A folded single coordinate axis

Dan Jacobson

[Image: Xinshe]

Let's say the City Council wants numbered streets, in both directions too, like Manhattan, but minus the nonsense. No, not letters and numbers. Just numbers. "No problemo!" we tell them.

Well, first, we cannot make the mistake Colombia, Alberta or Ontario, Canada, or even Hoopeston, IL, USA did, reusing numbers, thus needing to distinguish "11th Street" from "11th Avenue".

We intend to leave such worries behind, allowing even our most sleepy citizens to say:

A: Meet me on the corner of 11th & 61st.
B: 61st & 11th?
A: Yeah.

Naturally we also bid adieu to the North, South, East, West stuff encumbering the tired addresses of the past.

Hmmm, our valley is 25 blocks wide. We reserve the first 25 streets as north-south, and the rest as east-west.

Naturally house addresses, 100 per block, coordinate with street names, as shown in the image. 61st is 6100, etc.

In fact, let's directly use those grid values as our street names, like in Utah. Note these are indeed still "numbered streets", just as we promised the the City Council we would deliver.


[Image: Central Xinshe District, Taichung, Taiwan]

As we have already thrown N, S, E, W in the dumpster where they belong, the Utah style addresses for the four houses you see here would be simply "6051 1100", "1145 6000", etc. (But that's like ice cream bars and cones without sticks and cones, so it would be better to at least use "6051 1100 Road", "1145 6000 Road.")

Certainly we have arrived at the world's smartest two-component addresses. Alas we have made them overefficient. The public cannot understand them.

So let's go back and and try Taiwan-style numbered lanes, getting:

6051 Lane 1100
6158 Lane 1100
1145 Lane 6000
1042 Lane 6100

Full article: Xinshe Address planning.

We are very pleased, and that's the plan we are going to present to the City Council.

But wait. We also discovered something...

Single axis

When designing the Xinshe system, we discovered our "big road" (gold line in first image above) is in fact a coordinate axis with a sharp turn in it. It starts in the west (0), makes a sharp turn north (2500), and then continues on forever.

Using it enables us to dispose of the X and Y coordinate axes and rely instead on just a single coordinate axis.

Indeed such an abstraction allows us to anchor both north-south and east-west roads onto the same system,

6051 Lane 1100 Huanshe Rd.
1145 Lane 6000 Huanshe Rd.

whereas with straight axes we would need two different road names.

So what are the effects of using just a single axis on North American road names and addressing? Well, N, S, E, W have no meaning anymore and must not appear in addresses.

And as we know, nobody uses negative numbers in addresses. So with one axis that leaves only one direction for growth.

Yes, for North America, one simply can place the origin (0,0) off in one corner, and use just one quadrant, and reap all the benefits of simple addresses without bothering with the single axis concept.

But in North America the a grid covers the entire town. There is no field in postal addresses governing which grid you are using below "City: ____".

But in Taiwan there is. It is "Road: ____", and indeed, for the same town of Xinshe it was a lifesaver.

Appendix

Some of the following needs to be recycled...

  1. Evanston

    On any of the Evanston, IL static maps, on the right margin you see the coordinates 100-2900. Well let's continue them around the top, adding, oh, 3000 to each of the top coordinates. So now the top coordinates become 3100 to 6800. (And the bottom coordinates become 3100 to 5600.) We can describe what we did with:
    single2xy --fold 3000 --axis e2s

  2. Note none of this has anything to do with taxicab geometry. We are free to cut diagonally across vacant lots as the crow flies, nor do we concern ourselves about one-way streets.

    If saying "folded axis" is wrong, then call it "folded baseline".

    Yes, though we want to make odd and even house numbers, there is nothing stopping us from defining finer points: the sewer cover at 1100 and 6100 is at (1100.0000, 6100.0000), (same as (6100.0000, 1100.0000)).

  3. Properties

  4. In our Xinshe example here, in the X, Y world we are operating in the northeast quadrant. And in our single axis world, northwest. Of course for different towns we might pick employ different quadrants.

  5. We change the scale of our axis as it passes the "fold point", as we are dealing with rectangular blocks, not square. But still 100 house numbers per block.

  6. Skew axis

    Q: How might we describe the address grid of Highland Park, Illinois, USA , in use for many years already, using a single axis? Must the "fold" be 90°?

