Street grid pattern
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From: Chip Eastham on 19 May 2010 14:13 On May 19, 10:44 am, Ron Peterson <r...(a)shell.core.com> wrote: > Some cities have a rectangular grid pattern for their streets and in > the worst case, trips can be 40% longer than a straight line trip. > Other cities, have streets that follow the geography because of > mountains, streams, and lakes with some trips being several times as > long as a straight line trip. > > Is there a non-Cartesian grid pattern for a flat community that does > better for the worst case than the Cartesian grid pattern? > > Am I asking the right question? > > -- > Ron Hi, Ron: I'm sure this is a fertile area for posing challenging problems. One version of such problems starts with a fixed set of points in the plane and asks what the shortest total length needed to connect them all is. In one variant you are only allowed to put straight lines between the given points, but it is possible to reduce the distance further by introducing new endpoints. E.g. three equilateral points, say 1 unit apart so that the shortest connecting "roads" are two sides of the triangle for a total distance of 2 units. Instead we might put a point in the middle and have three line segments to that midpoint, for a collective distance of sqrt(3) units. I sort of remember that a lot of work has been done on finding where N points in the unit circle maximizes the minimum distance needed to connect all N points. regards, chip
From: Mike Terry on 19 May 2010 14:39 "alexy" <nospam(a)asbry.net> wrote in message news:ht17h0$6lb$1(a)news.eternal-september.org... > "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote: > > >"alexy" <nospam(a)asbry.net> wrote in message > >news:ht11oe$l3t$1(a)news.eternal-september.org... > >> Ron Peterson <ron(a)shell.core.com> wrote: > >> > >> >Some cities have a rectangular grid pattern for their streets and in > >> >the worst case, trips can be 40% longer than a straight line trip. > >> Actually, that is true only for a long trip. Worst case is a trip from > >> x street, midway between y avenue and y+1 avenue to the same place on > >> x+1 street. As the crow flies, it is 1 block, but by road it is 2 > >> blocks, a 100% longer trip. > >> > >> >Other cities, have streets that follow the geography because of > >> >mountains, streams, and lakes with some trips being several times as > >> >long as a straight line trip. > >> > > >> >Is there a non-Cartesian grid pattern for a flat community that does > >> >better for the worst case than the Cartesian grid pattern? > >> > >> With streets laid out in equilateral triangles, the worst case is a > >> 75% longer (actually sqrt(3)-1) trip, but unlike the square blocks, > >> the potential inefficiency doesn't decrease with distance. So the > >> triangles are better for short trips, but worse for long ones. > > > >This seems the wrong way round to me. Also the 75% seems wrong... > > > >For a triangular grid: > >- max inefficiency for short trip is 100% > > You are correct. I was thinking from the midpoint of one side to the > opposite apex. But the greater inefficiency is from points on two > sides, equidistant from the junction of their two sides (and not too > close to the other end of the street). > > >- max inefficiency for long trip is about 15% > > I don't get this. Assume the streets are laid out (using compass > points) 0-180, 60-240, and 120-300. If you want to go due east, you > are forced to take roads heading either 60 or 120 to make eastward > progress, and N-S roads to bring you back to the intended due east > course. How can you do that with only 15% inefficiency? You go either 60 or 120. Your intended direction is 90, so you are only travelling at 30 degrees "off course". 1/cos(30 deg) ~= 1.155 (i.e. about 15% overhead) > > > > >For a square grid, both are about 41% > > How do you get from 13th street, half-way between 1st and 2nd avenues, > to 14th street, half-way between 1st and 2nd avenues while walking > only 1.41 blocks? Hey you're right! Worst case for square grid is 100% overhead. But long distance worst case is about 41%, no? You are travelling at worst at 45 degrees from your intended direction, and 1/cos(45 deg) ~= 1.414. > > > >So the triangles are better for LONG trips, but worse for SHORT ones?? > > No, with your correction of my shart-trip inefficiency for the > triangle, I believe the triangular layout is worse for both. Square grid: Worst case : 100% Long distance worst case: 41% Triangular grid: Worst case : 100% Long distance worst case: 15% Basically, with the triangular grid you can travel closer to the direction you want in the worst case. Regards, Mike.
