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I am by no means a mathmatician. I guess you could say that I am a mathematically inclined individual, but never made anything of it until I was in my 30's and became a software engineer. Although I've never taken the math classes that I should have, the concepts still fascinate me, and I have a question.

I was watching an episode of Numb3rs and they stated what seemed like a fairly simple geometry/trigonometry question that I just could not wrap my brain around entirely.

They stated that with a few photos of the same location, by looking at the shadows caused by the sun, and knowing 1 distance in the pics and knowing the "exact" time stamp of the images, that they could calculate the exact (within a hundredth of a degree) location of the photos.

As I said before, I am no math expert, but wouldn't you also need 1 angle? Is what they said accurate? Could you calculate the angle based on the timestamps and the shadows given the height of the object that made the shadow?

Edit: I'm curious, If I posted 4 pictures (15 min apart, with accurate timestamps) with at least 1 item that was a known length, would some one take me up on the challenge of locating where the photos were taken?

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The sun moves over the time and its relative movement is different considering different places on earth. You can easily extract an angle if you know a distance from the picture too. – Listing Jul 9 '11 at 12:22
There exists a book Devlin K., Lorden G. The numbers behind NUMB3RS Although I do not think that htey deal with this particular question, it might be interesting for you, if you're a fan of that series. – Martin Sleziak Jul 9 '11 at 13:28
It's a modern twist on the Sherlock Holmes story ‘The Musgrave Ritual – Gareth Rees Jul 9 '11 at 20:40

Here is a sketch. Imagine that you've put a stick in the ground somewhere in the world whose height is known and you take pictures of its shadow at various times. Assume it has a shadow in each picture, and assume you can accurately deduce the length of the shadow from the pictures. Any given picture will tell you how far you are from the point on the Earth to which the Sun is closest at the time the picture is taken (the longer the shadow, the farther you are from that point; this should be some pretty straightforward trigonometry), so it narrows down your location to a circle (similar to a circle of latitude). This point moves as the Sun moves. If it moves enough, two pictures narrow down your location to the intersection of two circles, which should be two points. Three pictures should be enough to narrow down the location of the stick completely, in theory.

In practice I see some minor problems.

  • It's not completely trivial to deduce the length of a shadow from the length of a stick and a picture of the two; you need to be able to figure out what angle the picture was taken at relative to the slope of the ground. This should be doable if there are enough other things in the picture, but it sounds like a pain.
  • If there are errors in your measurements, your circles may not intersect unless you thicken them to account for error, and then they may intersect in more than one region, so it may be necessary to use more than three pictures.
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Two minor points: If the accuracy is supposed to be within one hundredth of a degree, you'll probably have to take into account that the Earth is a geoid, not a sphere, since the ellipticity is about $1/300$. And you need to be able to figure out at what angle the picture was taken relative to gravity, not relative to the ground, which might be sloped; that's going to be more difficult and might require the picture to contain liquid surfaces or plumblines :-) – joriki Jul 9 '11 at 15:51
I started writing an answer to this question an then stopped because I realized that it's actually quite complicated: I think it's possible to determine the location even without being able to determine the direction of gravity from the picture, but then you need to treat both the location and the direction of gravity as variables. Another approach would be to assume that the ground is horizontal and then do a best fit on the resulting scattered measurements. The resulting error might only be of the same order of magnitude as other errors made in processing the picture, anyway. – joriki Jul 9 '11 at 15:53
Hmm. Yes, I forgot that I was assuming the Earth was a perfectly round sphere... – Qiaochu Yuan Jul 9 '11 at 15:55
2 :-) – joriki Jul 9 '11 at 15:58

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