    [Image: SE corner of Highland Park, IL, USA]

    A: Well OK, we'll use our single axis to describe it. But we certainly wouldn't plan it ourselves this way, because it requires using negative numbers, which we would have avoided if we were in charge.

    Here we use only the northwest quadrant. Our single axis enters the picture at the bottom left of the picture, near -490 and proceeds east. It makes an acute turn to the northwest at 0, and finally leaves the map near +470.

    Here the fold does not occur at "25th Street", or "Lane 2500", but instead is at "0th Street", or "Lane 0". This means we will need to differentiate negative and positive.

    As of course in the world of house addresses there is no place for negative numbers, so a raw address "Negative 445 County Line Road" would translate to "445 West County Line Road".

    (But like all the other roads in Highland Park, there is only one County Line Road in the whole city, and all the addresses are West, so the West can be dropped with no confusion. Anyway, we have succeeded describing Highland Park's addresses with a single folded axis! We do not attempt to analyze Chicago though, since Chicago uses all four quadrants.)

    It is revealed that perpendicularity in our previous examples was merely a coincidence after all. In the general case only parallel lines exist, above shown one per 100 house numbers.

    No, we never fold a whole 180° ... etc.

  7. Parametric equations

    We can express our folded axis via parametric equations. E.g., ( x , y ) = ( fold − nlower , nhigher − fold).

    Indeed I made a (purely bijective) program single2xy to figure it out:

    ./single2xy --axis ⮤ --fold 2500 --x 1400 --y 3600
(n_lower,n_higher)=(1100,6100)
./single2xy --axis w2n --fold 2500 --x 1400 --y 3600
(n_lower,n_higher)=(1100,6100)
./single2xy --axis ⮤ --fold 2500 --n 1100 --n 6100
(x,y)=(1400,3600)
./single2xy --axis ⮤ --fold 2500 --n 6100 --n 1100
(x,y)=(1400,3600)

    As far as different scaling of the two sides of the fold, that is a later projection detail and doesn't affect the parametric equations. In other words e.g., 36th Street is 11 blocks away from 25th Street, no matter if each block is 660 feet or 330 feet, etc, and no matter if 25th Street points NNE or SSW, etc.

  8. Full transformation stack

    (This chart (not fully ordered correctly yet), is reversible. )

    1. Start with: 1042 Lane 6100 (or 1042 61th St.)

    2. Tricks like restoring the lost first block (0-99, north and south) in Oak Park IL or S. St. Paul, MN, USA.

      some things, like the truncated PLSS township, along Central St., Evanston, IL, are better handled via a ground control points stage.

    3. ./single2xy --axis e2n --fold 2500 --n 1042 --n 6100
(x,y)=(-1458,3600)

    4. Move a little back from the middle of the street.

    5. Translation and rotation. We move our map overlay map on top of the town map and rotate until it fits. Then we need to zoom in or out until it really fits.

    6. Now apply 1) Skew between X and Y axes; 2) Numbers per mile, km etc. (e.g., 800 north-south (400 on each side of street), 1200 east-west); 3) Meters per block (e.g., 200 e/w, 100 n/s); (Order of application doesn't matter(?).) Now we are matching UTM or some state plane coordinate system.

    7. Now unproject, say via inverse transverse Mercator.

    8. Finally produce output: longitude and latitude.

  9. Which parameters should my town choose?

    Perhaps it's time for a new numbering system for your town. One that won't get "messed up over time", "I promise".

    But note that unless confined by mountains, every place so far eventually develops beyond their original plans.

    Well, considering that all future growth can only be in one direction, don't waste that by picking a direction facing mountains. Grids don't make very good addressing systems in hilly areas in the first place.

    Also consider how many digits you will need. E.g., the (state wide) South Dakota system starts in the northwest corner (thus uses the southeast quadrant) at 100 East and 100 South. Well for the same three digits I would have named that starting point 100 East and 500 South! Indeed, as their state is less than 400 miles wide, the would not have any overlap, and could not worry about Street vs. Avenue. (North Dakota's Burkle system even adds the burden of quadrants.)

    So why did I put the fold at 2500 and not 5000 (same four digits long) above. Well I was certain that before the fold all I would ever need was 25 blocks. Imagine steep mountains beyond that.

  10. And beyond

    Who says we can only fold once? Let's say our single axis is rolling along and encounters another part of town with a differently oriented grid. Maybe it could bend a second time like "ㄣ" (or more times?) to encompass that part of town too.

