In Portland, I-405 is a loop just a mile west of I-5. Where else does a child Interstate stay so close to its parent?
Well, I-287 is combined with I-87 for several miles in New York. And I don't believe I-895 ever strays too far from I-95 in Baltimore, MD.
Quote from: ap70621 on February 17, 2012, 06:44:40 PMWell, I-287 is combined with I-87 for several miles in New York.
Yes, but it continues further south into New Jersey; as state that I-87 doesn't even touch.
In Philadelphia, PA & Gloucester City, NJ; I-676 doesn't stray too far from I-76.
In the Wilmington, DE area; I-495 doesn't stray too far from I-95.
In the Manchester, NH area; I-293 doesn't stray too far from I-93.
Quote from: ap70621 on February 17, 2012, 06:44:40 PM
Well, I-287 is combined with I-87 for several miles in New York. And I don't believe I-895 ever strays too far from I-95 in Baltimore, MD.
895 gets 2 miles away on the south side of the harbor.
Quote from: PHLBOS on February 17, 2012, 06:59:18 PM
Quote from: ap70621 on February 17, 2012, 06:44:40 PMWell, I-287 is combined with I-87 for several miles in New York.
Yes, but it continues further south into New Jersey; as state that I-87 doesn't even touch.
In Philadelphia, PA & Gloucester City, NJ; I-676 doesn't stray too far from I-76.
In the Wilmington, DE area; I-495 doesn't stray too far from I-95.
In the Manchester, NH area; I-293 doesn't stray too far from I-93.
I-676 is about 3 miles from I-76.
I-495 is about 2 miles away from I-95.
I-293 is farther than that.
Anything inside 1 mile?
I-580 at one point shares freeway with I-80 on a wrong-way concurrency.
as far as the largest distance from a 3di to its parent ... well, I-175 and I-375 only stray about a mile from their parent in Florida.
for a loop example, I-15 and I-215 are very close in Salt Lake City - about 3 miles apart at most.
I-395 in Baltimore doesn't get more than two miles from I-95.
I-277 in Charlotte is similar except it's a three-quarters loop.
Quote from: 1995hoo on February 17, 2012, 08:26:26 PM
I-277 in Charlotte is similar except it's a three-quarters loop.
True story, plus a couple more NC examples...
I-240 in Asheville doesn't stray too far from I-40, maybe a couple miles but I have never measured.
I-795 from Wilson to Goldsboro stays close parallel to I-95.
I-840 (Future) only exists as a one to two mile stub on the East and West side of Greensboro.
I-295 (Future) only exists as a couple mile stretch in Fayetteville, like G-Boro
I-140 in Wilmington only spurs 6 or so miles WSW from Wilmington
I-440 in Raleigh used to share pavement with I-40, but no longer.
Interstate H-201 stays within about two miles of Interstate H-1.
I-470 in Wheeling, WV doesn't get much more than a mile away from I-70.
Quote from: Duke87 on February 17, 2012, 09:09:08 PM
I-470 in Wheeling, WV doesn't get much more than a mile away from I-70.
Looks like a 1 1/2 at the widest betweent the two.
I-277 (in Akron) is 2 1/2 from I-77/76.
I-670 gets nearly 2 miles from I-70 before both diverge around Bexley (in greater Columbus)
There seems to be some confusion about whether we want the maximin or minimax.
St. Joseph MO is pretty small to have a 3-di in any event, so I-229 is real close to I-29.
In Miami, Fl I-395 only extends 1.2 miles from I-95. It is not even a loop, it just goes east for 1.2 miles and ends.
I-444 is close to I-244.
I-395 in MD only strays about 0.72 miles from I-95, but only because it's a short spur. I think what the OP is looking for is:
A = area enclosed by the 3di that meets its 2di twice (or an imaginary shortest-distance straight line completing the connection)
L = longest "shortest straight line" between 3di and 2di -- how far apart they are.
We want the largest value of A/L (I would actually go with A/L^2 since it's dimensionless).
I-980 in Oakland is about 1 1/2 miles from I-80, which I would guess is the closest 3di that doesn't actually touch its parent.
Quote from: kurumi on February 18, 2012, 02:02:30 AM
I-395 in MD only strays about 0.72 miles from I-95, but only because it's a short spur. I think what the OP is looking for is:
A = area enclosed by the 3di that meets its 2di twice (or an imaginary shortest-distance straight line completing the connection)
L = longest "shortest straight line" between 3di and 2di -- how far apart they are.
We want the largest value of A/L (I would actually go with A/L^2 since it's dimensionless).
