I present the following in hopes that you will be amused, as I was, by seeing something done that might seem "impossible" on first inspection. Perhaps also it may inspire you to undertake a similar exercise, and have the fun I had.
I'll present the information in four parts:
a) An overview
b) Details of what I did
c) Explanation of how I could have avoided several "cheats".
d) Using a spreadsheet for certain tedious calculations
Sadly, I fear there is a little bit of bad news at the end.
I was in Paris in February 2001. I took panoramic pictures of the view inwards from four points around the edge of a rough circle 4 km across (about 2.5 miles). (Just an ordinary camera).
When I got home, with just a little use of a professional's map (See "Avoiding cheats" for discussion of how this could have been avoided), I made a map of some Paris landmarks. With no special equipment, I managed to plot three landmarks to within about 30m of their actual locations. (30m is about 30 yards... The landmarks were more than 30m long!) Three more were plotted to about 50m of their actual location, one more within 70m, one more within 100m, and another was plotted... but my position was about 250m from the correct position. Two further landmarks were "half plotted".... I knew they were somewhere along a line on my map... but didn't know where along it.
All of the above on the first try, done after I got home from the trip. If I were in Paris and could retake some of the photos, a much better result would emerge.
I hope by now you've said "cool". If not- think about it: those landmarks were plotted on my map to within about a 70th of the distance I was looking across when I saw the landmarks from my four vantage points. I never visited most of them.
Just before going on: If you want to try this at home: You can repeat the exercise easily almost anywhere if you are willing to undertake a map of a smaller area. Not as exciting, but more manageable for a first attempt! All you need is three locations, preferably on the periphery of the area you want to map, and preferably well separated from one another. The photos must each show the places the other two photos were taken from, and some "features" to be plotted on the map. After 35 years, I still remember missing out on a tradition at my secondary school: A similar exercise was conducted annually, and the results achieved each year compared with the results from prior years. If you are going to start something like this at your own school, think carefully, and select some features that will remain fixed and visible in the years ahead.
So- How it was done, in detail:
A sheet of tracing paper was placed over a suitable map of Paris. (Suitable: Reasonable scale, included points of interest.)
The positions of the Eiffel Tower, Tour Montparnasse, and Sacre Coeur were marked. They will be called A, B and C respectively, from now on. These were the tree main places from which I took pictures. Using the professional map is NOT necessary... see "Avoiding Cheats", below. Lines were drawn between the three base positions.
The three lines made a triangle; the size of the angle in each corner was measured with a protractor. The angle in the corner representing the Eiffel Tower is (usual geometric convention) called BAC... i.e, it is the angle formed by the lines from B to A and thence to C.
Now... the explanation gets a little tedious here, but bear with me!
Let's consider plotting the location of St Suplice in SE Paris. Is is nearly directly north of B, and nearly on the line AC. We'll refer to it as point F. In practice, you would approach the work in a different sequence, but the following should make the principles clear.
First we will draw in angle CAF. On the photo taken from the Eiffel Tower (A), the distance from Sacre Coeur (C) to Tour Montparnasse (B) is 423 mm. Angle BAC (as measured on the map as drawn so far) is 84.5 degrees. On the photo, the distance from C to F is 295.7 mm. From these figures, you can work out that on the map, angle CAF should be 59.1 degrees. ((295.7/423)*84.5) Draw that line in on the map. You already have points A and C and line AC on the map. Thus there is only one place you can draw a second line to create angle CAF. (In practice, you would draw only the part of that line which falls near where point D should be, but there's no harm in drawing too much.) If you are reading this in a medium including the figures, see figure 1, the story so far. If not, you can imagine what we have so far, or you can sketch it from the following on a piece of paper at least 14 cm wide, 18 cm high: Put A at 2.2,6.9 (i.e. 2.2 cm from left edge, 6.9 cm from bottom edge). Put B at 7.4,2.0. Put C at 11.6, 15.0. The line showing the direction towards F from C starts at C and passes through 12.5,4.0. You can draw all of it without any harm.
(By the way... using a spreadsheet for the arithmetic REALLY helps! Ideas in part d)
So far, so good!
Now, by a similar logic, draw ACF. (Take a moment to get clear in your mind exactly what this line represents. Get clear the relationship the real world and the diagram (map).)
