Source
r/math•
Meta8

How can I measure distances on the surface of a sphere from an image? [Death Star]

Using only screenshots from the Star Wars movies, I'd like to measure two things:

  • The latitude of the center of the superlaser.
  • The diameter of the circle at the 'rim' of the superlaser.

Does anyone have any ideas of how I could do this from just an image? A few shots of the DS. The second one is the most useful, before the superlaser is almost straight in front.

Edit: Thanks for the help so far! I should have added: assume the Death Star has a diameter of 160 km.

Posted Jan 10, 2011, 12:42 AM UTC · edited

1424
jddes•

Here you go!

http://imgur.com/Ns1Bj

The latitude lines are spaced 1.8 degrees apart, while the ones perpendicular to the 'equator' (can't remember the name for those) are spaced 3.6 degrees apart. If you have trouble finding your information from this picture, ask and I'll give you the measurements (I don't know the diameter of the Death Star). It's not 100% accurate, but depending on what you want to do with that, I guess it will be enough.

EDIT: latitude lines are spaced 180 degrees / 100 intervals = 1.8 degrees/interval apart, as corrected by uorygl. Thanks!

8
Micro_Crabz•

looks like a disco ball

1
PhirePhly•

Now wait; what's the technical name for those things?

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Domian•

disco sphere!

10
PhirePhly•

ಠ_ಠ

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[deleted]•

[deleted]

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Meta8OP•

That is awesome! Thank you so much!

How did you do this?

Edit: Also, are you sure it's 3.6 degrees for the lines of latitude? How many lines of latitude are there?

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jddes•

I did it MATLAB, just drew a sphere on top of your screenshot, then adjusted the projection until it matched the death star. The sphere has 101 latitude lines, so 180 degrees/100 intervals = 1.8 degrees/intervals on latitude lines.

Glad you liked it!

Just curious what you want to do with that?

EDIT: latitude lines are spaced 1.8 degrees apart, not 3.6 degrees, as corrected by uorygl. Thanks!

1
[deleted]•

But latitude lines only cover 180 degrees. If you have a great circle going through both of the sphere's poles, it will intersect every latitude line twice.

2
sirphilip•

Without anything with known measurements in the image, it is impossible.

Also, I am fairly certain they shot those with a very small model.

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Meta8OP•

What if you know the diameter of the Death Star.

The actual value is disputed, but I'm assuming that it is 160 km in diameter.

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[deleted]•

There are several geek-tech books that describe the dimensions of the vessels, weapons, etc. of many Star Wars objects.

Essential Guide to Vehicles and Vessels

Essential Guide to Weapons and Technology

But, it might be in the planet and moon guide ...

2
killbot5000•

That's no moon. That's a space station.

2
harqalada•

That's no moon!

1
sirphilip•

I would use photoshop and draw a line between the things you want to measure. Then duplicate it end to end and see how many you can fit in the diameter. Divide 160 by how many you fit and that is how long the original line you drew is.

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IntrepidPapaya•

The distance across the projected image (the diameter) lies in a flat plane, so your method will accurately give you the length of an object in that plane. Two problems:

  1. If you're looking at an oblique view. The sphere of the Death Star works great because, no matter how you rotate it, the projected distance across it is the same. However, anything on the surface of that sphere will not necessarily have that same property. Just look at a globe, and watch how small a country gets as it gets close to the edge of your view of the surface. With a straight-on view of a small object on the sphere, this might not be a big deal. Otherwise, distances will be warped away from the center.

  2. Perspective. If that object is on the front of the sphere, then it's about 80 km closer than the diameter you're measuring. If it's on the back, then it's 80 km farther away. That change is going to make the object on the front look comparatively larger, and the object on the back look comparatively smaller.

2
mathboy0•

Lots of ugly inverse trig functions, with some proportions. Basically, you should be able to find the latitude of the center of the laser with an arccos, where you use the scale diameter in the picture. Then you do the same to find the latitude of the bottom of the rim (or the top), and from that and the overall radius, compute the radius of the rim.

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ElwoodDowd•

I think it IS the same model actually, just flipped left to right.

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dwchandler•

If you assume infinite focal length, treat the image as an orthographic projection (see inverse formulas). Then you can use pixel (x,y) to get (lon,lat). Use the center of the DS as center of image and that will go easier. Once you are able to get (lon,lat) of any pixel, the rest of pretty easy and depends on the radius of the DS.

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Meta8OP•

Ahh, thank you very much for this idea! I think I'll do that!

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tjrocha•
  • Trace the third photo on a sheet of paper
  • Draw a straight line from the estimated center of the DS through the center of the laser
  • Draw another line from the estimated center of the DS through the outermost rim of the laser
  • Measure the angle between the lines you drew using a protractor
  • Law of Cosines solving in terms of the DS' radius, (r)
  • Result: Radius of the laser's rim
2
JestersXIII•

Pretty sure without some other data as reference it'll be hard if not impossible. One way you might be able to do this is to find the speed at which an X-ring or Tie Fighter travels and take not about the point of entrance into the ravine in Star Wars and try to deduce the distance traveled from that point to the point where they pull out. It's extremely hard to make a pinpoint deduction but it might be possible.

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root45•

I would look at the second image and start drawing lines from the things you're interested in to the left edge. For example

  • Draw a line from the center of the superlaser to the left edge
  • Measure from the equator to that line along the edge to get an arc length
  • Measure the entire arc on the left edge from the equator to the pole
  • Take your first measurement and divide by the second to get a percentage
  • You can convert this into degrees for a latitude

You can do a similar thing for the diameter you want. Draw two lines over, measure the arc length and then find the percentage of the total arc length. Scale everything using a known diameter of the Death Star.

This is all going to be very approximate. I'm assuming the second picture is from straight on, which is probably not the case. But I think it'll be close.

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dufflad•

The length of a line on the unit surface of a sphere is directly related to the angle formed by intersecting planes. So if we take the equator as one plan intersecting the sphere and some other plane intersecting the origin then the angle formed between the two planes is equal to the distance between the two points on the surface of the sphere.

In other words look at a triangle. What I'm trying horribly to say is the side opposite an angle.

If you want to find out the distances between points on the Death Star you'd need to know the radius and some other information, such as points on the sphere or angle of intersecting planes.

Look more into spherical geometry, there is a lot I'm leaving out.

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m1k3st4rr•

The death star is fictional, so i'm not sure what level of accuracy you are looking for... but here's an estimate

Since the Death star is commonly mistaken for a moon, lets look at the size of some moons in our solar system. Some of these are under 100 km in diameter, some larger. Lets pick 100km for the diameter of our theoretical death star.

Using your second image, you can imagine rotating the death star 90 degrees to get an estimate on the latitude of the top and bottom of the 'rim', and average these to find the latitude. Lets say ~5 degrees for the bottom, ~40 for the top, giving 22.5 for the center. (You could actually measure this if you want).

The diameter is then the angle formed between opposite edges of the rim times the radius, so ~ pi/4*radius ~ 39km.

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