I think the OP was interested in the smallest L, but the other metrics are interesting. A/L would favor loop 3dIs with long distances between their ends (I-405 CA), and A/L² would favor loop 3dIs which enclose (with their parents) highly-elongated regions (I-470 OH-WV-PA). Personally I think W would have been a better letter for Kurumi's L variable, but whatever. Also, calculating A for Baltimore's I-895 could be interesting, as a strict mathematical approach would have the region northeast of the harbor cancel out part of the region southwest of the harbor...
There's nothing unstrict about using absolute values.
Okay fine, but absolute values inside integrals can be inconvenient sometimes. Maybe not in this case (since we'd be integrating numerically anyway) but sometimes. Anyway, most general-purpose expressions for calculating the area of a region don't have an absolute value inside the integral, because they expect the boundary to go in one direction around the whole thing. Common sense and awareness of this special case would fix the situation and it could still be "strict", yes. Maybe the word I was looking for was "naïve" or "automated".
I'm surprised I-790 hasn't been mentioned yet.
I-895 in New York is a spur of its parent and maybe is just over a mile from its sprouting.
Quote from: NE2 on February 17, 2012, 10:01:15 PM
There seems to be some confusion about whether we want the maximin or minimax.
Depends on how heavy the bleeding is. :-D
Quote from: DTComposer on February 18, 2012, 03:08:11 AM
I-980 in Oakland is about 1 1/2 miles from I-80, which I would guess is the closest 3di that doesn't actually touch its parent.
Aren't the 6th Street ramps off of I-280 in San Francisco about 2 blocks from the 5th Street ramps from I-80? I'd think that's even closer.
805/5 probably is the closest (about 4 miles at full width) for a loop 3di in California though.
First thing that comes to mind is I-670 in Kansas City MO/KS. Looks to be about 0.6 miles from I-70 at it's closest point.
What about 3di's that cross or run concurrent, like I-540 in Arkansas which has a 5 Mile duplex with I-40 ?
Quote from: US71 on February 23, 2012, 10:03:31 PM
What about 3di's that cross or run concurrent, like I-540 in Arkansas which has a 5 Mile duplex with I-40 ?
Yet ANOTHER reason to re-designate I-540 north of I-40 as I-49! :bigass:
Quote from: US71 on February 23, 2012, 10:03:31 PM
What about 3di's that cross or run concurrent, like I-540 in Arkansas which has a 5 Mile duplex with I-40 ?
I think someone mentioned I-87 and I-287 in NY as an example of that.
I can't remember. Are 40 and 540 both signed on the concurrency? Or is it only 40 that's signed?
Quote from: hbelkins on February 24, 2012, 10:23:19 AM
Quote from: US71 on February 23, 2012, 10:03:31 PM
What about 3di's that cross or run concurrent, like I-540 in Arkansas which has a 5 Mile duplex with I-40 ?
I think someone mentioned I-87 and I-287 in NY as an example of that.
I can't remember. Are 40 and 540 both signed on the concurrency? Or is it only 40 that's signed?
WB, both are signed. EB, only 40 is signed, except at Exit 7 (540 South)
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In NJ, I-95 (NJ Tpke) and I-295 are very close for a few miles. Further south on the outskirts of Camden, they aren't more than 1/4 mile apart, running parallel with each other. Of course by that point, I-95 has exited/vanished/etc.
Then of course in Trenton, the parent (I-95) turns into the child (I-295) NB, and vice versa for the other direction.
But we all knew that!
Quote from: shadyjay on February 24, 2012, 09:12:00 PM
In NJ, I-95 (NJ Tpke) and I-295 are very close for a few miles. Further south on the outskirts of Camden, they aren't more than 1/4 mile apart, running parallel with each other. Of course by that point, I-95 has exited/vanished/etc.
Then of course in Trenton, the parent (I-95) turns into the child (I-295) NB, and vice versa for the other direction.
But we all knew that!
But apparently we didn't all know the New Jersey Turnpike is
not I-95 south of Exit 6, which is where it and I-295 are at their closest.
EDIT: Apologies for the tone of my post yesterday. In my stupor, I clearly missed the last sentence of your first paragraph. Although I stand by my point that
I-95 and I-295 are only spectacularly close for the two miles from Exit 6 to Exit 7 on the Turnpike, at which point I-295 veers more northwesterly towards Trenton. The distance between the two is less than a mile for this segment though.
Does PRI-2 and PRi-22 count?
No.
The first one that came to my mind was I-90 and I-790 in New York. Also I-87 and I-587 in New York.
I-94 and I-794 in Wisconsin don't stray all that far apart, either.
