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I have using Mathematica functions that takes a Cartesian coordinate relative to the Earth (xyz) and converts it to a latitude, longitude, and altitude (lla). And here it is:

xyz2lla = First@GeoPosition@GeoPositionXYZ[#, "WGS84"] &

I'm using it to convert a satellite viewing line (line of sight or los) from xyz to lla. For instance, if I have a satellite at the following observation point (obs, meters) looking in the direction of look:

obs = {2.560453600382259, 5.245110323032143, -3.819772142191310} 1*^6;
look = {-0.233218833096895, -0.814561997858160, -0.531128729720249};

It has the line of sight:

obs+d look

The transformation xyz2lla is smooth, so I was hoping to use function interpolation:

f = FunctionInterpolation[{xyz2lla[obs + # look]} &[d], 
                          {d, 2000000, 3700000}
    ]

And while this works, and I get the following lat, lon, alt functions from it:

enter image description here:

enter image description here

enter image description here

I also get the following errors that I'm wondering why I get:

GeoPositionXYZ::invcoord: "\!\(\"{2.560453600382259*^6 - 0.233218833096895*d, 5.245110323032143*^6 - 0.81456199785816*d, -3.81977214219131*^6 - 0.531128729720249*d}\"\) is not a valid coordinate specification."

Thread::tdlen: Objects of unequal length in {-212500.,-70833.3,70833.3,212500.}^{} cannot be combined. >>

Thread::tdlen: "Objects of unequal length in {12.2667 +3.14159\ I,11.1681 +3.14159\ I,11.1681,12.2667}\ {}\\n cannot be combined."

FunctionInterpolation::nreal: Near d = 2.2125`*^6, the function did not evaluate to a real number.

Does anyone have any insight into these errors?

Also, how can I adjust the quality of the interpolation / how many points are investigated?

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1 Answer 1

up vote 2 down vote accepted

I don't think InterpolatingFunction is intended to work on vector functions. The doc page doesn't say anything about it. Try for instance

f = FunctionInterpolation[{x, x}, {x, 0, 6}]

Mathematica graphics

And

Table[f[x], {x, 0, 6, 1}]

returns

(*
==> {0., 0.9999652778, 1.998576389, 3., 4.004131944, \
4.994409722, 6.}
*)

So, no two dimensional output.

It's probably better to come up with three separate interpolating functions for each of the three components of the output.


Having said that, it doesn't look like this is the end of the problems.

xyz2llaPhi = GeoPosition[GeoPositionXYZ[obs + # look, "WGS84"]][[1, 1]] &;
fPhi = FunctionInterpolation[xyz2llaPhi[d], {d, 2000000, 3700000}]

yields

Mathematica graphics

Whereas

xyz2llaPhi[200000]

yields a nice numerical result:

(*
==> -34.86629487
*)

It looks like the combination of FunctionInterpolation and GeoPosition isn't a healthy one. Somehow, the index d is held unevaluated.

A workaround would be to generate a table of values and then use ListInterpolation.

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That is odd behavior from FunctionInterpolation as Interpolation can handle $\mathbb{R}\to\mathbb{R}^n$ functions just fine, and likely $\mathbb{R}^m\to\mathbb{R}^n$. Consider: Interpolation[{#, {#, #^2}} & /@ Range[0, 5]]. –  rcollyer Apr 15 '12 at 1:37

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