I was just looking at the Mercator-projection World Map on my wall here at work, and something occurred to me: Why does the distance between the parallels get larger as you approach the poles? The map I have is marked out in 15-degree quadrants, with the meridians equidistant. Why not make the latitude lines equidistant, as well, so that you have nice, neat squares? Seems like it would take care of some of those freaky “Greenland is bigger than Africa” distortions, too.
In an attempt to be helpful, I Googled to this site but I doubt it will help you! I found the opening para to be one of the least comprehensible things I’ve ever read on the web.
Good link. A deeper link inside yielded this:
So, good when combined with a compass. Since almost no one who buys such maps actually uses them for navigation, I wish they’d change the default cylindrical projection to a square grid. I’ll have to call the American Map Corp. and complain.
One of the great benefits of the Mercator Projection is that a straight line on the map corresponds to a straight line on the earth. If the parallels were equidistant this would no longer be the case.
Strictly speaking there are no straight lines on the globe; the closest you can get is a great circle. And some straight lines on the projection, like lattitude lines, are not great circles (except the equator).
Strictly speaking there are no straight lines on the globe; the closest you can get is a great circle. And some straight lines on the projection, like latitude lines, are not great circles (except the equator).
Well lets say sailing without turn the ship. Which I believe would be a great circle. As has been pointed out only a small subset of lines on the Mercator Projection are great circles. So it looks like I was flat out wrong. And the real benefit is what toadspittle found in the provided link.
WRONG. Eccept for the equator and lines running exactly north/south, great circles are not straight lines on a Mercator Projection.
>> Why not make the latitude lines equidistant, as well, so that you have nice, neat squares?
The reason is that the world is round, like a sphere, yes, really, so the meridians get closer together as you move away from the equator. So, at the equator, one degree of latitude is about equal to one degree of longitude but as you move away from the equator one degree of latitude remains sensibkly the same (60 nautical miles) while one degree of longitude is less and less until it is zero at the poles. You do not have to care about the size of Greenland. Thake a local map of anywhere in the mid-latitudes and you will see what I mean. Mercator has nothing to do with it. If i own a plot of land in Washington DC which is rectangular and 1’of latitude wide by 1’of longitude wide, I do not have a square piece of land as it is longer in the N-S direction.
And that is why maps look funny to you.
just want to point out that there are other projections even though I don’t recal any of their names. The one I do recall is called the Peterson (Peters?) projection which actually streaches out vertically the land at the equator and shrinks it at the poles. The idea behind it is even if the map isn’t exactly to form the land mass area would be accurate. (Alaska, Greenland would be squashed while Africa would be much longer then it normally appears)
As pointed out earlier, on a globe one degree of latitude is the same distance everywhere. The distance of one degree of longitude is a function of latitude. The distance of a degree (or any nmber of degrees) of longitude at latitude L divided by the distance of a degree (or any nmber of degrees) of longitude at the equator is equal to cos(L). We’ll call this the Longitude Distortion. The Latitude Distortion is 1 because a degree (or any number of degrees) of latitude is the same distance all over the globe.
Longitude Distortion/Latitude Distortion = cos(L) for a globe.
Now for a Mercator Projection we want a map that is a cylindrical projection that will allow us to keep our bearings when we set sail. On any cylindrical projection a degree (or any number of degrees) on the map (not on the land the map represents) is always the same everywhere on the map. (This means longitude lines are equally spaced.) Therefore Longitude Distortion = 1. The Mercator Projection is a special cylindrical projection because it allows us to keep our bearings. Why is this? Because Longitude Distortion/Latitude Distortion = cos(L) just like on a globe. Therefore Latitude Distortion for the Mercator Projection = 1/cos(L) or sec(L).
How does knowing this help?
toadspittle, look at your map again. Measure the distance between two lines of longitude. Divide by the number of degrees between the two lines. Call this number X measured in whatever units suit you. A latitude line at L degrees should be X*L/cos(L) units from the equator.
For bonus points, figure out how far the North Pole should be from the equator to better understand why it is not on the map.
Dunno what Mercator’s top priority was, but his map actually has two features of interest. They’re dependent on each other, but nobody has mentioned the second one.
For rhumb lines to be straight lines on the map, the north-south map scale has to increase the same as the east-west scale, as you move north or south away from the equator. The distance on the map from 10 degrees to 20 degrees latitude has to be a bit more than the distance from the equator to 10 degrees, and the distance from 20 deg to 30 deg is longer still. The object is to keep the east-west scale equal to the north-south scale at every point, so over small areas the map isn’t distorted.
How much longer? If we assume a spherical Earth, then we can draw a Mercator map of the world as follows:
Say one radian of longitude along the equator is one foot on the map. Then the distance on the map, in feet, from the equator to latitude L equals the arc sinh of the tangent of L. Or the arc tanh of the sine of L if you like that better. Both of those are the integral of the secant function, which is what we need.
That makes it easy to calculate rhumb-line distances and directions between two points. Not so easy on a non-spherical Earth, but still possible,
Think of a mercator projection as if there were a lightbulb at the center of the earth globe shining through the surface. Then wrap a big paper cylinder around it. Mercator’s map would be the result projected on that cylinder. As is obvious, you don’t get projection of the extreme pole areas because it would have to be an almost infinite tall (and deep) map.
Also, projecting higher latitudes exaggerates the size of Canada’s north and Greenland. (Greenland is tiny compared to Africa in real life; the USA is almost as big as Canada.)
There are plenty of options - See this:
As mentioned, a constant compass heading is a straight line on the Mercator, which for sailing in the latitudes below about 60° it’s not too distorted to use.
The projection chosen depends on the purpose of the map - some have roughly equal area, some have regular latitude intervals, etc.