  11. Conclusion

    There are all kinds of axes, helical axes, you name it. If we throw a kink into an otherwise straight axis, a bonanza of benefits are unleashed, including addresses even Captain Peter "Wrongway" Peachfuzz couldn't mess up.

    Anyways, we gathered up all the addressing ideas we encountered over the years, and wrapped a Taiwan numbered lane concept around it on two of four sides, and the folded single axis concept is what we got.

  12. Notes

    In Utah 3600 1000 East == 1000 3600 North. Using only one quadrant we could achieve 3600 1000 Street == 1000 3600 Street. We could go even further and say 3600 1000 == 1000 3600 but then our goal of fool proofing addresses for more reliable reporting and first responding starts to slip, as the general public won't know what we are talking about.

    Odd vs. even intersecting streets in Cuba. (But then how to match 100 numbers per city block?)

    Plus Codes, what3words etc. are beyond the scope of this article, and what the City Council asked us to do. Plus what if your phone gets stolen?

    One also notes that the even and odd sides of Lane 2500 persist as it makes its bend, and thus determine the even and odd sides of all the other streets in the system. We use this to our advantage to keep Xinshe's lanes numbered 6100, 6200,... and not 6101, 6201, etc.

    In our project, we are almost like these address system quadrants, except instead of the four quadrants you see for each city on their maps, there would be only one quadrant for each city, anchored at the fold in the "L" that we would create on the edge of such cities (that is if I were allowed to renumber all the houses.) Note our L's may overlap but never do the cities of the addresses they control.

    What about C, Z, U, S-shaped etc. roads? Well, these would sweep back and forth over the coordinates, giving identical twins and triplets addresses. So such roads would need to be "pulled straight" on the map prior to addressing. Hopefully any such segments aren't very long. In Highland Pl. Highland Pk. IL we observe the numbers increasing opposite of the grid, as it curls around.

    Q: "Considering today's (and tomorrow's!) technology, do cell phones and GPS make naming less important to find your way?" A: No way. While you are still fumbling with your cellphone, I with pep in my step, glide in my stride, am already on my way down the block, as the addressing / road naming system is so logical that I could even navigate it in my sleep.

  13. More math

    The Euclidean distance still works fine in my one-axis system.

    Are mine an unordered\ pair? I'm not sure yet.

    I looked up lots of coordinate systems and norms but couldn't find mine yet.

    Am I the first to employ a sharp turn in curvilinear coordinates' axis?

    Am I making an atlas of different cities' coordinate systems?

    I guess I am making a linear map, metric, lattice, dual, half a half plane? met...top..., ref..., closed linear subspace ?

    Note, even though I might skew (Highland Park), or stretch (rectangular, not square, city blocks, 100 numbers per side still), but I still don't allow non parallel red lines on my maps...

    Yes, we are using a Euclidean plane in linear algebra.

    We read (I have crossed out the things that I have removed the restrictions of),

    Later, the plane was described in a so-called Cartesian coordinate system, a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed [let's keep things positive for cities] distances from the point to two fixed perpendicular [skew OK too] directed lines, measured in the same [rectangular city blocks, but still 100 numbers per side OK too] unit of length. Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair (0, 0) [I yanked it...] The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin.

    The idea of this system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat, although Fermat also worked in three dimensions, and did not publish the discovery. Both authors used a single (abscissa) axis in their treatments, with the lengths of ordinates measured along lines not-necessarily-perpendicular to that axis. The concept of using a pair of fixed axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work.

    "any point specified by two parameters can be instead specified by one" is not what I am doing! I still choose to use two parameters: "1138" and "Lane 6100"

    Did I cook up a "space", met..., gen... , sim...?

    Lp...? Topological manifold is what one person said...

    Surely a uniform isomorphism.

    Somewhere in ?

    Note we are not naming or numbering blocks, but instead roads and houses.

    Did I make a new frame?

    By the way, I don't know math. All I know are streets and house numbers... So please communicate to me in the world of city addresses. That way I'll understand you. Thanks.

  14. Creation instructions

    An unordered list! Order doesn't matter.

  15. 3rd dimension

    Yes, there is a 3rd dimension, and it is simply floors: 12th Floor, 6158 11th St.; Room 1234, etc.

  16. See also

    Source code.

    North Dakota single axis.

    Maybe in Quebec Canada they really keep the avenue and street numbers separated...

Last modified: 2024-08-18 02:02:39 UTC