XYZ Coordinates to Earth Plot

[lostin3d]

Lost in 3D kind of explains my knowledge of advanced Excel. I've been trying to solve it on paper and can't get my head around the complexity. Hopefully some folks can help!
I have data from 5 satellites and I'd like to know how to calculate the position on Earth of the receivers coordinates, XYZ,
I know how to convert geodetic coordinates to Cartesian if necessary.

Please, can anyone help?


Satellite View.jpg
Satellite View Data.jpg
or Cartesian Coordinates:
ECEF COORDS X Y Z
SAT1 2926991.38342522 -3882838.58596944 4144244.87430255
SAT2 3268136.54260175 -4639752.26938107 2944623.86418239
SAT3 -347247.81198599 -5693759.79190739 2888355.61305460
SAT4 3864424.69757511 -2592871.37042505 4375986.84769265
SAT5 -1491397.01293895 -3254491.69640659 5285985.02139871

Looking for more classroom geometry as the assumptions don't include any real life error. Some assumptions:
•All four signals reached your GPS receiver simultaneously
•The waypoint is at sea level
•The Earth's atmosphere has no effect on the GPS signal
•The Earth is a perfect sphere, with a radius of 6370 kilometers
•The speed of light is 3.00E+08 meters per second
•Your GPS receiver has no time differential error"

[tony h]

You might find tis helpful. http://web.cecs.pdx.edu/~ssp/Reports/2006/Monaghan.pdf

And it also depends on whether you are after a classroom geometry answer or a realworld answer with all its additional problems.

[lostin3d]

Thx tony h.
I've update original post with the assumptions. Maybe I should have left any reference to Satellites out as I think solving this using the Pythagoras theorem and some other calculations in Excel is what 's needed. I could be very, very wrong; as you can tell, I'm lost.
I think it may be basic, what is the XYZ of a line intersecting a sphere; I know the radius of the earth and the total distance from 0,0,0 to the XYZ (of the Satellite). Don't know how to calculate the XYZ of the intersection.
I'm reading the pdf you indicated; trying to figure it out a bit.

[MrShorty]

It has been a while since I have done anything like this, but isn't it basically a vector addition problem?

There's the vector from the center of the earth to you (X)
There are the vectors from the center of the earth to each satellite (U(1), U(2), U(3),...)
There are the vectors from you to each of the satellites (V(1), V(2), V(3),...)
X+V(i)=U(i). With three components to each vector, you need data from at least 3 satellites to solve the resulting system of equations. If you have more, then it becomes more of a "best fit" type scenario where you find the X vector that provides the "best fit" of the vector summations.

It seems like it should not matter whether you express and solve the problem in spherical coordinates or Cartesian coordinates. All of the tutorials on vector addition I looked up to refresh my memory wanted to express the addition in terms of Cartesian coordinates, so it might be easiest to convert each known point to Cartesian coordinates, solve the vector additions to get your Cartesian coordinates, then convert back to spherical coordinates to get the data as in latitude, longitude, and "radius" (from which you should be able to back out elevation above the surface of the earth).

[tony h]

Ok. It is relatively simple then. It is just Pythagoras.

You have to think of the satellites as three points on a plane and the receiver as being the apex.

I could explain but the Khan academy is a much better place to go (I recommend them a lot. https://youtu.be/1UQ5IbihJNI

Hopefully this link will not only get you towards the answer but also improve your understanding - which I think is your intention. Putting the formulae into Excel should not be too difficult but if you get stuck ...

[lostin3d]

Great, let me watch some Khan video and see how to proceed. I think my head is spinning faster than the Earth.

[lostin3d]

OK, update.
I was able to convert to XYZ coordinates, thx. As I'm most familiar with Pythagoras I decided to solve some triangles, no problem.

Then, I then decided to cross-check some answers to be sure I was on the right track. Using the sin function at different angles I ended up with a set of 'negative' radians and angles.

Even I know the 3 angles of a right triangle should add to 180, the 90 plus the two opposite angles. If you ignore the negative value, the two opposite angles add up to 90 degree which is perfect for right triangle.

Question is, what do I do about the negative values for radians and angles? Is this an absolute value issue?

SAT EXB 1.JPG
SAT EXB 2.JPG

[MrShorty]

The meaning of the negative angles is going to depend on your spherical convention (https://en.wikipedia.org/wiki/Spheri...rdinate_system see the sections on definitions and conventions). Most of the time, I think geographic conventions assume longitude angles are measured from the prime meridian, and increase going east and decrease going west.

It seems that there is also the question of exactly how are you visualizing these triangles. It looks like you are working with the satellites alone at this point. For each satellite, I would expect the triangle to consist of:
1) The radius/hypotenuse from the origin to the satellite's position
2) The line going down from the satellite to the XY plane that is perpendicular to the XY plane.
3) The line from the origin in the XY plane to the point where the perpendicular line intersects the XY plane.

What do the angles and distances really mean in this triangle, and how are they related to the spherical coordinates you are given?

[lostin3d]

Thank you everyone for your input and responses. I've learned a lot.

I think I found the answers to my prayers! For the not-so-gifted math or excel users, like me; this website gives some excel triangle formulas, spelled out nice and easy.

I could always grasp the concept of my goal, however lacked the other skills to make it happen.

Check out this website: http://www.xlsdesign.com/trigonometry_formulas.html