Local vs regional road connections

[Torie]

Thanks for the math explanation.

Yes, "disfavored" or whatever term is used, will need to be defined. You know what I meant.

[muon2]

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

Definition: Local Connection. There is a local connection between two subunits within a county if there is a continuous path of public roads and ferries that allow one to travel between the two nodes without entering any other unit. Roads along the border of two units are considered to be in both units on either side of the border.

Definition: Regional Connection. There is regional connection between two counties or subunits in different counties if there is a continuous path of all season numbered state or federal highways or regularly scheduled ferries that allow one to travel between the two nodes without entering any other county, unless such highway in such other county is a qualifying nick cut path. If the node is not on a numbered highway, then the connection is measured from the point of the nearest numbered highway in the county to the node.

Definition: Connecting Path. There is often more than one possible path to connect two nodes. For both local and regional connections the connection between two units is considered to be the path that takes the shortest time as determined by generally available mapping software.

Definition: There is a  nick cut if a highway between the nodes of two counties enters another county without traveling through the node of such other county and its distance in such other county is one third of less of the distance of the length of the county measured from the two on its boundary that are farthest apart. The nick cut is a qualifying nick cut if there is also a local connection between the two counties connected by the highway with a nick cut.

The reliance on a particular fraction of the connecting path gives me pause. It isn't really consistent with the notion of paths. It's particularly at odds with the notion that the scale can float to suit the scale of the counties or townships (urban vs rural) in question. I have another thought that is more consistent and I think gets to much the same result.

Definition: Regional Path. A regional path between two units is a continuous path of all-season numbered state or federal highways or regularly scheduled ferries that allow one to travel between the two nodes. If the node is not on a numbered highway, then the path is measured from the point of the nearest numbered highway in the county to the node.

Definition: Nick Path. A nick path is a regional path between two counties or subunits in different counties that are otherwise locally connected. The nick path may traverse counties other than the two counties to be connected. The nick path cannot be part of any valid path between the counties to be connected and any of the other counties traversed by the path.

Definition: Regional Connection. There is regional connection between two counties or subunits in different counties if there is a regional path between the two nodes that enter no other unit than those connected by the path, or if there is a nick path between the two nodes.

Examples. Consider these maps to show only numbered state highways. Assume that any contiguous counties are locally connected.

Let's start with this map. We agreed that the connection from Burr to Calhoun was not a link for local connections.



With the definition above Burr remains regionally unconnected to Calhoun. The path that passes through Agnew provides the path from Agnew to Calhoun, so it's not valid as a nick path. It also provides a path from Agnew to Burr, and though that's not the connecting path (it's not the shortest), it's still a valid path from Agnew to Burr.

Next consider this map. We agreed that the connection from Clay to Dewey was not a link for local connections. As before we'll now consider these to be counties, and there's a local road with a bridge that connects Clay and Dewey.



Here the highway that passes through Adlai on the way between Clay and Dewey is not a path to the node in Adlai. Since we've assumed that Clay and Dewey are otherwise locally connected, the regional path from Clay to Dewey through Adlai qualifies as a nick path. Thus as counties Clay would be regionally connected to Dewey.

[muon2]

To show an application of my nick path definition, I'll look at NY. This is the map based on direct regional connections.



Here are the contiguous county pairs in western NY not directly linked with a regional path.

Allegany-Livingston; locally connected, no nick path.
Steuben-Ontario; locally connected, no nick path. NY-21 crosses through a bit of Yates south of Canandaigua Lake, but that's also on a path from Ontario to Yates.
Steuben-Schuyler; locally connected, and there is a nick path. NY-414 cuts across a corner of Chemung and that stretch is not part of a direct regional path to Chemung from either county. Note that NY-226 does not qualify, since it goes into Yates and overlaps the path from Yates to Schuyler on NY-14A.
Yates-Seneca; not locally connected.
Chemung-Tompkins; locally connected, no nick path. The potential nick paths all are part of other direct paths.

[Torie]

I don’t understand the sentence underlined in red. I thought a numbered highway needed to go into the node jurisdiction. If not, is the only requirement is that the numbered highway enter the county in question, and then can use local roads to get to the node thereafter?

