# the theory of parallel parking

It turns out that parallel parking is surprisingly simple.
In theory.

I first encountered this as a homework problem in general relativity. Graduate level.
Since I believe it is still assigned as a homework problem, I will merely highlight the key points, and not provide the solution.

Consider the infinite real two dimensional plane (or a Mall parking lot, after closing).

You are in a vehicle, which can be idealised as a rectangle, width a, length b, with four wheels. For simplicity, assume only the front wheels turn, on a rigid axle, and the rear wheels provide traction displacing the vehicle.

Then the entire degree of freedom of your vehicle is well described by two differential operators:
R, the infinitesimal rotation of the front wheels, and
D, the infinitesimal displacement of the wheels, translating the vehicle.

Clearly, R and D do not commute: “RD” does not lead to the same effect as “DR”

Equally clearly, this is adequate to cover the entire plane, through repeated operation of R and D in some succession.
There are a number of interesting properties, notably the vehicle is not oriented on the plane, and by assumption R and D can have either sign.

Now consider the “parallel parking problem”:

b
———-
| |
| | a
———-

———— _______________
| |
| | W
——————
L

Through a succession of R, and D operations, the vehicle can in general be translated into the “space” limited by W >= a, and L >= b

Clearly there is some minimum L=Lmin for which this is possible.
Calculating Lmin, is left as an exercise for the reader.
For any L’ > Lmin, parallel parking the vehicle is trivial, requiring merely some algorithmically determined sequence of R and D operations, depending on the initial conditions only.

QED.

1. #1 Sean Carroll
December 23, 2007

I first saw the problem in William Burke’s differential geometry book, although I think he gives credit for it to someone else.

2. #2 Charles
December 23, 2007

The problem is due to Nelson, as far as I can tell. I first encountered it in Rossman’s “Lie Groups” book, and in fact I did a fairly detailed post about this on my blog (though I made a mistaken claim, which I left in and was corrected in the comments). Normally I don’t post links to myself, but something funny seems to be going on and the link to see old posts is missing, so here’s my post on parallel parking:
http://rigtriv.wordpress.com/2007/10/01/parallel-parking/

3. #3 Paul
December 23, 2007

You assume L >= b! I used to be able to park my Civic on the streets of Chicago in L < b... Of course this required the use of a third infinitesimal 'bumper contraction' operator C.

4. #4 Paul
December 23, 2007

Dang … I meant “in L .lt. b.”

5. #5 Dean W. Armstrong
December 24, 2007

I just see the problem as getting out of a tight parking spot, just with time reversed (a la antimatter).

6. #6 Dr Skepto
December 27, 2007

Here is a hint from an experimentalist: parallel parking need not be “collisionless”…