Which is why, until they started using 3D to render a globe in the zoomed-out view, all internet maps (Google Maps, etc.) used Mercator. All other rectangular map projections have some sort of distortion at small scales–mostly either unequal scaling or some kind of shearing. Neither one is desirable when you zoom into your town and want to see a normal-looking map.
But the availability of advanced graphics caught up and they mostly do this thing where they blend a globe into a flat map as you zoom in.
Though the OP (zombie as it is) has a very easy explanation: if the map had equal distance between parallels, it would have been called equirectangular, not Mercator.
It actually wouldn’t, and I don’t know why every single thing that talks about map projections uses that illustration in connection with the Mercator. You could certainly make a map projection that way, and it would have some qualitative traits in common with the Mercator, like the poles being infinitely far away, but it’s not actually the same in the details.
Sounds like the OP (24 years ago) was interested in the Plate Caree projection.
And as I recall, for medium-small scale surveying using GIS applications, a lot of applications use UTM, or Universal Transverse Mercator.
Basically the same as Mercator, but rotated 90° so what would be the equator is a line of longitude around the earth. Divide the earth into 60 zones, each with its own UTM and coordinate system, so each particular UTM zone spans 6°. Thus, you never travel too far from where the projection distorts reality.
Prior to computer based navigation systems, navigation was done on flat charts with simple tools: ruler, dividers. Distance on the chart needed to match distance in reality in a useful manner. Also angles on the chart need to translate to headings.
The projection chosen gets you near to these objectives in a manner that matches your use case. Three main use cases: air, sea, and land.
At sea there are scant landmarks and opportunities for navigational fixes are few. But you are travelling slowly relative to the large distances . A chart where you can measure a heading and then steer to it is a good design. You will traverse a Rumb line, which isn’t quite the shortest path, but a sequence of Rumb lines will be pretty good. Sextant, clock, compass, charts. Good to go.
Air, and you are covering distances fast. Prior to GPS your average aviator used a lot of lesser radio based navigation aids. Non-directional beacons littered the landscape providing sources that you could get a bearing to with useful accuracy. A few such fixes, and you had position. Closer in to civilisation and systems such as DME and VOR could guide you in. A useful map projection lets you plot back bearings from NDBs to triangulate your position and do so on a scale useful to flying. So you want to preserve angles, aka use a conformal mapping. Since you are travelling fast, it would be nice to have great circle courses easily found. The Lambert Conformal Conical Projection manages this. You can read the required course to steer off a straight line drawn on the map as a set of headings creating a piecewise set of Rumb lines fitting the great circle. (Straight lines on the map cross meridians at different angles.) Layered onto navigation is calculation of wind triangles, to give you the final heading steered. The projection for the map may choose the best location for the line on the map to touch the Earth, improving fidelity for the region depicted. Maps of adjacent regions cannot be joined at their edges. You don’t care.
On land, you are travelling short distances at moderate speeds. There is stuff everywhere and you probably want to know exactly what is around you. You can reasonably navigate with only the map in your hand and looking at the landscape, but usually would have compass ruler and dividers. You would really appreciate it if the map was as close to a scaled down match for the territory as possible in every direction, no matter where on the Earth you were. So maps are based on more locally relevant projections. The simplest is to rotate the Mercator projection so that the contact between the cylinder and Earth runs up the meridian you are located on. Transverse Mercator. You are assured of reasonably sensible behaviour and fidelity in all directions. But only on a local scale. The further away from the chosen meridian the worse it gets. So you eventually move to another map. This makes large scale maps infeasible.
Finally, you might demand that you can determine location in your map part region by measuring distances on a rectilinear grid, not with angles. So you can define a plane cutting the Earth that presents you with a locally flat Earth. You measure location by distance on a grid relative to the middle of your chosen plane. There is some distortion of distances, they are only accurate where your plane cuts the Earth. But it is pretty good. Down sides are that you have lots of regions, maps don’t join at the edges between regions, locations have mutually inconsistent locations in the areas where mapping regions overlap, and locations are a mouthful to convey. But if you are marching about the landscape or measuring and documenting the landscape on human scales, for very human activities, very handy.
I feel like I should also mention polar projections. To make one of those, you put a plane tangent to the Earth at some point (which might be the North or South Pole, but doesn’t need to be), and then draw lines from the center of the Earth through the surface and the plane. These have the disadvantage that they can only show one hemisphere (the edge of the hemisphere would be an infinite distance away), and of course you get increasing distortion the further away you are from the point of tangency, but they have the very significant advantage that any great circle on the globe will be mapped to a straight line on the map, and vice-versa.
So to plan out a great circle route, you’d first find a polar projection map which includes both your starting point and your destination, and draw a straight line on that map. Then, you’d find a bunch of waypoints on that straight line. Then, you’d switch to a Mercator map, and plot all of those same waypoints on it, and then you’d draw straight lines between the waypoints on the Mercator map. Then you’d find the compass bearings of all of those lines on the Mercator map, and use the first bearing until you got to the first waypoint, and then switch to the second bearing, and so on.
If you doubt Chronos’ correction, just look at a Mercator map of the world. If you did the lightbulb thing, the distance on the resulting map from the equator to 45 degrees latitude would be less than the distance from 45 to 65 degrees. Which, on an actual Mercator map, it isn’t.
The latitudes actually do get very slightly further apart as you go north (or south) from the equator. Not nearly enough to answer the OP, though. The reason is that the earth is not a sphere, but an oblate spheroid. I once read a book (can’t recall its name) that the government of France once sponsored two teams, one to measure the length of a degree of latitude in some place like Equador and the other to do the same in northern Norway or Sweden. I believe the difference was something like a km, out of a degree or about 450 km. One group had to contend with the weather, while the other ran into political problems.