Chord Star 3: Remote Radii

Suppose you find some big curved thing out in the world. Some things are curved more tightly than others. But how much more? How can we put a number on how tightly curved something is?

One way is to figure out the radius of curvature. The smaller the radius, the tighter the curve. (Would you tell students this at the beginning? Of course not. But I can’t describe how this can work without giving things away. So consider this a report on my own investigation.)

Let’s apply what we learned two posts ago. To review, we found out that if you pick a point inside a circle, and run a chord through it, the point divides the chord into two segments. The lengths of those segments are inversely proportional, that is, their product is a constant—it’s the same no matter which chord you pick.

Then, last time, we saw how that product varies with the point’s distance from the center.

Let’s see how we can use this to measure radii of curves out in the world. The cool thing is that we can do this remotely. Unlike most radii in school geometry, we can figure out the radius of curvature without ever finding the center of the circle.

The picture above is a hint. If that’s enough for you, don’t read further! Go do it!

Let’s find some arcs. I noticed that in my neighborhood, there is a block where the curbs seem to have different curvatures as they go around corners. In the picture (from Google Earth) you can see that the curb at A looks tighter than the one at B. You can imagine where the center of A is, but for B—it might be inside the house! Let’s measure the radii of curvature anyway.

Here we go: in my basement, I found a piece of cabinet bracket left over from the kitchen remodel. It was almost exactly 50 inches long. The key thing is not the length (though long is good) but that it was straight. A yardstick would work for many applications.

I marked the center of this piece of metal. So each side is 25 inches. Now the insight: This could be a chord. If the center is “the point,” the product is 625 (square inches, though I don’t currently see how to find a useful area this corresponds to).

If I take this piece of metal and lay it as a chord on a large piece of arc—the photo is at curb A—I can measure the perpendicular distance (s for stub) from the middle of this chord out to the arc.  That little stub is on a diameter of the circle; the stub distance times the rest of the diameter will be 625.  That is, . At this point, we should test our thinking: if the curve is tight, that gap in the middle, the “stub,” should be big. So this formula should give us a smaller radius as the stub, s, increases. (This is true as long as s < 25—showing and understanding that is interesting in itself.)

Onward! The picture at the beginning of this post is a close-up of that little ruler in the setup on curb A. If you squint you can see that the stub distance s is about 2 inches. So , which leads to a diameter of about 315 inches, or a radius of about 13 feet.

Here is the stub at curb B:

Here, the stub distance is about 1 inch, which yields twice the diameter: about 26 feet. Both of these distances are completely reasonable.

This has been a quick taste of the investigation. A bunch of questions should occur to you, for example:

• How accurate is this technique?
• How can you check it?
• Where else can I find big arcs besides curbs?
• How long a stick do I need to measure the radius of the track at school?
• If a curve is changing its radius, what answer do we get?
• How bad is the approximation d = 625/s? (And why would we even think of it?)
• Suppose I want to measure the diameter of a big tank. I can’t get inside. How can I use the same technique if I can’t place my stick as a chord?
• How does this relate to the old how-far-can-you-see-from-the-cow’s-nest problem?
• Is it OK to use the empirical discovery about products if you could never come close to proving the relevant theorem in geometry?

I’d love to hear more from you all. Let me know.