Imagine you’re riding your bicycle along a straight stretch of road. Up ahead you see that the road curves sharply to the right. You’re really flying along so you lean into the curve and keep pedaling until you emerge onto another straightaway.

Fortunately, you make it to your morning math class in the nick of time. Unfortunately, your teacher asks you to get out a map and draw a diagram of the exact path that your bicycle took. You know this guy is looking for a fair degree of geometrical precision so you take out your compass and straightedge. You look at the bright side, it could have been a word problem.

Here’s your diagram:

“That’s a nice drawing of the road, now please add a dashed line to indicate the exact path that your bicycle took.”

Well that’s still not too bad, you stayed in your lane along the side of the road so it’s easy to draw in a new line parallel to the edge.

“Nice. What’s the radius of that curve on your path?”

Oh that’s how it’s going to be. Fine, you understand how to take a measurement off of the map and use the scale to figure out the actual dimensions. Take the measurement, push a few buttons on your calculator, the inner edge of the road has about a 200 foot radius, add another 10 feet to get to my bike lane and there we go, 210 feet. You write that along the dashed curve.

“That’s the path you took on your bicycle?"

"Yes."

"Impossible. How did you go from riding straight to turning on a 210 foot radius?”

He’s trying to throw you, but it isn’t that early in the morning, no sirree. You did the homework, you know the vocab.

“The straight line is tangent to the curve at both the start and the end, so the transition is smooth.”

“It looks smooth, but think about it. While you were riding straight ahead your handlebars were perpendicular to the frame of your bicycle. In the middle of the curve, they were angled to the right. Do you remember snapping your handlebars between those two positions instantaneously at the point of tangency?”

“Er, no.”

“You must have changed from going straight to turning on a 210 foot radius over some small period of time. What was the radius of that straight line?”

“It doesn’t have a radius, it’s a straight line.”

“What was the radius when you first started to turn?”

Now that was interesting to think about. It went from no radius to some very large radius – perfectly smoothly!

“I guess you could think of the radius of the straight line as being infinite.”

“Ah. It sounds like as you were moving forward your turning radius was changing from infinity down to 210 feet.”

“Yeah, then it changed back up to infinity as I came out of the curve.”

“Good. A path that follows a changing radius from a common center like that is called a spiral. Since there’s enough room in the width of the road for your bicycle (or a car) to follow its own spiral path without leaving a lane they usually don’t bother building them in. But railroads are different, the steering is all done by the rails. If they don’t follow carefully planned spirals in and out of curves they’ll get slammed the way you would have if you’d snapped your handlebars from one position to the other at the point of tangency.”

Jeff Allen