Monday, October 23, 2023

The $1.50, Ultra Compact, Field Expedient Rangefinder

I wanted to see if I could DIY a stadiametric rangefinder. But, before I could start crafting I needed to understand the math behind this distance estimation technique.

One clue as to how stadiametric rangefinding is calculated, comes from the unit of measure found in military spotting scopes: the mil. Mil stands for milliradian and is a measure of angular distance. From this, I inferred that I was going to be digging into trigonometry. I braced myself some complex math. When I finally untangled the how this all works, I found the process to be delightfully simple. In fact, we can trivially derive it.

Decoding the Math

Consider this crude drawing of me standing in front of an object:

The object is ad (actual-distance) away from me, and is ah (actual-height) tall. Assume that I know ah, and am trying to figure out ad. Now imagine I grab a ruler and hold it up in front of me. The picture now looks like this:

The length of my arm is od (observed-distance), and I can read off of the ruler how tall the object appears to me, oh (observed-height).

The magic, if you will, is that these measurements are in proportion. That is:

od   ad
-- = --
oh   ah

Intuitively, this makes sense: how tall the object appears to me is directly related to how close I stand to it. But, there's no reason to trust your gut here: the mathematical principle in play is similar triangles and it's more than worth your time to read up on it.

As with any proportion, if you know three values you can calculate the fourth. This suggests that there's nothing to actually build to create this type of rangefinder. I simply need a ruler I can hold out in front me. As luck would have it, the wallet sized fresnel lens I carry in my man-bag has both inch and centimeter markings on it. By the principles above, I should be able to calculate distances in the field using it. Let's try this out.

Let's Try It

I situated myself on the edge of my driveway and peered down to a bank of traffic lights in the distance. My goal: find the distance to the traffic lights.

I held up the wallet sized magnifier, and read .7 centimeters off of it. Keep in mind, all I'm interested in is the ruler on the magnifier; I'm ignoring the magnification itself.

Using a sewing style measuring tape, I measured the distance from my face to my outstretched arm. That came to 66cm. I also Googled around to figure how out how tall a bank of traffic lights may be. The consensus was that each light was 12". So I used: (12" x 3) + 4", or 40", as my estimate. I now had all the information I needed to fill into the formula above:

(ad / ah)      = (od / oh) ; the formula
(ad / 40")     = (66cm / .7cm) ; known values plugged in
(ad / 101.6cm) = (66cm / .7cm) ; inches to cm
ad             = 101.6cm x (66cm / .75cm) ; solve for 'ad'
ad             = 9,579cm ; divide and multiple values
ad             = 95.79m ; convert cm to m

According to the math above, the stop light was about 96 meters away. I then pulled up Google maps, switched to satellite view, and measured the distance on the map. It came out to a shockingly close 92 meters. I'm amazed that my estimate was so close to what I measured in Google Maps. Still, I suppose the math is sound, so I can't really complain.

A Little Field Prep

While there may be nothing to build to do this kind of range finding, there is important information to know. To make performing these calculations in the field easier, I created a text file containing some common unit conversions, as well as some sizes of standard objects (thanks range-r-card for inspiration!). Here's what in the file:

Arm length: 66cm

To centimeters
inch x 2.54
feet x 30.48
yard x 91.44
meter x 100
kilometer x 100,000
mile x 160,934

From centimeters
cm / <factor above>

traffic light: 12" (30.48cm)
person: 5'9 (175.26cm)
avg SVU: 1.9m (190cm)
door: 2m (200cm)
semi truck: 3.5m (350cm)

I printed this information out as a cheat sheet and did a bit of poor-man's laminating by covering it with packing tape. I then tucked it in to the same slot that holds the fresnel lens.

And just like that, I'm ready to do some field expedient distance estimation. And best of all, the setup was free and doesn't add any bulk to my gear.

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