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Main Question or Discussion Point
About 30 years ago (long before GPS!) we pitched our tent in a mountainous wilderness area. I took a photograph of the mountain with our tent in the foreground (attached). We now, for nostalgic reasons, would like to pitch our tent again on exactly the same spot, or at least as close to it as possible. I have a rough idea where the spot is to within several hundreds of meters, but I would like to do better than that. Could you perhaps please assist with that?
There are a number of well known peaks on the photo whose GPS coordinates and height above sea level are known. It should be therefore possible that the various angles that I could draw on a copy of the photo with lines from peak to tent to another peak would provide enough information to set up a number of equations to solve for the two unknowns i.e. the x and y coordinates of the tent position.
One assumption that I have to make is an estimate of the height of the tent above sea level in order to get the difference in height of the various peaks above that of the plane of the tent. That should not introduce a major uncertainty since I have a good topographic map with 20 meter contours and the terrain is reasonably flat over the broader area of where the site is likely to be.
One can then construct a number of right angled triangles with each perpendicular triangle passing through the tent and a peak (see figure). The one side of a given triangle is the height of the tip of the peak, the hypotenuse is the line from the peak tip to the tent. The third side is the line from the tent to the base of the peak as projected onto the horizontal plane of the tent. The lengths of some of the sides can be expressed in terms of the unknown tent coordinates.
The angles that are measurable from the photo are the angles between the various hypotenuses. Application of the cosine rule should then relate the value of the cosines of the angles on the photo to the lengths of the sides of the relevant triangles that contain the unknown coordinates.
Unfortunately I find that my equations are quadratic terms divide by the square root of the product of quadratic terms, which I cannot solve analytically. That would call for numerical methods.
Furthermore, an error is probably introduced in my approach because the assumption is that the angles as measured on the photo are the true angles as if the camera was in a position directly above the tent pole. In fact, I was standing ten or more meters away from the tent. Hopefully that is a small distance as compared to the thousands of meters to the peak tips. That then may or may not influence the size of the measured angles thus introducing an error.
Do you perhaps see a different way of how to tackle the problem of solving for the unknown coordinates so that I could enter the solution into my GPS and walk straight to the spot? It should be a piece of cake for people familiar with the maths of the far more complex situation of how GPS satellite info is derived.
To get around that uncertainty one might rather draw the lines towards the actual camera position on the photo and then place the tent about 10 meters from there after solving for the camera position. However, where on a photo is the point of the position of the camera? Presumably it is off the photo, but where? I could have taken an identical photo more than a kilometre away if I had a very long telephoto lens. Where would you then place the camera on the photo relative to the tent? It seems that the focal length of the camera must somehow then enter into the discussion as well.
Thank you for your attention.
Regards.
Wim
There are a number of well known peaks on the photo whose GPS coordinates and height above sea level are known. It should be therefore possible that the various angles that I could draw on a copy of the photo with lines from peak to tent to another peak would provide enough information to set up a number of equations to solve for the two unknowns i.e. the x and y coordinates of the tent position.
One assumption that I have to make is an estimate of the height of the tent above sea level in order to get the difference in height of the various peaks above that of the plane of the tent. That should not introduce a major uncertainty since I have a good topographic map with 20 meter contours and the terrain is reasonably flat over the broader area of where the site is likely to be.
One can then construct a number of right angled triangles with each perpendicular triangle passing through the tent and a peak (see figure). The one side of a given triangle is the height of the tip of the peak, the hypotenuse is the line from the peak tip to the tent. The third side is the line from the tent to the base of the peak as projected onto the horizontal plane of the tent. The lengths of some of the sides can be expressed in terms of the unknown tent coordinates.
The angles that are measurable from the photo are the angles between the various hypotenuses. Application of the cosine rule should then relate the value of the cosines of the angles on the photo to the lengths of the sides of the relevant triangles that contain the unknown coordinates.
Unfortunately I find that my equations are quadratic terms divide by the square root of the product of quadratic terms, which I cannot solve analytically. That would call for numerical methods.
Furthermore, an error is probably introduced in my approach because the assumption is that the angles as measured on the photo are the true angles as if the camera was in a position directly above the tent pole. In fact, I was standing ten or more meters away from the tent. Hopefully that is a small distance as compared to the thousands of meters to the peak tips. That then may or may not influence the size of the measured angles thus introducing an error.
Do you perhaps see a different way of how to tackle the problem of solving for the unknown coordinates so that I could enter the solution into my GPS and walk straight to the spot? It should be a piece of cake for people familiar with the maths of the far more complex situation of how GPS satellite info is derived.
To get around that uncertainty one might rather draw the lines towards the actual camera position on the photo and then place the tent about 10 meters from there after solving for the camera position. However, where on a photo is the point of the position of the camera? Presumably it is off the photo, but where? I could have taken an identical photo more than a kilometre away if I had a very long telephoto lens. Where would you then place the camera on the photo relative to the tent? It seems that the focal length of the camera must somehow then enter into the discussion as well.
Thank you for your attention.
Regards.
Wim
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