gCentroid (rgeos) R vs. Actual Centroid (in python

Summary: I thought that using gCentroid in R would return the centroid of a group of points, however I realised that for some reason it actually returns the geometric mean and not the centroid

I wanted to replicate a centroid calculation I did in R:

gCentroid {rgeos}

The centre of mass of these points:

34.7573,    -86.678606  
38.30088,   -76.520266  
38.712147,  -77.158616  
39.704905,  -84.126463

... using the r-script ...

require(rgdal)
require(rgeos)

no_am_eq_co <- "+proj=eqdc +lat_0=0 +lon_0=0 +lat_1=20 +lat_2=60 +x_0=0 +y_0=0 +ellps=GRS80 +datum=NAD83 +units=m +no_defs"
wgs84 <- "+proj=longlat +datum=WGS84"

df <- as.data.frame(list(c(34.7573, 
                           38.30088, 
                           38.712147, 
                           39.704905),
                         c(-86.678606,
                           -76.520266,
                           -77.158616, 
                           -84.126463)))

df$Name <- "points_A"
colnames(df) <- c("lat", "lon", "Name")

# FROM: Coordinates are geographic latitude/longitudes
coordinates(df) <- c("lon", "lat")
proj4string(df) <- CRS(wgs84)

# TO: Project into North America Equidistant Conic
df <- spTransform(df, CRS(no_am_eq_co))

# Get centroids
ctrs <- lapply(unique(df$Name), 
               function(x) gCentroid(SpatialPoints(df[df$Name==x,])))
ctrsout <- setNames( ctrs , unique(df$Name ) )

# Create data frame 
df <- do.call(rbind, lapply(ctrsout, data.frame, stringsAsFactors=FALSE))
coordinates(df) <- c("x", "y")
proj4string(df) <- CRS(no_am_eq_co) 
df <- as.data.frame(spTransform(df, CRS(wgs84)))
names(df) <- c("longitude", "latitude")

print(df$latitude)
print(df$longitude)

Came to:

37.94873834, -81.18378815

I constructed the following example in python - I wanted to replicate the calculation, using:

import numpy as np
from pyproj import Proj, transform

# Using: http://www.spatialreference.org/ref/esri/102010/ we get the Proj4js format
na_eq_co = "+proj=eqdc +lat_0=0 +lon_0=0 +lat_1=20 +lat_2=60 +x_0=0 +y_0=0 +ellps=GRS80 +datum=NAD83 +units=m +no_defs"
wgs84 = "+proj=longlat +datum=WGS84"

def proj_arr(points,proj_from,proj_to):
    inproj = Proj(proj_from)
    outproj = Proj(proj_to)
    func = lambda x: transform(inproj,outproj,x[0],x[1])
    return np.array(list(map(func, points)))

def get_polygon_centroid(polygon):
    #https://en.wikipedia.org/wiki/Centroid#Centroid_of_polygon
    pol = np.array(polygon)
    if np.any(pol[-1] != pol[0]):
        pol = np.append(pol,[pol[0]], axis=0)
    pol_area = get_polygon_area(pol)
    x = pol[:,0]
    y = pol[:,1]
    Cx = np.sum((x[:-1] + x[1:]) * ((x[:-1] * y[1:]) - (y[:-1] * x[1:]))) / (6. * pol_area)
    Cy = np.sum((y[:-1] + y[1:]) * ((x[:-1] * y[1:]) - (y[:-1] * x[1:]))) / (6. * pol_area)
    return np.array([Cx, Cy])

def get_polygon_area(polygon):
    pol = np.array(polygon)
    x = pol[:,0]
    y = pol[:,1]
    return np.sum( (x[:-1] * y[1:]) - (y[:-1] * x[1:]) ) / 2 

def get_polygon_mean(polygon):
    pol = np.array(polygon)
    x = pol[:,0]
    y = pol[:,1]
    return np.array([np.mean(x),np.mean(y)])

