You have the right formula to start with, but you’re not simplifying it correctly.
\Delta = \sqrt {x^2+R^2} - R
\Delta = R\sqrt {1+x^2/R^2} - R (factoring out R^2 and removing it from the radical)
\Delta \approx R (1+x^2/(2R^2)) - R (binomial approximation for x^2/R^2 << 1)
\Delta \approx x^2/(2R) (simplifying)
You can be certain that it can’t be just x/R, because the units don’t work out on that.
It’s probably not curvature of the earth that affects your street grid.
Assuming you are in the US, it’s much more likely that the streets were laid out based on land surveys from 150 years ago, when the land was divided into square sections, each one mile wide. The section lines were supposed to be parallel to each other, but in actuality, they never are.
Exactly.
The history of surveying in the US, and the manner in which boundaries were defined is pretty fascinating. Currently there are 135 state planes defined; obviously larger states needing more than one plane. The planes usually run on county lines, and try to evenly subdivide parent states. They don’t take anything more into consideration. Boundaries between these planes is one place where you will see discontinuities.
However as @chappachula notes, a huge amount of the US is based upon the original Public Lands Survey System (PLSS) performed in 1785 onward. This survey was needed in a hurry to allow the new federal government to get out of debt by selling off land. The surveying defined square mile blocks using nothing more sophisticated than a metal chain and a compass. Issues with the spherical shape of the Earth were really just bodged over. Each square stated in the south-eastern corner and guys with the chain just measured them out. Especially to the north of the US, this meant that squares were not square at all. However the surveyed boundaries still take precedence today, even if errors were made. So a great deal of the US is laid out with lines of very old provenance. Cities and roads follow the path laid out by a couple of guys with a chain and a compass. Within the PLSS regions were referenced to 37 different datums. Not all of the US was surveyed by the PLSS, those states that had pre-existing surveyed land boundaries retained them.
Altitudes are interesting as well. In 1927 the US established a national vertical datum based upon an estimate of the geoid. For want of better accuracy, a farm in Kansas contains the reference zero. Work with early satellites allowed some refinement of the geoid, but it wasn’t really until the advent of WGS84 and GPS that there was any large scale move. When GPS became available it became possible to significantly further improve estimates of the geoid. WGS84 is not a static thing, and continues to be updated, right now it is accurate to about 2cm.
Then you get maps based on UTM, which are much larger planes, and these are defined relative to the global lat-long coordinate system, basically giving you a 10x10 degree rectangle. But where the Earth is projected onto a plane. Because the discrepancy at the edges is so great it is usual to continue a regional map into the next rectangle staying with the rectangle the map started with, so you can end up with multiple different coordinates in UTM for the same point. MGRS (the military system) uses the same projection, but uses a different way of encoding the coordinates. As I noted earlier, each plane actually intersects the geoid in a circle, in an effort to even out the distance distortions.
Maps and coordinate systems based on projection onto a plane have the nice property that coordinates can be expressed with a constant distance step on a rectilinear grid. So you typically have Eastings and Northings (where, no matter where you are on the Earth, Northings increase as you go north, and Eastings increase as you go east.) Over short distances this is great. But it breaks down at some point dependant on the projection.
No matter how good a job the surveyors did, the section lines can’t be parallel to each other, because of the curvature of the Earth.
It’s pretty subtle, and varies by latitude. Imagine a city with a 20-mile north/south spread. Now imagine two north/south section lines 5280 feet apart at the southern end of the city. Assuming the city is at 40 degrees latitude, if the section lines follow meridians, then at the northern extend of the city the section lines will be 5257 feet apart - a difference of just 22 feet. If your city is at 60 degrees latitude (e.g. Juneau, AK), then the difference is 46 feet, and if your city is near the equator (e.g. Nairobi, Kenya), then the difference is negligible.
Sections at the north end of a city being half a percent smaller than those at the south end wouldn’t necessarily be a big deal… but now apply that to a whole state. In a state with a 200-mile north-south spread, we’d be talking about a several percent difference, enough that the landowners would care. So the surveyors of each state didn’t make the sections line up exactly: On a regular interval, as they’re heading north, they’ll have an adjustment line, a line of latitude along which they re-measure the widths of the sections. And so, unless you’re right at the east edge of the state, those new section boundaries won’t line up with the old ones.