On the photo from Sacre Coeur (C), from B to A is 104.3mm long. Angle BCA is 31.0 degrees. On the photo, from B to F is 128.9 mm, so angle ACF must be 8.4 degrees. Draw the line from C towards F that creates angle ACF on the map. You only need the bit of the line near where it crosses AF. In theory, where these lines cross is the location of St Suplice (F). (See figure 2)
In a perfect world, we'd be done with plotting the location of F. As it isn't a perfect world, a check is useful.
To make the check, I turned to the third panoramic view, the one taken from Tour Montparnasse. By the same procedures, I determined that angle ABF should be 87.8 degrees. When drawn in on the map, that line fell very close to where the other two lines towards F crossed. (Figure 3) Hurrah! Initial result confirmed!
Making things even better and including adjacent areas....
Plotting the location of other landmarks was completed by the same method. Unfortunately, not all of the landmarks were visible in every photo. Happily, I was also able to take a panorama from the top of the Arc de Triomphe. It's location on the map could be determined in the same manner as St Suplice's (F's) location was determined.... if l'Arc had appeared in at least two of my other panoramas. It didn't, so I again cheated, and used the professional map to place l'Arc on my map. (The one photo I did have resulted in a line that went to the right place!) Once it was placed, I could draw lines from it towards landmarks. In some cases this gave me a further check on already plotted points, (figure 4) in others it provided the crucial second line, so that something could be located by where the lines crossed. (In one disappointment, two lines were not enough: One ran from SE to NW while the second ran from NW to SE. This was point O, which is a church on the line between B and I. Oh well. If my picture from the Eiffel tower hadn't been taken at night, the church would show in that, and I could plot it on my map. Next time!!
So! You have what you need to make the maps. Once you've mapped the first area, you can "grow" your map outwards with measurements taken from points determined in the first phase.
A simplified version of my map appears in figure 5. Figure 6 is a collection of blow ups of the crossings of my lines. Note the scale. Where there's a small dot, that is where the feature is, according to the professional map.
A word about taking the panoramic photos: don't set your camera on an extreme wide angle setting... some distortion creeps in at the edges. (If you have a camera that only has one setting (not an SLR), then incorporate significant overlap between frames.) In 35mm SLR terms, a 50mm lens is okay, but go to about 80 or 10 if cost isn't a big deal, and you want a good result. Be sure to get some overlap between each frame!! Put another way: I used about one frame for 20 degrees of view.
c) Avoiding "cheats":
Don't worry- this is fairly simple, after what you've already made your way through.
The big "cheat" in what I actually did was made necessary because I didn't know the size of the following angles: ABC, BCA, and CAB. If I had known those angles (more on this in a moment), I would have drawn my map as follows....
Sketch, while in Paris, the ROUGH relationship between A, B and C.
To do map of the A, B, C in this instance: Put a point on my paper near the bottom edge, roughly in the middle, and call it B. Put a second point on the paper about the paper's width up from B, about 3/4s of the way across the page. Call it C.
If B and C are in the "wrong" locations, the only consequence is that some of the landmarks you want to plot will be off the edge of the page.
You know the size of ABC (more on this later). You map has points B and C on it. Draw in line BC. Draw the line from B that completes angle ABC. There's only one place it can go and make ABC the right size. (Assuming you do it on the right side of BC.)
You know the size of BAC. Draw the line from A that completes angle BAC.
A is where BA and CA cross! As you know angle ACB, you can use the angle made by the lines to check that all is well.
One little detail: You won't know the scale of your map. I'd recommend cheating and using a professional's map to determine how far apart you place B and C, so that your map will have a sensible scale. The alternative is to measure the distance between two landmarks on your map and in the real world, and derive the scale from that. You need to measure a big distance to get an accurate result.... not something that is easy to do in most realistic circumstances.
So- how do you know the three angles? Of course you only NEED ABC and ACB, but if you're going to measure those, measuring the third is no big deal.
If you have a camera with a fixed focal length (i.e., no zoom), you can proceed as follows:
Put camera on tripod or table. Get a friend to walk out, say, 20' from the camera, and left until he/ she is at the edge of the photo. Get friend to push a stick into the ground there. Get friend to walk to the right until at the other edge of the photo. Push stick into ground. Measure angle! (Doing it accurately will be a challenge!) From now on, you know that the full width of photos from that camera represents x degrees. Of course, this assumes that whoever processes your films uses exactly the same portion of the negative each time......
