Math on projecting lat/long to a small map

Factual Questions
21

Different projections serve different aims. Mercator was more for navigation, the polar latitudes are obviously distorted. (In your typical schoolroom map, Canada looks a helluva lot bigger than the USA - it’s not reall that much bigger). There are conic projections for better depiction of near-polar regions, there’s the orange-peel map for more accurate depiction of surface area without distortion, etc. Depends on your aim.

If all you want is to display county data, and only for the continental USA, a sort of cylindrical projection is probably sufficiently undistorted.

A degree of Longitude at 49 degrees lat. is 45.5 miles; at 25 degrees lat (bottom of Florida) a degree of longitude is 62.73 miles. A degree of lat is pretty much the same in both cases, about 69 miles. The average for longitude in this range is about 54 miles.

If you just translate lat/long to x, y and use the intermediate value for longitude ( each lat degree translates into 61 pixels, each degree of long translates into 54 pixels, scaled appropriately. You map will be slightly stretched at the top and slightly squashed at the bottom, but not more than a typical Mercator projection.

So let’s say the map for is for 65W to 125W long (60 degrees), 25N to 50 N (25 degrees) (round numbers, sorry, clips off the Florida Keys. Lower right is 0,0. lets say this is 1000 pixels high, so 1 degree lat is 1000/25=40 pixels vertical.
By the same scale, if 40 pixels is 61 miles, 54 miles would be 35.41 pixels.
(Note 25º x 61 miles is 1525 miles). A rectangle 1 degree each side would look like on the map scale 61 miles high and 54 miles wide, so 40 pixels high and about 35.4 pixels wide in the map image.
your map would be 60º x 35.4 = 2,144 pixels wide.

A point A(aº b’ c") , D (dº e’ f") lat and long would be - translate to decimal - A = a + b/60 + c/3600 and B = (d+e/60+f/3600)

Translate these to pixel coordinates x,y - lower right is 0,0 pixels, 25º lat 65º long (and x is vertical, y is horizontal)
x= (A-25)*40
y=(B-65)*35.4

So the top left corner of the map is x=(50-25)*40 =1000, y=(125-65)*35.4 = 2124pixels somewhere in the middle of Vancouver Island.

Assuming I can still do math.
if you want the pixels to number left to right, I leave this as an exercise for the reader.

If you have a different goal in a map - undistorted area, or true distances - well that doesn’t work well for significant chunks of a spherical surface. All maps are a compromise.

22

OK then. Decimal degrees. Is this data in some sort of spread sheet or delimited text file a .csv?

Is there another value that you want to carry with the point that is created? That could be anything. Home value, Average rainfall. Whatever.

If you could cut and paste a few dozen value pairs even in a response to this, I should be able to import it to excel, delimit it and produce some locations on a map of the USA.

I could test it for you and see if it’s possible with free software. It probably is.

23

Using lat long pairs as x,y coordinates results in what’s called the Plate-Carrée projection, though today many of us might also refer to it as unprojected data. For areas only a couple of degrees of latitude high, you can cheat by squeezing the resulting map to the cosine of the latitude. Before GIS was entirely available to my mapmaking efforts, I sometimes used this trick. For Chicago, for example, at 41º North, I’d squeeze the map to be 75% of horizontal but 100% vertical.

However, this is not a good practice for the entire continental US, which deserves to be displayed in a conic projection.

24

Good point. (Plate-Carrée looks “squished” in my latitudes, to most eyes, without the sort of correction you outlined.)

But none of that matters, if the OP insists on using the original map of apparently unknown projection. If that’s the case, I don’t see how georeferencing — rubber-sheeting that map into one with known projection, then easily plotting the lat-long points into that — can be avoided.

Also…it was you, Mr. Downtown, to whom I was referring as our resident GIS expert, in an earlier post (I mistakenly wrote “Mr. Dibble”).

25

The map you’re interested in is a conic projection, what’s most familiar for maps of the continental US. It’ll be a tricky matter of trial and error to get your lat-long points onto this one properly. You have to know the central meridian and two reference parallels just to start doing the complex math.

But these days there are lots of ways to map a list of points—especially if you already have the lat-longs. You can Google “alternatives to Fusion Tables” to be led to several of them. https://mapalist.com/ is one.

Or you can PM me the list and I’ll throw them onto a very similar map of counties for you. If one already has the software installed, it’s a pretty trivial task these days.

26

As you see from all the replies - the main question is - what is important in this map?
Does it have to have exact (as exact as possible) area? Distances? Compass angles?
Or is it just for comparison or demonstration - “here is beer consumption by county” - so it just has to be recognizably shaped so people can identify regions.

My suggestion to use suitably scaled lat.,long as x,y is the first projection on this list - equirectangular. One-degree lines are squares, which approximately matches reality near the equator.
The only alteration I have suggested was using 37.5 Lat as the baseline rather than the equator, and scale appropriately.

27

To depict “beer consumption by county”, you need a cartogram, not a projection.

28

You can start from this map of US counties (zoom in to the US), and search for the long/lats in the search bar. Each one you find can be added to the Map Notes, and when you zoom out you should be able to see all of them.

If you just want to plot coordinates onto a county map of the US, it’s easiest to just let someone else take care of the projection for you. Why bother doing the math yourself?

29

Right. When the OP started by saying they wanted to use “this map,” did they really mean that very map — in which case, its projection would be important, and if unknown, it would have to be he referenced/rubber-sheeted to a base map with a known projection — OR did they actually mean “I want to use the DATA that is SHOWN in this particular map?”. Probably, the latter — and most replies in this thread have dealt with that. In other words, easier to make a new map with two layers: the lat-long points, and the underlying data displayed in the original map — which the OP will have to write into a table, and join (using GIS or otherwise) to an empty polygon file (easy to acquire) with the boundaries of the units used in the data (counties? States?).

Georeferencing as I described it is normally only used when the original map truly is the important document, typically Some old, historic paper map.

30

Depend whether you are emphazing quantity or geographic location.

We’ve seen this with, for example, voter maps. Who won which district (by color) gives a somewhat distorted picture because one metro area is a small blip which contains as many votes as several central states.