The word “any” that I underlined, suggests that there can be more than one regional path between two counties. If that is the case, then the direct red line that I drew representing a regional highway connection node C to node B would not be a nick cut, because part of it also serves as a regional path from node A to node B, albeit a circuitous one and its use of the highway in question for a bit was not built to connect node A to node B (which is the point).  And the second red line from node C to the right does use the shortest direct regional path from A to B (assuming there is always only one qualifying regional path, to wit, the shortest one). Do we really want to exclude that?

My recollection is that I made the test the number of the state highway. I drew two red lines on your map, which represent additional numbered state highways in addition to what you drew (the local politicians are good a pork barreling, so thus the plethora of state highways). If the right red line and the vertical black line for the last bit of the path to node B are the same number, then it is a qualifying nick path. If not, it is not. In the nicked county, or the county where the regional connection is in issue, if we don't have the same number as the state highway at the point where it enters the nicked county, it is not a valid nick path. If the number is the same, then you do.  And by that test, if the path from node C to node D in your second graphic has a different highway number for the final portion in Dewey going north, then you would also not have a valid path there as well.

And how about that rural NC connection issue in the western part of the state, where we had a numbered highway that traversed the whole county as a nick cut without traveling through the node of that “nicked” county. We wanted to exclude that, so we did the distance metric. If that path did not use a highway that served in part as a, or the (depending on the resolution of the above), regional connection from the node of the nicked county to the node of some other adjacent county, then it would count as a regional connection?

So we have quite a bit here to think about.

 

[muon2]

The underlined sentence I can answer directly. I'll give thought to the other parts.

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

The node is a specific point not a jurisdiction. For a county it is the place where the county government meets. A couple of years ago I tried to use the whole county seat, but I ran into problems when chops took a bite out of the seat and I tried to score plans with those chops. From a software perspective a single point is preferred to a polygonal shape as well.

Government buildings will be on roads so there is always a clear definition of a local path between two nodes. Regional paths can only use state and federal highways. Government buildings may not directly be on a state highway, so I need to place the seat of government on some state or federal highway to create a path. If there are two regional paths between a pair of county nodes then I need to know the reference point to determine which regional path is the shortest connecting path in case there are chops of that county. I doubt the ambiguity of lengths comes up much if at all, but the software still needs a reference point for measurement.

[jimrtex]

The underlined sentence I can answer directly. I'll give thought to the other parts.

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

The node is a specific point not a jurisdiction. For a county it is the place where the county government meets. A couple of years ago I tried to use the whole county seat, but I ran into problems when chops took a bite out of the seat and I tried to score plans with those chops. From a software perspective a single point is preferred to a polygonal shape as well.

Government buildings will be on roads so there is always a clear definition of a local path between two nodes. Regional paths can only use state and federal highways. Government buildings may not directly be on a state highway, so I need to place the seat of government on some state or federal highway to create a path. If there are two regional paths between a pair of county nodes then I need to know the reference point to determine which regional path is the shortest connecting path in case there are chops of that county. I doubt the ambiguity of lengths comes up much if at all, but the software still needs a reference point for measurement.

What if a city hall is not within the corporate limits?

[muon2]

The underlined sentence I can answer directly. I'll give thought to the other parts.

Definition: Node. A node is a reference point for a geographic unit. For a political unit the node is the primary place of government for that unit.

The node is a specific point not a jurisdiction. For a county it is the place where the county government meets. A couple of years ago I tried to use the whole county seat, but I ran into problems when chops took a bite out of the seat and I tried to score plans with those chops. From a software perspective a single point is preferred to a polygonal shape as well.

Government buildings will be on roads so there is always a clear definition of a local path between two nodes. Regional paths can only use state and federal highways. Government buildings may not directly be on a state highway, so I need to place the seat of government on some state or federal highway to create a path. If there are two regional paths between a pair of county nodes then I need to know the reference point to determine which regional path is the shortest connecting path in case there are chops of that county. I doubt the ambiguity of lengths comes up much if at all, but the software still needs a reference point for measurement.

What if a city hall is not within the corporate limits?

That's a situation that we've encountered (VA county seats in independent cities, township halls in incorporated city limits). There will have to be either an exception or clarification rule. I think that should wait until we have some other broader rules pinned down. The general rules may suggest the best way to deal with these exceptions.

