> Napier’s main motivation was to find an easier way to do multiplication and division.
> Next, mathematicians decided to combine these tables. If you wanted to multiply trigonometric functions, you could find the values in a trigonometric table and then convert them to logarithms.
Actually, Napier's 1614 Mirifici Logarithmorum Canonis Descriptio contains tables of −10⁷ ln(sin x/10⁷) . Non-trigonometric log tables appeared later.
My resulting map is on the last page here: https://adelie.antarkt.is/aron/l0020b.pdf
EDIT: Apparently it was one of my GIS courses, not the cartographic one, which were more color theory and labels.
One bad thing about the projection is that great circles are curved, so my straight lines should have been curved as well.
So we need a map projection that weighs distances by political resistance. :)
https://www.flightradar24.com/ makes it look like most flights are direct (too far south for ANC, too far north for HNL), but I couldn't quickly find any way to filter out cargo from pax.
PS. Having just seen https://www.youtube.com/watch?v=5hdUKZLQIuQ I have to ask if you all really chew snus? What savages — tobacco is for schnupfen. https://www.youtube.com/watch?v=oOa1cz7O3kU
If you are creating your own global-extent map at a limited range of scales, most of the OSM tooling for example PostGIS lets you specify a custom projection like http://shadedrelief.com/ee_proj/
I had the privilege of chatting with one of the developers there who works on their projection engine (which is unmatched in the industry, as far as I know). To contribute to it, one effectively has to be at the graduate level in mathematics, low-level programming (especially numerical methods), and geography at the same time.
There was one subtle bug that I had in the web interface where at the US level, clicking on the wrong place in the spash screen loaded, instead of the clustered dealerships at that level, instead added a pin to the map for every single dealership in the country. It took about 10 minutes to render on 2007 hardware but was really pretty when it was done. I no longer have the screenshot unfortunately.
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.
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.
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.
You can find resources for the State Plane Coordinate System online.
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 and datum.
Geodesy and geomatics are pretty huge areas of applied science that go far beyond the scope of a blog post.
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.
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.
The maps then eventually record what was actually built (measured by satellite nowadays).
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 :)
Otherwise, you will end up with 90-degree intersections that look squished, which is unhelpful.
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: . This is what it looks like for a map projection that does not preserve angles (shapes): . 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.
Can someone more mathematically literate than me shed any light on whether it matters? I guess it's still useful even if the integral is undefined at certain points off the edges of the map.
To slightly misquote the great Richard Feynman, "It turns out that it's possible to sweep the infinities under the rug by a certain crude skill." :)