Mike
I think we can go ahead and just throw in all of the shortest 3dis automatically. :-) But I think the spirit of the question would be, of those 3dis that aren't necessarily the shortest, and in particular among loops and bypasses, which ones never get particularly far from their parent?
To put it another way: what is the shortest maximum distance between a 3di and its associated 2di that isn't achieved primarily as a result of the 3di's overall shortness? (The original example is I-405 in Oregon, to which I might add I-670 in Kansas City.)
In that case, I suppose I-790 might count, since its maximum distance from I-90 is less than its total length. The same can't be said, however, for I-587.
I think that's the question here; otherwise the subject line would have been "Shortest 3di?" and we all know that's pretty easy to look up!
I think this can be expressed as:
what is the interstate with the smallest ratio a/b of:
a) shortest crow-flies distance back to its parent as averaged over every point on its length
b) its overall length
I think that mathematically rigorous definition agrees well with intuition. It gives a spur like I-375 in Florida a result equal to approximately 1/2, while something like I-405 in Oregon comes out to be well less than 1/2.
this might be interesting if extended over many miles. what is this ratio if we treat I-94 as a loop of I-90, for example?
also, what is the "most greatly deviating" 3di, for which this ratio is the largest? I think in the case of a spur route extending directly perpendicularly away from its parent, the ratio of 1/2 is an absolute maximum - and some spur routes like I-175 and I-375 in St. Petersburg nearly perfectly meet this criterion, but what about for a loop route? I-435 in Kansas City comes to mind, as does I-495 in Boston.
Connecticut: I don't think I-384 from East Hartford to Bolton gets too far south of I-84, once it leaves at Exit 59 in East Hartford.
Massachusetts: I-391 isn't all that long in the Chicopee area (just north of Springfield).
Maine: I-395 only goes a few miles east from I-95 in Bangor, heading over the Penobscot River into Brewer.
Quote from: agentsteel53 on March 20, 2012, 01:53:16 PM
also, what is the "most greatly deviating" 3di, for which this ratio is the largest? I think in the case of a spur route extending directly perpendicularly away from its parent, the ratio of 1/2 is an absolute maximum - and some spur routes like I-175 and I-375 in St. Petersburg nearly perfectly meet this criterion, but what about for a loop route? I-435 in Kansas City comes to mind, as does I-495 in Boston.
I-275 in IN—KY—OH probably scores rather high for a full circumferential...
By the way, a perfect circular beltway bisected by a perfectly straight parent would score 1/π², or about .1; a semicircular loop, twice that; a quadrant of a circle (spur) twice
that.
interesting. I had a post a minute ago that said that your answer does not seem intuitively correct... but upon looking back at it, you are right. this is because the beltway counts "twice" for the same length of mainline cut off, compared to the loop road, so indeed the beltway's average is half that of the loop.
this was not something which I had considered originally, as to me it leads to the pathology of being able to thread multiple beltways through two points which intersect the mainline (imagine concentric ellipses of increasing obliqueness with one dimension remaining fixed), which brings down the average as the number of ellipses tends to infinity and the number of through roads remains one.
so, then, I offer the caveat that the mainline cuts into segments the beltway under consideration. there are two segments of I-435 to be considered, then, in my view.
Quote from: agentsteel53 on March 20, 2012, 05:58:26 PM
interesting. I had a post a minute ago that said that your answer does not seem intuitively correct... but upon looking back at it, you are right. this is because the beltway counts "twice" for the same length of mainline cut off, compared to the loop road, so indeed the beltway's average is half that of the loop.
I'm not sure I get your little explanation there, but I explain it like this: the full beltway has the same average distance
a from its parent as the half-circle loop, but its total length
b is greater, so the score
a/b is less. Either way, I agree that full circumferentials should be divided at their parent to be compared fairly under this metric.
yep, we're approaching the same intuitive result from slightly different angles - that to have the metric be fair, the paths should be split by the mainline.
btw your math (2/pi^2 for the loop road) is right. it's a bit shocking to see how rusty I am at integral calculus: took me about 15 minutes to get the answer! :ded:
the real question is, can we prove that the score approaches 1/2 as the spur road gets shorter and shorter of a semicircular segment and approaches zero length?