[muon2]

Is this the example from western NC?

And then, at the other extreme from the UCC urban area exception issue,  we have the Clay County "nick cut" link.  State highway 64 as it traverses through Clay County from Cherokee to Macon just misses the county seat of Clay, Hayesville, and there are no other urban areas in Clay County. Yet the "nick" traverses the length of the whole county. That is a no go. That is not a nick. I am not sure what rule to fashion to deal with that. That the length of the nick highway cannot be more than a third of the length of the county measured from the two points thereof that give the longest length? Or something like that? And by measuring the length of the highway for this ratio, the rule would tend to penalize twisty state highway nick cuts that twist and turn over rugged terrain, which is a good thing I would think. An absolute highway distance parameter not based on a ratio might not work too well, given counties vary a lot in size.

My definition handles this case. US-64 is on a path from both Hayesville (Clay) to Murphy (Cherokee) and Hayesville to Franklin (Macon). Either or both disqualify it by definition. So it can't be used as a nick path from Murphy to Franklin.

[Torie]

Is this the example from western NC?

And then, at the other extreme from the UCC urban area exception issue,  we have the Clay County "nick cut" link.  State highway 64 as it traverses through Clay County from Cherokee to Macon just misses the county seat of Clay, Hayesville, and there are no other urban areas in Clay County. Yet the "nick" traverses the length of the whole county. That is a no go. That is not a nick. I am not sure what rule to fashion to deal with that. That the length of the nick highway cannot be more than a third of the length of the county measured from the two points thereof that give the longest length? Or something like that? And by measuring the length of the highway for this ratio, the rule would tend to penalize twisty state highway nick cuts that twist and turn over rugged terrain, which is a good thing I would think. An absolute highway distance parameter not based on a ratio might not work too well, given counties vary a lot in size.

My definition handles this case. US-64 is on a path from both Hayesville (Clay) to Murphy (Cherokee) and Hayesville to Franklin (Macon). Either or both disqualify it by definition. So it can't be used as a nick path from Murphy to Franklin.

That is where "a" versus "the" is benign. But in other cases its malignant. You are using "a" as a substitute for a distance metric.

[muon2]

Is this the example from western NC?

And then, at the other extreme from the UCC urban area exception issue,  we have the Clay County "nick cut" link.  State highway 64 as it traverses through Clay County from Cherokee to Macon just misses the county seat of Clay, Hayesville, and there are no other urban areas in Clay County. Yet the "nick" traverses the length of the whole county. That is a no go. That is not a nick. I am not sure what rule to fashion to deal with that. That the length of the nick highway cannot be more than a third of the length of the county measured from the two points thereof that give the longest length? Or something like that? And by measuring the length of the highway for this ratio, the rule would tend to penalize twisty state highway nick cuts that twist and turn over rugged terrain, which is a good thing I would think. An absolute highway distance parameter not based on a ratio might not work too well, given counties vary a lot in size.

My definition handles this case. US-64 is on a path from both Hayesville (Clay) to Murphy (Cherokee) and Hayesville to Franklin (Macon). Either or both disqualify it by definition. So it can't be used as a nick path from Murphy to Franklin.

That is where "a" versus "the" is benign. But in other cases its malignant. You are using "a" as a substitute for a distance metric.

Then we need to find a case where it is malignant. The distance metric gives the wrong results in the artificial example below. The distance metric would have a link from Burr to Calhoun when we agreed that there should not be one.

Let's start with this map. We agreed that the connection from Burr to Calhoun was not a link for local connections.



With the definition above Burr remains regionally unconnected to Calhoun. The path that passes through Agnew provides the path from Agnew to Calhoun, so it's not valid as a nick path. It also provides a path from Agnew to Burr, and though that's not the connecting path (it's not the shortest), it's still a valid path from Agnew to Burr.

And the distance metric might fail to find the nick cut in this other example, especially if I move the node for Clay to the SE close to the border with Adlai. It is counter intuitive that as the nodes get physically closer the ability to use a nick path diminishes, since the fraction in the intervening county increases.

Next consider this map. We agreed that the connection from Clay to Dewey was not a link for local connections. As before we'll now consider these to be counties, and there's a local road with a bridge that connects Clay and Dewey.



