You can use any model aircraft equipped with a camera to measure distances on the ground. This is great for hobbyists interested in making their own maps, or if you’re just curious what the distance between two far apart objects is. With a little trigonometry (a gasp is heard throughout the room), all you need is to measure one angle and one distance. I’ll walk you through the math, it’s not actually that hard, and the end result is more than worth it.
What You Need
To do this project, you’re going to need a few materials:
- An RC helicopter, or RC airplane with 3 channel control or better – You’re going to need a stable platform to take pictures from. Helicopters are useful because you can hover in one position, but airplanes are cheaper. You might already have an airplane or helicopter around, but if you don’t, the Multiplex Easystar works well for this project. Note that to make a remote shutter function, more than 3 channels are needed.
- A lightweight camera, with remote shutter operation – You need to be able to remotely trigger the camera. This can be done a number of ways, but the simplest and most economical is to attach a servo to the top of your camera with rubber bands. Make it so that when you throw a spare channel switch on your transmitter, the servo arm will move and press down the shutter switch. As with any mod, feel free to create your own solution.
- A large protractor
- Some string and a weight – Clay works well as a weight, I’ll explain why you need it shortly.
- A drinking straw – You’re going to need this to build a sight for the protractor.
- Measuring Tape – You need to know the altitude of your aircraft, and to do this, you need to measure it’s distance from you. Landmarks with a known distance can also be used – consult any street map with a scale for these.
- A notebook / paper / pencil
- A friend to help – You can’t fly your aircraft and make measurements at the same time. Take a friend along to help you, and ask them to record the measurements.
The Math Involved
Now we come to the hard part – a little bit of math. You don’t really need to understand all these derivations, feel free to simply use the formula that I’ll give you. The problem is this: given an aerial picture, how can we figure out the scale?
We need the altitude of the airplane to figure out the image scale. This can’t be done directly, so we measure the angle (a), the distance (d), and use them to compute the height (h). This is a right triangle, and there are some handy trig functions that apply. In this case, we use the tangent function, which gives the ratio of the opposite and adjacent sides. This is expressed as follows:
By taking the tangent of the angle a, and multiplying by the distance (d), we get the height (h). Here’s the formula that you would use:
So how do you get the angle? It’s simple: take the protractor and tape a piece of drinking straw to the flat bottom. Then attach a piece of sting to the bottom center of the protractor, so that it dangles straight down the 90 degree mark. Use the ball of clay to make a weight at the bottom of the string. Now, when you tilt the protractor, the string will measure the angle. Be careful though: the angle with respect to the ground is not what’s read directly off the protractor scale. Reading the difference between the indicated angle and 90 degrees will give you the angle you need.
So why do we care about the altitude? Well, it turns out that the ratio of the altitude and the focal length of the camera is the image scale! You can find out the focal length of your camera by reading it’s manual. This is usually expressed in millimetres, so convert it to whatever unit you have used to measured the distance, and thus the altitude in. Let’s put that all in a convenient formula:
And that’s the only formula you need. Just measure the distance, and the angle, know the focal length, and you’re done.
How to Do the Measurements
All that math was fun, but how do you actually measure distances using this method? I’ll illustrate with an example:
Suppose that you’ve just gone out to a field, and want to measure the distance between a tree and a building. The first step is to find a landmark you can fly over with a known distance. Using a measuring tape or map, you find that a nearby hill is 50 feet from where you’re standing. With a friend ready to measure and write down the angle, you launch your airplane and fly over the nearby hill, taking several pictures. Your friend sights the model aircraft through the protractor – straw device built earlier, and finds the angle to be 75 degrees.
Using a calculator, you find the altitude to be:
After landing, you download the pictures and print them full size, with no scaling. Your camera’s user manual reports that the focal length is 152 mm (millimeters). Converting this to feet is easy, just type it in Google or multiply by the number of feet per millimetre. 152 mm is 0.48 feet, so you plug that into the formula we derived earlier and obtain the following:
This means that distance measured on the picture is 0.000257 times as big as the real distance. You’re almost done: using a ruler, you measure the distance between the tree and building on the image to be 1 inch. Converting this to feet (because we want the distance between the building and tree in feet), gives a distance of 0.0833 feet. Now, multiplying this by 1 divided by the picture scale gives a final answer of 324 feet.
And that’s it – you’ve just measured the distance between two objects using nothing more than a RC aircraft, camera, and a little trigonometry. Just keep in mind that you have to know the distance between the airplane and you accurately for this to work – always take pictures right on top of the marker with a known distance.
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