Ever notice that there are some streets in a city, where every side street that crosses them doglegs as it does so? That’s because streets are often built along the edges of sections, and those streets were on the adjustment lines.
This map of Hamilton County shows how the section lines start to wander up and down as you head north towards Forest Park and Fairfield in Butler County. I’m told the original surveyors from the 1700s had a bad surveying chain that caused errors to add up the further they went. Several of the roads up that way follow these wonky lines, and there’s plenty of other idiosyncrasies that I have no explanation for other than pre-industrial surveying is hard, and at some point mapping a square grid on a sphere just needs to cry uncle.
I am always amazed at how conversations on SDMB evolve. We’ve gone from large buildings to suspension bridges and now discussing the history of land surveying and how it affects urban planning.
The gist I am getting from the conversation is that the curvature of the Earth is not enough to really worry about except in unusual structures. Even in a very large building like the Tesla factory in Austin, the effects of the round planet would fall within the margin of error for just assembling the pieces of the building. Exceptions would be structures for scientific experiments where “straight” needs to be really, REALLY “straight”, not merely level with the Earth.
Thank you to all for a lively conversation.
It is rather amusing that at a certain scale you have to decide if you want your building to be either level or flat, since it can’t be both. I’d imagine that except for those rare scientific situations you’d want to stick to level, lest you start introducing drainage problems, desks where pens and balls roll away, and where level/straightness can only be measured with laser surveying tools instead of a store-bought water level or plumb bob.
One type of structure that hasn’t been mentioned is canals or aqueducts, where water must flow by gravity over long distances, often while losing as little elevation as possible.
Water flow needs to follow the gravitational field, with sufficient slope to make the water actually flow. So essentially, by definition, it is defined by the geoid (at least at sea level, at different altitudes, at least in principle, the field isn’t quite the same as the geoid with an altitude related offset, but is likely as near as matters.)
This exemplifies the range of different ways you can define the shape of the Earth. The geoid is a very useful basis for most purposes, but it isn’t the only way you might define the Earth’s shape. The geoid locally changes with local density changes in the crust and mantle, whereas a different shape estimate might say, average the surface of the Earth, and get you a different shape. Both are curved, but at different scales and different ways of measuring have local differences in shape. Even defining the centre of the Earth can be done in multiple ways and give you different locations.
Another amusing fact - if you have a flat frictionless of any size, the center of which is normal to gravity, objects placed off center will oscillate, with the same period as an object in a low earth orbit, or at the end of an earth radius pendulum or in a frictionless straight tunnel.
…embedded in a gravitational field equivalent to that at the Earth’s surface, but uniform over scales larger than the Earth. If you can find such a gravitational field anywhere, let me know.
I keep it near the devices that can dig straight tunnels thousands of miles deep, and maintain a vacuum in them.
On the other hand, the elevation difference is 0 inches per 40,000km.
Oh, sure, this is all textbook-land. But since your other examples involved an (approximately, at least) spherically-symmetric inverse-square gravitational field, such as we actually do have in the vicinity of Earth, I thought it worthwhile to point out the distinction.
Understood and appreciated. Thanks.
I didn’t watch the LIGO video, but I did read this last night:
“The pipes are so long – nearly two and a half miles – that they have to be raised from the ground by a yard at each end, to keep them lying flat as Earth curves beneath them.”
LIGO gravity wave observatories use folded laser beams to measure the distance between stationary objects (mirrors) at each end of two tunnels at right angles. Because gravity waves cause the space between the stationary objects to grow and shrink, this detects them. Note, it’s not the objects moving – they get further and closer because the space itself is changing. The tunnels are constructed of 12’ diameter concrete pipes with various inner constructions for vacuum and whatnot, the pipes appearing to rise up at either end by a yard. The laser beams travel in straight lines, whereas the Earth’s surface curves. I wouldn’t have used the phrase “lying flat” as the author does, because “flat” is ambiguous in this context.
This quote is from “The Billion Year Wave” by Nicola Twilley, in The New Yorker.
I think this makes LIGO a very large building that had to account for the curvature of the Earth.
Wouldn’t the tubes have to curve slightly due to Earth’s gravity?
Brian
No more so than the laser beam. Sounds like a good exercise to calculate exactly how many nanoradians that particular effect will have.