Here the highway that passes through Adlai on the way between Clay and Dewey is not a path to the node in Adlai. Since we've assumed that Clay and Dewey are otherwise locally connected, the regional path from Clay to Dewey through Adlai qualifies as a nick path. Thus as counties Clay would be regionally connected to Dewey.

[Torie]

[muon2]

Now I'm confused. I thought the uniform highway number was a different standard than the maximum 1/3 of distance in the cut. At least they were two different proposals when you and jimrtex were debating this over NC.

Moving C closer to D does not change the length in A, but it does change the fraction of the path in A. If I put the nodes for C and D on the same numbered highway right at the border with A, the fraction in A would be the majority of the length. Yet if I move C and D farther apart I can eventually get the fraction in A under 33% and it becomes a link. That is counter intuitive to me.

[Torie]

Now I'm confused. I thought the uniform highway number was a different standard than the maximum 1/3 of distance in the cut. At least they were two different proposals when you and jimrtex were debating this over NC.

It must be both the same highway number and fit within the length parameter.

Moving C closer to D does not change the length in A, but it does change the fraction of the path in A. If I put the nodes for C and D on the same numbered highway right at the border with A, the fraction in A would be the majority of the length. Yet if I move C and D farther apart I can eventually get the fraction in A under 33% and it becomes a link. That is counter intuitive to me.


The fraction is the length of the nick as the numerator and the length of the nicked county as the denominator. The length of the road outside the nicked county is irrelevant. If I am missing something, please put up a graphic.

[muon2]

Now I'm confused. I thought the uniform highway number was a different standard than the maximum 1/3 of distance in the cut. At least they were two different proposals when you and jimrtex were debating this over NC.

It must be both the same highway number and fit within the length parameter.

Moving C closer to D does not change the length in A, but it does change the fraction of the path in A. If I put the nodes for C and D on the same numbered highway right at the border with A, the fraction in A would be the majority of the length. Yet if I move C and D farther apart I can eventually get the fraction in A under 33% and it becomes a link. That is counter intuitive to me.


The fraction is the length of the nick as the numerator and the length of the nicked county as the denominator. The length of the road outside the nicked county is irrelevant. If I am missing something, please put up a graphic.

I don't know what the underlined means. I don't recognize it from a previous proposal. jimrtex had referenced the length of the entire path as a denominator IIRC.

I'd still like to see an example where you feel my proposal creates an untenable result. From my perspective, it would not bother me to have no nick paths, but I'm trying to accommodate those who see a value in them.

[Torie]

Now I'm confused. I thought the uniform highway number was a different standard than the maximum 1/3 of distance in the cut. At least they were two different proposals when you and jimrtex were debating this over NC.

It must be both the same highway number and fit within the length parameter.

Moving C closer to D does not change the length in A, but it does change the fraction of the path in A. If I put the nodes for C and D on the same numbered highway right at the border with A, the fraction in A would be the majority of the length. Yet if I move C and D farther apart I can eventually get the fraction in A under 33% and it becomes a link. That is counter intuitive to me.


The fraction is the length of the nick as the numerator and the length of the nicked county as the denominator. The length of the road outside the nicked county is irrelevant. If I am missing something, please put up a graphic.

I don't know what the underlined means. I don't recognize it from a previous proposal. jimrtex had referenced the length of the entire path as a denominator IIRC.

I'd still like to see an example where you feel my proposal creates an untenable result. From my perspective, it would not bother me to have no nick paths, but I'm trying to accommodate those who see a value in them.

The length of the nicked county is the length of the line drawn between the two points on the county's boundary that are farthest apart. That is the denominator. The numerator is the length of the road while in the nicked county. I have used this metric since the NC Clay County discussion. I have described in our discussion here my issues with your approach. My approach focuses on whether the highway is really designed to link the two nodes that happens on the way to nick another county.

[muon2]

I don’t understand the sentence underlined in red. I thought a numbered highway needed to go into the node jurisdiction. If not, is the only requirement is that the numbered highway enter the county in question, and then can use local roads to get to the node thereafter?

