49 ms·

100 years to solve an integral: The history of the Mercator map

widforss 6y ago
I did a course on cartography a couple of years ago, and one of the better assignments was to find a suitable projection and make a proper map describing a comparison in distances between two points. The projection I used was of course Two-Point Equidistant Projection (the only possible one). My resulting map is on the last page here: https://adelie.antarkt.is/aron/l0020b.pdf 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.
- 082349872349872 6y ago
  LLA: the optimal hub — at least in summer. (and modulo transpacific routes?)
  - widforss 6y ago
    The LLA freight proposal was given by the teacher in the assignment, but I guess there is a real benefit. Isn't Anchorage preferred over Honolulu for most trans-pacific flights?
    - 082349872349872 6y ago
      Hej. Sorry, I have no idea. https://www.flightradar24.com/ 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 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 https://www.youtube.com/watch?v=oOa1cz7O3kU
      - widforss 6y ago
        Haha, if you drive EPA you most definitely use snuff, in the mouth.
    - goodcanadian 6y ago
      Yes, Anchorage is a major freight hub. Apparently, fourth busiest in the world: https://en.m.wikipedia.org/wiki/Ted_Stevens_Anchorage_International_Airport https://en.m.wikipedia.org/wiki/Ted_Stevens_Anchorage_Intern...
  - Pinus 6y ago
    I remember there was much talk in the local press around 25 years ago about how Kallax/LLA was to be the next big freight hub. It went as far as extending the runway to accommodate bigger planes (making it the longest runway in Sweden, if Wikipedia is to be believed). As far as I know, the freight boom failed to materialize, largely because Russia were not very keen on allowing flights through their airspace. So we need a map projection that weighs distances by political resistance. :)
NickM 6y ago
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.
- widforss 6y ago
  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.
  - azepoi 6y ago
    Some use a local Lambert conformal conic projection.
    - widforss 6y ago
      Interesting! I guess it is preferable for countries stretching in the east-west direction, instead of north-south?
      - legerdemain 6y ago
        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/4MapBasics/StatePlaneCoordinateSystem.htm https://www.edc.uri.edu/nrs/classes/NRS409509/Lectures/4MapB... [2] https://en.wikipedia.org/wiki/Geoid https://en.wikipedia.org/wiki/Geoid [3] https://en.wikipedia.org/wiki/Geodetic_datum https://en.wikipedia.org/wiki/Geodetic_datum
mci 6y ago
> In 1614, John Napier introduced logarithms. > 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) [0]. Non-trigonometric log tables appeared later. [0] https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34187/31151005337641.pdf#page=25 https://jscholarship.library.jhu.edu/bitstream/handle/1774.2...
- the_origami_fox 6y ago
  Yes you're right. It has to do with how he derived the approximation formula for the natural logarithm. He needed a function y=sin(x) for his log(y) calculations. But I am not sure when the log(f(x))) tables for the other trigonometric functions came about. As far I understood, initially a single log(y) table sufficed.
  - mci 6y ago
    Well, I suppose Napier presaged the concept of a single source of truth. Aren't other log-trig tables a waste of paper if you can look up log(cos x) under log(sin(90°−x)) and quickly calculate log(tan x) as log(sin x)−log(cos x)?
krick 6y ago
BTW, is there still no online map that would use some better projection (i.e. anything but [Web-]Mercator)? I mean, there is Google Earth, of course, but it has too much visual effects added, to use it as a go-to tool, as I use OSM or Google Maps. But mercator makes large scale distance and area comparison absolutely unintelligible, and I would rather much prefer to be able to use Kavrayskiy VII/Natural Earth or something like that with OSM. No way it could be too computationally expensive in 2020, right? (I mean, once again, we do have Google Earth.)
- bdon 6y ago
  I don't think there is an area-preserving projection that can work for all the scales online map services are designed for. 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/ http://shadedrelief.com/ee_proj/
  - krick 6y ago
    If I'm creating my own map I can kinda draw whatever the fuck I want. I need a web-map (preferably OSM-based) that I can use as a web-map, but with a projection less fucked up than Mercator, of which I can name at least 10 in addition to 2 I already named. Also, web-mercator generally is not a problem on small scales, so there isn't really a problem of alternating between projections depending on scale. (Which is not really necessary too, since other projections are generally more or less ok even on small scale. You can totally use globe view on google maps on city-scale and barely notice any difference at all.)
    - elliekelly 6y ago
      Perhaps it was unintentional but the tone of your comment comes across as over the top hostile in response to someone who was just trying to help you.
- TimTheTinker 6y ago
  Esri's ArcGIS Online supports over 100 projections, and they're always working on more. 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.
rm445 6y ago
The author tells the story of typical undergraduate instruction for the integrals of tan(x) and sec(x). I would have thought that such a setting would have included that these were improper integrals, because of the infinities in the functions. i.e. if you evaluate the definite integral for any particular interval, it will give you the right answer except if you've gone through the part where the graph goes up to plus infinity and back through minus infinity. 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.
- deleted 6y ago
  [deleted]
- the_origami_fox 6y ago
  It matters here in the sense that to show the poles, you need to stretch the Mercator map to infinity, because the function sec(x) is undefined at 90 degrees. This is clearly absurd, so map makers get around the problem by cutting the top of the map off, usually around the 85 degrees parallel. I chose not to include this detail in the article. 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." :)
sova 6y ago
Buckminster Fuller invented a map projection called the Dymaxion [1], imagine peeling an orange peel as one intact piece and placing the North pole at the very center. 1 https://en.wikipedia.org/wiki/Dymaxion_map https://en.wikipedia.org/wiki/Dymaxion_map
- shard 6y ago
  The irregular shape of Dymaxion always bothered me. I much prefer the Cahill–Keyes projection: https://en.wikipedia.org/wiki/Cahill%E2%80%93Keyes_projection https://en.wikipedia.org/wiki/Cahill%E2%80%93Keyes_projectio...
pedrocr 6y ago
In modern shipping is this used or do ships just follow the shortest distance between points? Perhaps GPS has made that simple?
- tgb 6y ago
  Even back "in the day" the routes weren't actually following those lines. Here's a neat animation of the routes many ships took (from their logbooks) starting in the 1750's: http://sappingattention.blogspot.com/2012/04/visualizing-ocean-shipping.html http://sappingattention.blogspot.com/2012/04/visualizing-oce...