Better... but not always possible... but will be okay with zoom lenses, too:
Make up a "T" out of string. Make the top 20 feet long, and the stem 20 feet long. In your panoramas, at least once for each zoom setting, include two of your friends in the photo.... holding the ends of the top of the T, with it horizontal, with the base of it's stem at the camera. (Figure 7) The distance on the photo between the tips of the top of the "T" will represent a 58 degree arc.
Lastly, with different strengths: By hook or crook, you measure the angle from your vantage point between two landmarks in the panorama. From that, back at your desk with your photos, you can work out everything else you need. A sophisticated tool/method is explained at www.arunet.co.uk/tkboyd/mm1.htm. Alternatively, use, rotated 90 degrees, the tool used by early navigators to determine the elevation of celestial bodies above the horizon. It is a stick, say 3 feet long, with a second stick, say 2 feet long, attached to make a cross... but the cross piece must be able to slide back and forth along the first. The long stick is placed on your cheekbone. (Care, please!! Don't get your eye poked!) Then the short stick is slid in an out until it is at the right place to appear to just "touch" the two things you are trying to measure the angle between. From the length of the crosspiece, and how far it had to be from your eye, you can calculate the angle.
Better yet: Devise your own solution!
In a single worksheet, for each panoramic photo, you can set up a small table. I trust you'd label the elements of your worksheet, unlike what I present below. I left out the labels in hopes that the row numbers/ column letters would be more clear. These are actual figures from my Paris map, so you can use them to check the calculation parts of your worksheet.
A B C D
1 104.3 31.0
3 0.0 0 G Notre Dame
4 15.3 4.5 J Pantheon
5 55.0 16.3 F St. Suplice
6 79.6 23.7 B Tour Montparnasse
7 183.9 54.7 A Eiffel Tower
8 215.0 63.9 K Spire
A1: Distance on photo from B to A, mm
B1: Size of angle BCA
Column A3 to A8: Measurements from photo, in mm, all distances from position of Notre Dame on photo. You probably make things easier for yourself if you list the landmarks from left to right. Your spreadsheet should allow you to insert a row when something has to be added after your first attempt at the data entry.
Column B3 to B8: These figures calculated. Enter the formula once in B3, then use copy/paste to enter B4 to B8. You'll need to use "absolute" references to A1 and B1. In Quattro, this is done as follows:
Column C: You enter these simply by typing in the letters assigned to the various landmarks.
Here's a chance to use a nice spreadsheet feature known as a lookup table. Somewhere on the worksheet, lets say starting at E4, you type out ONCE the relationship between ID letters and the features they stand for....
4 A Eiffel Tower
5 B Tour Montparnasse
6 C Sacre Coeur
7 D Invalides dome
8 E Ferris Wheel
Once the lookup table is done, you only need to enter the letters of features in the other tables on the worksheet. Next to the letters, in column D, with a formula, the spreadsheet can fill in the longer description of that landmark. In Quattro, the formula for D3 would be:
This says: "Find the value from C3 in the first column of the block E3F16 and return whatever is in the 1st column to the right of it."
I hope you get some fun from this, if only as a "spectator". Who would have thought that simple photos could be enough? I am going to try to work up a classroom pack with simulated photos so that the calculating and plotting exercise can be enjoyed without all of the overhead of the picture taking. Send an email if you would be interested. I would also be happy to send you the figures referred to in the text, if you will send me a postal address for them to go to.
Sadly, I must give you a little bit of bad news, here at the end of this page.
When I tried again, in different circumstances, to make a map from photos... it didn't work! I don't know what I did differently the second time. Was it just beginner's luck the first time?? I don't think so. I think the principle is sound.
I suspect that what I did wrong may involve the focal length of the lens I used for taking the picture. Any lens, at any zoom setting, introduces certain distortions into the image recorded. I suspect, that if you want to use the photos as above, there is a "right" zoom setting to use.
Answers on a postcard, as they used to say. An email would be equally welcome. The Paris exercise really did go as described!
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