Re: [math-fun] Jaywalker geometry
Marc LeBrun <mlb@well.com> wrote:
It sounds fascinating, but I'm really having trouble picturing this. Are the grid points at the intersections of these narrow-most roads?
Yes.
Are the jays walking "diagonally" along them or across them or what?
Diagonally along them. Each straight section of road is a long narrow rectangle, and the jaywalker takes the diagonal of that rectangle if he's turning left at one end of it and right at the other. (Okay, it's ambiguous which side of the road he starts and ends his trip on. Might as well compromise on the middle as Gareth implicitly does.)
You imply the shortest path from 0,0 to 2,2 goes thru 1,1.
Right.
But why? Given some arbitrary path from A to B, how do we compute its length? Can you help by providing some worked examples or the like? Thanks!
Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
The total distance is some unpleasant thing involving square roots of quadratics in h, but e.g. with h=0.001 it's about 3.994 whereas the path that goes (0,0) - (2-h,h) - (2,2) has length about 3.998.
It's really quite straightforward. It the road has length 1 and infinitesemal width w, the diagonal is sqrt(1^2 + w^2), which equals 1 + w^2/2. Proof by the FOIL method: (1 + w^2/2)^2 = 1 + w^2/2 + w^2/2 + w^4/4) = 1 + w^2. (The fourth power of an infinitesimal is vanishing in comparison with the square of the infinitesimal.) It the road has length L and infinitesemal width w, the diagonal is sqrt(L^2 + w^2), which equals L + w^2/2L. Proof: (L + w^2/2L)^2 = L^2 + Lw^2/2L + Lw^2/2L + w^4/(4L^2)) = L^2 + w^2. Since we're starting and ending in the middle of the width, it makes more sense to use Gareth's half-width h than my full-width w. You can use the same equations with w replaced by h, since you just have narrower rectangles. It's past my bedtime. I'll finishing working it out in detail tomorrow or Tuesday if someone doesn't beat me to it.
This is the first I've seen Gareth's assumtion confirmed as the desired intent, i.e., that the starting and ending locations are in fact in the centers of street intersections, rather than one of the street corners as in any real-world situation. Given this unintuitive definition, I would suggest that the term "jaywalking" is misleading, since even under the most extreme defintion of jaywalking, these paths do not qualify. Perhaps some term that does not involve walking would make more sense. Tom Keith F. Lynch writes:
Since we're starting and ending in the middle of the width, it makes more sense to use Gareth's half-width h than my full-width w. You can use the same equations with w replaced by h, since you just have narrower rectangles.
A more apt term might be the "Theseus distance", after the Greek hero Theseus, who used a ball of string to keep track of his path through the Minotaur labyrinth. The length of the string used corresponds to the desired path length, and in this case starting in the center is intuitive. Tom Tom Karzes writes:
This is the first I've seen Gareth's assumtion confirmed as the desired intent, i.e., that the starting and ending locations are in fact in the centers of street intersections, rather than one of the street corners as in any real-world situation.
Given this unintuitive definition, I would suggest that the term "jaywalking" is misleading, since even under the most extreme defintion of jaywalking, these paths do not qualify. Perhaps some term that does not involve walking would make more sense.
Tom
Keith F. Lynch writes:
Since we're starting and ending in the middle of the width, it makes more sense to use Gareth's half-width h than my full-width w. You can use the same equations with w replaced by h, since you just have narrower rectangles.
participants (2)
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Keith F. Lynch -
Tom Karzes