Quote from: agentsteel53 on March 20, 2012, 06:16:06 PM
the real question is, can we prove that the score approaches 1/2 as the spur road gets shorter and shorter of a semicircular segment and approaches zero length?
it does. dang, this took me even longer to do. if we let x be the portion of the semicircle "built" (x==1 is beltway, x==1/2 is loop road), our integral to calculate the sum of all the infinitesimal distances is:
from theta=0 to theta=2*pi*x, integral of r*abs(sin(theta))*dtheta.
knock off the absolute value because 0 < x < 1/2 for this exercise, and set r == 1 for simplicity; evaluate to get -cos(2*pi*x) - -cos(0), as the sum. write that more simply as 1 - cos(2*pi*x)
to get the average, divide by the total arc length 2*pi*x, and then to get our score, divide again by total arc length.
so we need to figure out lim(x->0) (1 - cos(2*pi*x))/((2*pi*x)^2)
let z = 2*pi*x for simplicity; as x->0, z->0 as well, so we can swap out the limit to z->0.
this is of the form 0/0, so derivative of numerator and denominator yields sin(z)/(2z)
this is more usefully expressed as 1/2 * sin(z)/z as this is a well-known limit. (proving why lim(z->0) sin(z)/z == 1 is a bit beyond my recollection at this time!)
1/2 * 1 == 1/2. QED.
All of this math is literally giving me a headache.
The Inner Loop in Rochester is a 3/4 loop off of I-490 (1/4 is multiplexed, technically). NOT COUNTING the multiplexed part, which really isn't signed as such anyway, the average distance would be roughly 2/9 if it were considered a square. This is a strange but fun exercise...
Quote from: agentsteel53 on March 20, 2012, 06:29:28 PM
to get the average, divide by the total arc length 2*pi*x, and then to get our score, divide again by total arc length.
That's why I think this metric is a little weird, because we're actually dividing by the length squared. Then again, didn't some of the previously-discussed metrics divide by the square of the length, too?
Quote from: agentsteel53 on March 20, 2012, 06:29:28 PM
this is of the form 0/0, so derivative of numerator and denominator yields sin(z)/(2z)
Yes, L'Hopital's Rule is quite useful.
Quote from: agentsteel53 on March 20, 2012, 06:29:28 PM
proving why lim(z->0) sin(z)/z == 1 is a bit beyond my recollection at this time!
I forget the actual proof, but I know it involves the Sandwich Theorem.
Quote from: vtk on March 20, 2012, 11:58:39 PM
Quote from: agentsteel53 on March 20, 2012, 06:29:28 PM
proving why lim(z->0) sin(z)/z == 1 is a bit beyond my recollection at this time!
I forget the actual proof, but I know it involves the Sandwich Theorem.
This is also L'Hopital's Rule, innit?
Quote from: NE2 on March 21, 2012, 12:36:02 AM
Quote from: vtk on March 20, 2012, 11:58:39 PM
Quote from: agentsteel53 on March 20, 2012, 06:29:28 PM
proving why lim(z->0) sin(z)/z == 1 is a bit beyond my recollection at this time!
I forget the actual proof, but I know it involves the Sandwich Theorem.
This is also L'Hopital's Rule, innit?
Yes, apparently that also applies. I think I must have learned about the Sandwich Theorem before L'Hopital's Rule, or maybe the former is used to prove the latter? This is stuff I learned a decade ago, and typing that really makes me feel old.
Hmmm, I don't think I ever learned the 'sandwich theorem' per se - it was just "duh, of course that's true". I mean Archimedes used the bloody thing.
Quote from: vtk on March 20, 2012, 11:58:39 PM
That's why I think this metric is a little weird, because we're actually dividing by the length squared. Then again, didn't some of the previously-discussed metrics divide by the square of the length, too?
I think it makes sense because the score ends up as a dimensionless figure, meaning that we can scale the world and the score remains the same. thus, we can compare something like I-90/I-94 from Wisconsin to Montana, to something like I-10/I-610 in New Orleans, and have the relative results be meaningful.
QuoteQuote from: agentsteel53 on March 20, 2012, 06:29:28 PM
proving why lim(z->0) sin(z)/z == 1 is a bit beyond my recollection at this time!
I forget the actual proof, but I know it involves the Sandwich Theorem.
Wikipedia calls it the squeeze theorem, which now I seem to vaguely recall my calculus teacher calling it that as well:
http://en.wikipedia.org/wiki/Squeeze_theorem
from there, we get that we can prove the sin(x)/x limit by using the inequality cos(x) < sin(x)/x < 1 for the two bounds.
I don't remember - how do we prove that cos(x) < sin(x)/x as x approaches 0?
Quote from: agentsteel53 on March 21, 2012, 11:08:33 AM
I don't remember - how do we prove that cos(x) < sin(x)/x as x approaches 0?
Start with tan(x)>x (I don't remember how best to prove this, but I think you can use the fact that the derivative of tan(x) (which is sec^2(x)) is greater than or equal to 1, to the curve is above a line with derivative equal to 1).
