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Comment by NickM | Hacker News Reader

If you are looking for directions in a city, what matters most to you is that the roads look correct. This is why the Mercator map is used.

Whoa, wait a second...is this saying that roads which look straight on a map are actually not straight, in the sense that they're not actually following the shortest path between two points?

It's slightly appalling to me to think that people all over the world are building crooked roads so that they appear straight on a Mercator map, and then using Mercator maps so that existing roads continue to look straight.

I guess it probably doesn't make much difference on a local level, where the distances are relatively short, but there's still something a little bit horrifying about it, if I'm understanding all of this correctly.

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Any physical infrastructure, such as roads, is going to be distorted by local topography, which furthermore may not be entirely consistent from year to year (land can creep forward or hillsides erode somewhat). This deviation will tend to enormously outweigh the errors caused by map projection.

Another note is that legal boundaries tend to be defined on the basis of actual surveyed boundary markers, not on the logical definition of the line. So the US-Canada border, while originally defined as exactly 49°N, the actual border is actually generally about a few hundred feet south of that line. These surveys were generally conducted by people running literal chains in a constant bearing for some distance (maybe several hundred yards at most) and then taking another reading of their position to correct the line. Of note is the Mason-Dixon Line, which was precise enough to note that the errors in their surveying had a systematic error to them--which was realized to be the physical mass of the Appalachian Mountains ever-so-slightly deviating plumb lines from vertical.

Also note that historical boundaries were more often specified as "reference point and bearing" (or along line meridians/parallels, which amounts to the same thing) then "from point A to point B". In this regard, the straight line on a Mercator map is more accurate than the geodesic shortest-distance-between-two-points.

Then there's the history of the Tennessee-Kentucky border (and that weird jog in the otherwise straight line). https://www.tngenweb.org/campbell/hist-bogan/surveyor.html

You know that bump in Minnesota where it goes up into Canada? It was a map making fuck up:

https://en.wikipedia.org/wiki/Northwestern_point_of_the_Lake...

The most important thing for a map at the local level is that it be conformal - that is, shape-preserving.

If the angle between two roads is 38 degrees, the angle should be 38 degrees on your map. Mercator is conformal.

Area distortions aren't a big deal, because on a local level, they only have the effect of scaling the map by an overall factor. You have to zoom in more to see features of a given size at the equator than at high latitudes, but that's not such a big problem.

Another nice feature of Mercator is that it's cylindrical, so when you zoom out, the map fits on a rectangular screen.

Many countries (I know for certain Sweden, Norway and the US) use Transversal Mercator for local maps. The UTM zoning system makes the error as small as possible in the local area.

Some use a local Lambert conformal conic projection.

Interesting! I guess it is preferable for countries stretching in the east-west direction, instead of north-south?

Every US state consists of one or more zones that use specific map projections for best representation. The choice goes significantly further than just the choice of projection family.

You can find resources for the State Plane Coordinate System online.[1]

For example, the transverse Mercator projection intuitively maps from a sphere onto a cylinder oriented east-west (rather than pole to pole). But since we can vary the relative position of the sphere (such as which longitude is closest to the line perpendicular to the surface of the cylinder), we actually define a family of projections.

Another variable is the model of the Earth's surface used for the mapping. For example, the web Mercator projection maps from an ideal sphere, but that is not sufficient for many more precise applications. See geoid[2] and datum.[3]

Geodesy and geomatics are pretty huge areas of applied science that go far beyond the scope of a blog post.

[1] https://www.edc.uri.edu/nrs/classes/NRS409509/Lectures/4MapB...

[2] https://en.wikipedia.org/wiki/Geoid

[3] https://en.wikipedia.org/wiki/Geodetic_datum

>Whoa, wait a second...is this saying that roads which look straight on a map are actually not straight, in the sense that they're not actually following the shortest path between two points?

The Gaussian curvature of Earth is small enough that for distances below 50 miles and locations south of the Arctic Circle, it really doesn't matter.

But the nice thing about the Mercator map is that it shows a constant bearing (Δlat/Δlong) as a straight line. The only projection which shows great circles as straight lines is the gnomonic projection, but this is limited to showing a very small region before distortion becomes unacceptable.

I believe some highways are purposely built slightly crooked (like an S) over long flat stretches, to keep drivers more engaged and less likely to fall asleep. I'm not sure if there are any that are built that way to look straight on Mercator projection.

Of course, there are large parts of the world where the population density isn't worth doing such a thing, and you have quite long roads that are perfectly straight from the driver's point of view.

Citation?

It's an interesting question but the precise straighteness of roads would more likely be derived from local ground surveying (e.g., optical) which places the markers from which construction commences.

The maps then eventually record what was actually built (measured by satellite nowadays).

This is not what I meant XD You want roads in real life that are straight to look straight on a map. Actually, most global map projections do not meet this simple requirement. For an extreme case, think of how compressed everything is at the top of Lambert's map.

Very long roads are not truly straight because they follow the curvature of the earth, but I was not referring to those. In any case, you're more likely to notice winding around local topology and local elevation changes on long road trips.

Also see roywiggins answer :)

Mercator preserves compass angles. So a road that goes northwest/southeast will always run at a 45 degree angle to horizontal. This preserves the appearance of city grids, and means that if the intersection is as a particular angle, it will look like it on the map.

Otherwise, you will end up with 90-degree intersections that look squished, which is unhelpful.

Aren't there distortions the further north/south you go? At the limit, a 90 degree angle with the corner at the pole would map to a half box shape. I'm having problem visualizing what happens right before the limit though.

There are increasing scale distortions as you go farther from the equator, but angles (and thus shapes) are always conserved.

Tissot's indicatrix is a way of visualizing both scale and shape distortions on maps. It shows how equal-radius circles drawn on the Earth are distorted by a given map projection. This is what it looks like for the Mercator projection: [1]. This is what it looks like for a map projection that does not preserve angles (shapes): [2]. The latter projection would produce very misleading road maps far away from the equator. Intersections would get squished in the North-South direction, so the angles between roads would be wrong.

1. https://en.wikipedia.org/wiki/Tissot%27s_indicatrix#/media/F...

2. https://en.wikipedia.org/wiki/Tissot%27s_indicatrix#/media/F...

The angle preservation does break eventually, right at the poles. But it's an abrupt break, it goes from "perfectly preserved" in the region around the poles to "totally broken" at the poles.

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