The word “any” that I underlined, suggests that there can be more than one regional path between two counties. If that is the case, then the direct red line that I drew representing a regional highway connection node C to node B would not be a nick cut, because part of it also serves as a regional path from node A to node B, albeit a circuitous one and its use of the highway in question for a bit was not built to connect node A to node B (which is the point).  And the second red line from node C to the right does use the shortest direct regional path from A to B (assuming there is always only one qualifying regional path, to wit, the shortest one). Do we really want to exclude that?

My recollection is that I made the test the number of the state highway. I drew two red lines on your map, which represent additional numbered state highways in addition to what you drew (the local politicians are good a pork barreling, so thus the plethora of state highways). If the right red line and the vertical black line for the last bit of the path to node B are the same number, then it is a qualifying nick path. If not, it is not. In the nicked county, or the county where the regional connection is in issue, if we don't have the same number as the state highway at the point where it enters the nicked county, it is not a valid nick path. If the number is the same, then you do.  And by that test, if the path from node C to node D in your second graphic has a different highway number for the final portion in Dewey going north, then you would also not have a valid path there as well.

And how about that rural NC connection issue in the western part of the state, where we had a numbered highway that traversed the whole county as a nick cut without traveling through the node of that “nicked” county. We wanted to exclude that, so we did the distance metric. If that path did not use a highway that served in part as a, or the (depending on the resolution of the above), regional connection from the node of the nicked county to the node of some other adjacent county, then it would count as a regional connection?

So we have quite a bit here to think about.

 

If I understand this, you would have either of the red highways become a nick path. I would exclude them, since either of them becomes the connecting path (ie shortest) between A and C. To me that's true even if the highway was explicitly built for a connection from C to B.

Can I ask if any of the county pairs I listed in NY causes grief? I want to make sure we aren't arguing over something so artificial that it need not control the rules.

[Torie]

If I understand this, you would have either of the red highways become a nick path. I would exclude them, since either of them becomes the connecting path (ie shortest) between A and C. To me that's true even if the highway was explicitly built for a connection from C to B.

Yes, either would be a nick path assuming their length in the nicked county is a third or less of the nicked county's maximum length.

Can I ask if any of the county pairs I listed in NY causes grief? I want to make sure we aren't arguing over something so artificial that it need not control the rules.

There are 50 states. I think my rule hews to the common sense metric. It makes sense. I don't think yours does. It does not focus on whether there are really direct links between nodes (not interrupted by going through another node), and intended to be so. Sure, this will not be a very common occurrence. Neither are Indian Reservations. But it is important.

[muon2]

We shall have to disagree here until we can find a real world case that clearly divides us and is intolerable by one method or the other. You may find your metric to be common sense, but it is a metric wholly new to my definitions of paths and erosity. I only invoke distance when comparing two established paths between the same points for the purposes of determining fragment connections.

The strength of my definitions is that paths have no reliance on real distance which allows them to naturally scale in size and scope when they enter urban areas. Without scaling erosity breaks down as a measure as do many other tests of compactness. By using distance only to compare paths between the same points, the scale is preset by those points, not by the paths.

[Torie]

Isn't taking a ratio of distance in a nicked county, naturally scaling? Yes, to me, common sense, trumps introducing "novelty" to your system. That is never going to change.

[muon2]

Isn't taking a ratio of distance in a nicked county, naturally scaling? Yes, to me, common sense, trumps introducing "novelty" to your system. That is never going to change.

The ratio could be truly scaling if the densities were generally uniform. One problem with many scale-independent measures of compactness is how they are tripped up by urban concentrations. For example one standard test is to compare the area of the smallest circle that encloses a district with the area of the district (Reock measure). That test scales with the size of the district. Yet, if the district has large regular rural areas combined with erose areas in an urban area the district scores well. I am similarly concerned that a nick passing through an urban area misses features that would be irrelevant when in cuts a rural area.

[Torie]

Isn't taking a ratio of distance in a nicked county, naturally scaling? Yes, to me, common sense, trumps introducing "novelty" to your system. That is never going to change.

The ratio could be truly scaling if the densities were generally uniform. One problem with many scale-independent measures of compactness is how they are tripped up by urban concentrations. For example one standard test is to compare the area of the smallest circle that encloses a district with the area of the district (Reock measure). That test scales with the size of the district. Yet, if the district has large regular rural areas combined with erose areas in an urban area the district scores well. I am similarly concerned that a nick passing through an urban area misses features that would be irrelevant when in cuts a rural area.