Quote from: NE2 on March 21, 2012, 12:20:44 PM
Quote from: agentsteel53 on March 21, 2012, 11:08:33 AM
I don't remember - how do we prove that cos(x) < sin(x)/x as x approaches 0?
Start with tan(x)>x (I don't remember how best to prove this, but I think you can use the fact that the derivative of tan(x) (which is sec^2(x)) is greater than or equal to 1, to the curve is above a line with derivative equal to 1).
I'm sure there must be a clever proof that cos(x)<sin(x)/x without relying on derivatives. Unfortunately I often have trouble remembering clever proofs.
Who'd have thought we'd derail a thread so badly over calculus?
Quote from: vtk on March 21, 2012, 12:39:49 PM
I'm sure there must be a clever proof that cos(x)<sin(x)/x without relying on derivatives. Unfortunately I often have trouble remembering clever proofs.
yes, I feel like using L'Hopital's rule would be circular reasoning. I don't know the validity here of using Taylor series for sin(x) and cos(x) (as those are generated using derivatives), though doing so does make the proof very easy:
x cos(x) < sin(x)
substitute Taylor series...
x(1-x^2/2!+x^4/4!...) < x-x^3/3!+x^5/5!...
x - x^3/2! + x^5/4! ... < x - x^3/3! + x^5/5! ...
0 < x^3(1/2!-1/3!) - x^5(1/4!-1/5!) + x^7(1/6!-1/7!) ...
as x approaches 0, the most significant term becomes the x^3 term, implying 0 < x^3/3. since x is positive, this is true.
Quote from: NE2 on March 21, 2012, 12:20:44 PM
Start with tan(x)>x (I don't remember how best to prove this, but I think you can use the fact that the derivative of tan(x) (which is sec^2(x)) is greater than or equal to 1, to the curve is above a line with derivative equal to 1).
yep.
tan(x) > x
sin(x)/cos(x) > x
cos(x)/sin(x) < 1/x
cos(x) < sin(x)/x
so if we assume that we know that tan(x) > x and sin(x) < x as x approaches 0 from the positive side, we can use the squeeze theorem to prove that lim(x->0+) sin(x)/x = 1, and probably also lim(x->0+) tan(x)/x = 1 but I will not attempt to prove the second here.
to prove lim(x->0-) sin(x)/x = 1 and lim(x->0-) tan(x)/x = 1, we probably do the exact same proof with certain signs flipped.
Quote from: vtk on March 21, 2012, 12:39:49 PM
Who'd have thought we'd derail a thread so badly over calculus?
I'm surprised I remember most of this. apart from the infinite series stuff, which I use as back-of-envelope calculations all the time, I had not used most of this since high school and college.
Quote from: US71 on February 24, 2012, 10:59:33 AM
Quote from: hbelkins on February 24, 2012, 10:23:19 AM
Quote from: US71 on February 23, 2012, 10:03:31 PM
What about 3di's that cross or run concurrent, like I-540 in Arkansas which has a 5 Mile duplex with I-40 ?
I think someone mentioned I-87 and I-287 in NY as an example of that.
I can't remember. Are 40 and 540 both signed on the concurrency? Or is it only 40 that's signed?
WB, both are signed. EB, only 40 is signed, except at Exit 7 (540 South)
(https://www.aaroads.com/forum/proxy.php?request=http%3A%2F%2Ffarm3.staticflickr.com%2F2634%2F4073936870_95a7bca751_z_d.jpg&hash=cf1d54ec817b129c82cdcb6e757acd1fc136aa2a)
That sign on the left could qualify under the Worst Of Road Signs thread. Each set of numbers on the sign shields goes higher than the other.
....and its Clearview (yecch!!!).
Quote from: agentsteel53 on March 21, 2012, 12:58:28 PM
Quote from: vtk on March 21, 2012, 12:39:49 PM
Who'd have thought we'd derail a thread so badly over calculus?
I'm surprised I remember most of this. apart from the infinite series stuff, which I use as back-of-envelope calculations all the time, I had not used most of this since high school and college.
This thread demonstrated to me why I topped out in college mathematics at right around this level. Once I started attempting 400-level classes (diff-eqs, etc.), I just totally hit a wall.
Quote from: bugo on March 20, 2012, 06:53:15 PM
All of this math is literally giving me a headache.
Yeah, my head hurts. I had all that stuff in high school and promptly forgot it.
Quote from: deanej on March 22, 2012, 04:21:12 PM
Quote from: agentsteel53 on March 21, 2012, 12:53:41 PM
tan(x) > x
Let x = 0.
tan(0) > 0
0 > 0
No, no, no, lim(tan(x)) as x->0 = 0. I knew a girl named Lim and she was not tan.
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