like? – NorthSon Nov 16 '21 at 09:36

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