Many superlatives describe the border between Canada and the United States. It’s the longest non-militarized border on the planet. It touches three different Oceans (Atlantic, Pacific, Arctic) plus the Great Lakes. It extends 8,891 kilometres (5,525 miles). While impressive, this isn’t about any of that. No, I’m more interested in the extremes in the opposite direction, the tiny ones.
View Larger Map
Look at any map of the two nations and one will notice two very long straight segments. The first one forms a vertical line between Yukon and Alaska along 141 degrees west longitude. The second one forms a horizontal line from the Strait of Georgia to the Lake of the Woods, half way across the North American continent, at 49 degrees north latitude. Or do they?
These two territorial behemoths — the second and fourth largest nations on earth — established the International Boundary Commission to maintain their peaceful boundary. This commission handles border demarcation, places and maintains the physical markers, removes underbrush to keep the border visible, and generally works to prevent territorial disagreements from occurring before they can arise. It’s a noble mission.
A Regular reader of the Twelve Mile Circle, "Greg" wrote a comment back in July 2009:
The International Boundary Commission says on their website that the two closest boundary markers between the US and Canada are 46 cm apart, IIRC. Where on Earth (well, in North America) is that?
I replied at the time with the following comment:
I started looking at the boundary coordinates at [the International Boundary Commission website] http://www.internationalboundarycommission.org/products.html#maps but I started going cross-eyed. The answer would be somewhere in those data sets. It shouldn’t be too difficult to set up a spreadsheet to figure that out, so now I have a project for a rainy day.
The rainy day arrived. More accurately, it was a snowy day. Either way I had plenty of time while I was trapped indoors the last couple of days to ponder Greg’s question further.
First, let’s dispel a misconception before I dive into the coordinates and mathematics. The border doesn’t actually include long, straight segments. 141 West and 49 North? forgetaboutit. The border is actually composed of thousands of very tiny individual segments cobbled together. On land the lines run physically between monuments. On water they run virtually between turning points. A typical segment, and I’m pulling this pseudo-randomly from the data set, runs approximately 1.8 kilometers between 48.999978 N, -109.713375 W and 49.000039 N, -109.673353 W. One would expect both of those latitudes to be 49.0 N if the border were one long line, but it is not.
That’s a typical length by the way. I loaded the Boundary Commission’s coordinates into a spreadsheet and it produced a file with 11,501 rows. That’s more than eleven thousand separate lat/long coordinates used to define an 8,891 kilometer border. Thus, Greg’s question about the 46 centimeter segment doesn’t sound so far-fetched anymore, now does it? It should only a matter of going through the spreadsheet, sorting it by length and finding that shortest segment.
Ultimately I couldn’t find it. However, I narrowed it down to several possibilities.
View Border Length Extremes in a larger map
I had a file with every lat/long coordinate along the Canadian – American border so what was the big deal? Couldn’t I simply calculate the distance between the points on the spreadsheet and find the shortest one? Theoretically, yes. Formulas exist to determine Great Circle distances between points on a sphere but the earth isn’t a perfect sphere and the basic, least complicated formulas decrease in accuracy as the points move closer together.
One of those methods is the Haversine formula, a specialized application of spherical trigonometry used in navigation. There are more complicated variations on this theme but I don’t have the inclination to decipher them. This is math I thought I’d never need when I attended Junior High School back in the Dark Ages — very shortsighted of me at the time — so my skills are rather limited in this field today.
The BlueMM website was a godsend. The author was kind enough to translate the formula into something that could be dropped directly into an Excel spreadsheet.
=ACOS(COS(RADIANS(90-Lat1)) *COS(RADIANS(90-Lat2)) +SIN(RADIANS(90-Lat1)) *SIN(RADIANS(90-Lat2)) *COS(RADIANS(Long1-Long2))) *6371
Simply replace Lat1, Lat2, Long1 and Long2 with cell references and it’s good to go. Replace 6371 (radius of the earth in kilometers) to 3959 to calculate distances in miles. Copy the formula 11,500 times and create the distance between each of the tiny segment along the Canada – USA border. I don’t know the shortest one for certain but each of the instances displayed on the map above calculate mathematically to a distance of less than a meter. I’m going to guess the 46 centimeter distance between two monuments falls within the eastern cluster but I may never know unless someone from the International Boundary Commission happens to stumble across this page.
View Border Length Extremes in a larger map
It wasn’t much easier finding the greatest length along the border. Most of those on land appeared to be along the Yukon – Alaska boundary but even those rarely stretched ten kilometers or more. The longest distance between two monuments on the southern border appeared to be these two straddling Missisquoi Bay, an arm of Lake Champlain near the geo-oddities known as Fort Blunder and Alburg.
I didn’t find the final answers but if I ever want to create a border map I have everything I need to generate an xml file and read it into a webpage. Maybe I’ll save that for the next rainy/snowy day.