# Finding the centroid of a shape

So I wanted to determine the geographical center of Lorain County Ohio. I didn’t find it online at first so I decided to calculate it myself. I have various maps at hand from the Ohio Dept of Transportation printed out that I mark up with various things. I picked a printout about 16" high and cut it out carefully then used the old school method of establishing it with plumb lines. Despite the grade school feel of this method it is an actual geometrically correct method.

http://www.metroparks.cc/parks.php

It’s a nice shape for this method with 8 different corners to hang it from. I used a thin pin and hung it on the wall and let it stabilize then hung a plumb bob made of a very thin thread with a washer on the end and traced the relevant sections onto my paper map. They all converged with excellent precision at one point. I used various landmarks and a ruler to transfer this point onto a Google map to determine the latitude and longitude. I got 41.2998, -82.1571, great!

But then I did manage to find it listed on this page:

https://www.anyplaceamerica.com/directory/oh/lorain-county-39093/

They claim it is at 41.3149, -82.1561

This is more or less on the same longitude but one mile north in latitude compared to my measurement. On my paper map that comes out a half inch further north but on the same vertical line.

I’m trying to figure out what I did wrong? If I were to bias my paper map to make it work out I would have to add about a 1 inch strip along the shoreline to pull the centroid north a half inch. I doubt that my ODOT map is off that much.

Dennis

Anyplace is using spherical coordinates, you are using a flat projection of a curved surface.

I thought of that and you are probably right. But trying to visualize it it seems that the spherical method would come up with a different (greater) area, but that area would be distributed equally in all directions from the centroid. As I mentioned to my colleagues, apparently Huntington Township weighs less then I thought.

Dennis

A possibility is that the two methods are using different map projections. In particular, if one of them uses a Mercator projection, then you’re basically mapping longitude lines to vertical lines on a grid and latitude lines to horizontal lines. On the page, the distances along the “top” and “bottom” of the grid; but on the Earth, the distance along the “top” of the grid is less than the distance along the “bottom” (assuming north is up and you’re in the Northern Hemisphere.)

This means that the Mercator projection effectively gives extra weight to the more northerly areas, which would skew the centroid towards the north compared to a projection that represents the areas more faithfully. What’s more, this effect wouldn’t really skew the centroid to the east or west all that much, so long as the area wasn’t too asymmetric between east and west. Lorain County isn’t perfectly symmetric, but it’s symmetric enough.

That is correct. Except my paper/plumb method apparently skewed the centroid towards the south and I can’t find anything that should do that.

Dennis

A Peters projection would pull the centroid southwards (the areas are all correct, but not their distances from the center), but at any latitude other than the equator would produce a very visible distortion, squishing everything north-south. I dunno, maybe a scaled Peters projection? One doesn’t usually need to worry about what projection one is using, on such a small scale.

Did you try other maps? I took screen shots of the maps on both of your links and also from Google maps. When the three maps are scaled to the same width, the map in your first link is a different height than the other two maps.

The location of the coordinates you gave for the centroid on Google maps looks like it might be based on the entire county, including the portion of it that is in Lake Erie.

Just out of curiosity…why is it important to find the centroid of a county?
I can see why knowing the centroid of an irregular shaped solid,( say , a cam-shaped part of a machine) is important to a mechanical engineer.
But a county is just an arbitrary shape drawn by politicians.

just curious.

Me too. That’s all! I know the center of the USA is well known and the locals have a marker, I think. Maybe I’ll paint a giant white cross hairs there so we can see it from the air.

Manlob might be right, perhaps they use some underwater area. That is exactly what it seems like.

Dennis