I don't understand how your comment is relevant to my point, and I don't understand your last sentence at all. What does "misses features that would be irrelevant" mean?

Sure an erose elongated county might allow for a longer nick. So what? That is hardly a big deal. The metric works well enough for government work as it were.

[muon2]

Isn't taking a ratio of distance in a nicked county, naturally scaling? Yes, to me, common sense, trumps introducing "novelty" to your system. That is never going to change.

The ratio could be truly scaling if the densities were generally uniform. One problem with many scale-independent measures of compactness is how they are tripped up by urban concentrations. For example one standard test is to compare the area of the smallest circle that encloses a district with the area of the district (Reock measure). That test scales with the size of the district. Yet, if the district has large regular rural areas combined with erose areas in an urban area the district scores well. I am similarly concerned that a nick passing through an urban area misses features that would be irrelevant when in cuts a rural area.

I don't understand how your comment is relevant to my point, and I don't understand your last sentence at all. What does "misses features that would be irrelevant" mean?

Sure an erose elongated county might allow for a longer nick. So what? That is hardly a big deal. The metric works well enough for government work as it were.

If it's about what works well enough, then I claim mine is simpler to test. The way I defined it, as long as the nick doesn't have a state highway fork in the county that takes you to that county's seat it's a valid nick. I don't need to measure anything. The example from Burr to Calhoun has a fork to Agnew. So it's not a nick for me.



As we know these nicks are relatively rare. I'd still like to see where my model gives an untenable result in the real world.

[Torie]

As described with my two red lines above, the result as to what is a valid nick, and what is not, appears to be very arbitrary. If a road happens to tie into another road going that goes between two notes, even with a highly circuitous route, with another route more direct, you lose the validity of the nick, and if not, you have one, even if highly circuitous.

[muon2]

As described with my two red lines above, the result as to what is a valid nick, and what is not, appears to be very arbitrary. If a road happens to tie into another road going that goes between two notes, even with a highly circuitous route, with another route more direct, you lose the validity of the nick, and if not, you have one, even if highly circuitous.

Circuitous yes, arbitrary no. We have many circuitous connections defined directly between counties. Among these are ones that jimrtex has complained about, yet they remain for me (and I think you). There's nothing arbitrary about saying that a nick path can't already be on a direct path in the nicked county. It's saying that nick paths arise out of unusual situations where a highway passes through a third county in a truly incidental way.

Conversely using a 1/3 distance standard would be circuitous no, arbitrary yes. It tends to lead to more direct nick cuts so they would be presumably less circuitous. The use of 1/3 is itself arbitrary as it is not based on any comprehensive study of real geography. Look at the work that jimrtex put into UCCs to come up with a cutoff within MSAs that was not arbitrary.

[Torie]

As described with my two red lines above, the result as to what is a valid nick, and what is not, appears to be very arbitrary. If a road happens to tie into another road going that goes between two notes, even with a highly circuitous route, with another route more direct, you lose the validity of the nick, and if not, you have one, even if highly circuitous.

Circuitous yes, arbitrary no. We have many circuitous connections defined directly between counties. Among these are ones that jimrtex has complained about, yet they remain for me (and I think you). There's nothing arbitrary about saying that a nick path can't already be on a direct path in the nicked county. It's saying that nick paths arise out of unusual situations where a highway passes through a third county in a truly incidental way.

Conversely using a 1/3 distance standard would be circuitous no, arbitrary yes. It tends to lead to more direct nick cuts so they would be presumably less circuitous. The use of 1/3 is itself arbitrary as it is not based on any comprehensive study of real geography. Look at the work that jimrtex put into UCCs to come up with a cutoff within MSAs that was not arbitrary.

Well I don't agree that the third rule is arbitrary, and most counties are not that erose anyway. I guess we are at a dead end here. What's arbitrary is involving highways that happen to connect in some circuitous way between nodes, to invalidate a direct nick cut, or allowing nick cuts that avoid that that themselves are circuitous and not designed to connect nodes. We just are not going to agree on this one.