From: Rod Speed on 19 May 2010 14:50 Ron Peterson wrote: > Some cities have a rectangular grid pattern for their streets and > in the worst case, trips can be 40% longer than a straight line trip. But thats only with the shortest trips, so isnt as significant as it looks. And it isnt 40% either. > Other cities, have streets that follow the geography > because of mountains, streams, and lakes with some > trips being several times as long as a straight line trip. Yes, mostly with lakes and big rivers etc. > Is there a non-Cartesian grid pattern for a flat community that > does better for the worst case than the Cartesian grid pattern? Yes, a higher density cartesian grid, smaller blocks. > Am I asking the right question? Nope.
From: alexy on 19 May 2010 15:07 "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote: >"alexy" <nospam(a)asbry.net> wrote in message >news:ht17h0$6lb$1(a)news.eternal-september.org... >> "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote: >> >- max inefficiency for long trip is about 15% >> >> I don't get this. Assume the streets are laid out (using compass >> points) 0-180, 60-240, and 120-300. If you want to go due east, you >> are forced to take roads heading either 60 or 120 to make eastward >> progress, and N-S roads to bring you back to the intended due east >> course. How can you do that with only 15% inefficiency? > >You go either 60 or 120. Your intended direction is 90, so you are only >travelling at 30 degrees "off course". 1/cos(30 deg) ~= 1.155 (i.e. about >15% overhead) Well, if you want to go 1 mile east, I agree that after traveling on a 60-degree road for 1.155 miles, the eastward component of your travel will have been 1 mile. But you will be .58 miles north of the point you were aiming for. Travel down the N-S road to your target, and you will have traversed 1.73 miles to go one mile due east. >> >> > >> >For a square grid, both are about 41% >> >> How do you get from 13th street, half-way between 1st and 2nd avenues, >> to 14th street, half-way between 1st and 2nd avenues while walking >> only 1.41 blocks? > >Hey you're right! Worst case for square grid is 100% overhead. > >But long distance worst case is about 41%, no? You are travelling at worst >at 45 degrees from your intended direction, and 1/cos(45 deg) ~= 1.414. > >> > >> >So the triangles are better for LONG trips, but worse for SHORT ones?? >> >> No, with your correction of my shart-trip inefficiency for the >> triangle, I believe the triangular layout is worse for both. > >Square grid: > Worst case : 100% > Long distance worst case: 41% > >Triangular grid: > Worst case : 100% > Long distance worst case: 15% I still think the 15% should be 73%. > >Basically, with the triangular grid you can travel closer to the direction >you want in the worst case. But your "correction" is along a road that has no component contributing to your intended direction of travel, while for the square, you are making progress even on the correction leg. I think tiling the city with hexagons may actually be the most efficient, but I haven't tried to put any numbers to paper yet on that. (Enough trouble just getting it right on squares and triangles!) -- Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently.
From: alexy on 19 May 2010 15:16
"Rod Speed" <rod.speed.aaa(a)gmail.com> wrote: >Ron Peterson wrote: > >> Some cities have a rectangular grid pattern for their streets and >> in the worst case, trips can be 40% longer than a straight line trip. > >But thats only with the shortest trips, so isnt as significant as it looks. If a city is laid out on a N-S / E-W grid, how can you move in a northeasterly direction without traveling sqrt(2) times the straight-line distance between the points. > >And it isnt 40% either. I assumed he was rounding for sqrt(2) - 1 > >> Other cities, have streets that follow the geography >> because of mountains, streams, and lakes with some >> trips being several times as long as a straight line trip. > >Yes, mostly with lakes and big rivers etc. > >> Is there a non-Cartesian grid pattern for a flat community that >> does better for the worst case than the Cartesian grid pattern? > >Yes, a higher density cartesian grid, smaller blocks. Won't make any difference, assuming we are talking about idealized zero-width streets and point intersections. Obviously, with real-world street widths, you can save a little by walking diagonally up each block, and this effect would improve with smaller blocks, but I took his question to be more the idealized case. >> Am I asking the right question? > >Nope. Well, why don't you help him frame the question you would like to answer? -- Alex -- Replace "nospam" with "mail" to reply by email. Checked infrequently. |