def run_test(points):
    points = points[:,::-1] #Flip-axis (so that longitude x-axis, latitude y-axis)
    points_proj = proj_arr(points,wgs84,na_eq_co)

    centroid_proj = get_polygon_centroid(points_proj)
    mean_proj = get_polygon_mean(points_proj)

    centroid = proj_arr([centroid_proj],na_eq_co,wgs84)
    mean = proj_arr([mean_proj],na_eq_co,wgs84)
    return (centroid[:,::-1][0], mean[:,::-1][0])

if __name__ == '__main__':
    my_points = np.array([[34.7573,-86.678606],
                       [38.30088,-76.520266],
                       [38.712147,-77.158616],
                       [39.704905,-84.126463]])

    test = run_test(my_points)
    print("Centroid calculation: {0}\nMean calculation {1}".format(test[0],test[1]))

From this I get:

37.72876321 -82.35113685

Not:

37.94873834,-81.18378815

With a bit more digging I added a function give me the geometric mean:

Centroid calculation: [ 37.72876321 -82.35113685]
Mean calculation [ 37.94873834 -81.18378815]

I realised that for some reason the gCentroid seems to be calculating the geometric mean not the feature centroid (I have added a mean function, which you can see matches the R-result)

Edit:

I thought that perhaps the reason was: since I had a grouping of points, instead of fitting a random polygon through them - like me in the example - or even a convex hull and then taking the centroid of that, the command would default to a mean calculation if the data-type was 'point'. So I explicitly passed it a polygon:

x = readWKT(paste("POLYGON((-6424797.94257892  7164920.56353916,
                  -5582828.69570672  6739129.64644454,
                  -5583459.32266293  6808624.95123077,
                  -5855637.16642608  7316808.01148585,
                  -5941009.53089084  7067939.71641507,
                  -6424797.94257892  7164920.56353916))"))

python_cent = readWKT(paste("POINT(-5941009.53089084  7067939.71641507)"))
r_cent = gCentroid(x) 

plot(x)
plot(r_cent,add=T,col='red')
plot(python_cent, add=T,col='green')

Where the python centroid is:

centroid = get_polygon_centroid(np.array([[-6424797.94257892,  7164920.56353916],
                                             [-5582828.69570672,  6739129.64644454],
                                             [-5583459.32266293,  6808624.95123077],
                                             [-5855637.16642608, 7316808.01148585],
                                             [-6424797.94257892, 7164920.56353916]]))

And then plotted the centroid of this in red (-5875318 7010915) and then the centroid on the same polygon (using python) in green (-5941009 7067939) and the simple mean (-5974304 7038880) in blue:

It turns out that: if a group of 'Points' are supplied, then instead of guessing a polygon through the points or producing a convex hull - the command automatically gives you the mean of the projected co-ordinates.

However, if you supply a polygon then you get a centroid (the same as the python script) - in my python example I was missing one co-ordinate:

centroid = get_polygon_centroid(np.array([[-6424797.94257892,  7164920.56353916],
                                             [-5582828.69570672,  6739129.64644454],
                                             [-5583459.32266293,  6808624.95123077],
                                             [-5855637.16642608, 7316808.01148585],
                                             [-5941009.53089084,  7067939.71641507],
                                             [-6424797.94257892, 7164920.56353916]]))
#polygon closed
#[-5875317.84402261  7010915.37286505]

So running this R-script:

x = readWKT(paste("POLYGON((-6424797.94257892  7164920.56353916,
                  -5582828.69570672  6739129.64644454,
                  -5583459.32266293  6808624.95123077,
                  -5855637.16642608  7316808.01148585,
                  -5941009.53089084  7067939.71641507,
                  -6424797.94257892  7164920.56353916))"))

python_cent = readWKT(paste("POINT(-5875317.84402261  7010915.37286505)"))
r_cent = gCentroid(x) 

plot(x)
plot(r_cent,add=T,col='red', pch = 0)
plot(python_cent, add=T,col='green', pch = 1)

Everything matches nicely: