Two paths, or one?

It all started when I was riding my bicycle in to work last week.

As I completed a left turn out of an intersection, it hit me: would my rear tire ever become perfectly in line with the front tire?  Mathematically it wouldn’t, would it?  Wasn’t that some kind of asymptote?

Here’s my bike finishing the left turn:


…and the position of its wheels over time (time proceeding leftwards in this case ;) :


Don’t Give Me That

And don’t give me “You’re going to joggle the handlebars and the rear tire will cross that asymptote.”  Because if that happens (as I admit it will (repeatedly if you ride like I do!)), you will have merely shifted the asymptoticity to the other side — the rear tire will be approaching from positive instead of negative infinity!

Concentric Circles

We didn’t solve that one, but while I was explaining my question to some co-workers last week and drawing on my white board, I drew the paths of my bicycle’s wheels as I had performed the left turn:


But someone took issue with the two separate paths, saying they should be one path.

If It Weren’t For The Lean

This seemed easy to address: of course they trace out two different paths, because (in the extreme case) if you turn the handlebars perpendicular to the rear wheel, the rear wheel doesn’t go at all:


Yes, said someone — but it’s different when you’re leaning — or else the rear tire must do some skidding.

Hmm.  Leaning?  Skidding?  Now I could envision it both ways.   There’s a discussion about this out there, but I got lost in the details. So…

I brought in my bicycle and we did an experiment in the parking lot, last Friday.

Which was it, gentle reader?  Two paths, or one?


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