I did a course on cartography a couple of years ago, and one of the better assignments was to find a suitable projection and make a proper map describing a comparison in distances between two points. The projection I used was of course Two-Point Equidistant Projection (the only possible one). My resulting map is on the last page here: https://adelie.antarkt.is/aron/l0020b.pdf 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.

082349872349872 6y ago
LLA: the optimal hub — at least in summer. (and modulo transpacific routes?)
- widforss 6y ago
  The LLA freight proposal was given by the teacher in the assignment, but I guess there is a real benefit. Isn't Anchorage preferred over Honolulu for most trans-pacific flights?
  - 082349872349872 6y ago
    Hej. Sorry, I have no idea. https://www.flightradar24.com/ 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 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 https://www.youtube.com/watch?v=oOa1cz7O3kU
    - widforss 6y ago
      Haha, if you drive EPA you most definitely use snuff, in the mouth.
  - goodcanadian 6y ago
    Yes, Anchorage is a major freight hub. Apparently, fourth busiest in the world: https://en.m.wikipedia.org/wiki/Ted_Stevens_Anchorage_International_Airport https://en.m.wikipedia.org/wiki/Ted_Stevens_Anchorage_Intern...
- Pinus 6y ago
  I remember there was much talk in the local press around 25 years ago about how Kallax/LLA was to be the next big freight hub. It went as far as extending the runway to accommodate bigger planes (making it the longest runway in Sweden, if Wikipedia is to be believed). As far as I know, the freight boom failed to materialize, largely because Russia were not very keen on allowing flights through their airspace. So we need a map projection that weighs distances by political resistance. :)

NickM 6y ago

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.

widforss 6y ago
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.
- azepoi 6y ago
  Some use a local Lambert conformal conic projection.
  - widforss 6y ago
    Interesting! I guess it is preferable for countries stretching in the east-west direction, instead of north-south?
    - legerdemain 6y ago
      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/4MapBasics/StatePlaneCoordinateSystem.htm https://www.edc.uri.edu/nrs/classes/NRS409509/Lectures/4MapB... [2] https://en.wikipedia.org/wiki/Geoid https://en.wikipedia.org/wiki/Geoid [3] https://en.wikipedia.org/wiki/Geodetic_datum https://en.wikipedia.org/wiki/Geodetic_datum

mci 6y ago

> In 1614, John Napier introduced logarithms. > 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) [0]. Non-trigonometric log tables appeared later. [0] https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/34187/31151005337641.pdf#page=25 https://jscholarship.library.jhu.edu/bitstream/handle/1774.2...

the_origami_fox 6y ago
Yes you're right. It has to do with how he derived the approximation formula for the natural logarithm. He needed a function y=sin(x) for his log(y) calculations. But I am not sure when the log(f(x))) tables for the other trigonometric functions came about. As far I understood, initially a single log(y) table sufficed.
- mci 6y ago
  Well, I suppose Napier presaged the concept of a single source of truth. Aren't other log-trig tables a waste of paper if you can look up log(cos x) under log(sin(90°−x)) and quickly calculate log(tan x) as log(sin x)−log(cos x)?

krick 6y ago

BTW, is there still no online map that would use some better projection (i.e. anything but [Web-]Mercator)? I mean, there is Google Earth, of course, but it has too much visual effects added, to use it as a go-to tool, as I use OSM or Google Maps. But mercator makes large scale distance and area comparison absolutely unintelligible, and I would rather much prefer to be able to use Kavrayskiy VII/Natural Earth or something like that with OSM. No way it could be too computationally expensive in 2020, right? (I mean, once again, we do have Google Earth.)

bdon 6y ago
I don't think there is an area-preserving projection that can work for all the scales online map services are designed for. 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/ http://shadedrelief.com/ee_proj/
- krick 6y ago
  If I'm creating my own map I can kinda draw whatever the fuck I want. I need a web-map (preferably OSM-based) that I can use as a web-map, but with a projection less fucked up than Mercator, of which I can name at least 10 in addition to 2 I already named. Also, web-mercator generally is not a problem on small scales, so there isn't really a problem of alternating between projections depending on scale. (Which is not really necessary too, since other projections are generally more or less ok even on small scale. You can totally use globe view on google maps on city-scale and barely notice any difference at all.)
  - elliekelly 6y ago
    Perhaps it was unintentional but the tone of your comment comes across as over the top hostile in response to someone who was just trying to help you.
TimTheTinker 6y ago
Esri's ArcGIS Online supports over 100 projections, and they're always working on more. 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.

rm445 6y ago

The author tells the story of typical undergraduate instruction for the integrals of tan(x) and sec(x). I would have thought that such a setting would have included that these were improper integrals, because of the infinities in the functions. i.e. if you evaluate the definite integral for any particular interval, it will give you the right answer except if you've gone through the part where the graph goes up to plus infinity and back through minus infinity. 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.

deleted 6y ago
[deleted]
the_origami_fox 6y ago
It matters here in the sense that to show the poles, you need to stretch the Mercator map to infinity, because the function sec(x) is undefined at 90 degrees. This is clearly absurd, so map makers get around the problem by cutting the top of the map off, usually around the 85 degrees parallel. I chose not to include this detail in the article. 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." :)

sova 6y ago

Buckminster Fuller invented a map projection called the Dymaxion [1], imagine peeling an orange peel as one intact piece and placing the North pole at the very center. 1 https://en.wikipedia.org/wiki/Dymaxion_map https://en.wikipedia.org/wiki/Dymaxion_map

shard 6y ago
The irregular shape of Dymaxion always bothered me. I much prefer the Cahill–Keyes projection: https://en.wikipedia.org/wiki/Cahill%E2%80%93Keyes_projection https://en.wikipedia.org/wiki/Cahill%E2%80%93Keyes_projectio...

pedrocr 6y ago

In modern shipping is this used or do ships just follow the shortest distance between points? Perhaps GPS has made that simple?

tgb 6y ago
Even back "in the day" the routes weren't actually following those lines. Here's a neat animation of the routes many ships took (from their logbooks) starting in the 1750's: http://sappingattention.blogspot.com/2012/04/visualizing-ocean-shipping.html http://sappingattention.blogspot.com/2012/04/visualizing-oce...