```source("http://macosa.dima.unige.it/r.R")    # If I have not already loaded the library
---------- ---------- ---------- ---------- ---------- ---------- ---------- ----------
S 07 Centroid (and Barycenter) # With centerPol(x,y) I have the centroid of the polygon "x,y". See above on the left.
BF = 3; HF = 3; x = c(1,4,6,1); y = c(0,1,6,4); PLANE(0,6,0,6)
polyC(x,y,"green"); BOX(); Line(1,4, 4,1,"red"); LIne(1,0, 6,6,"blue"); centerPol(x,y)
#  3.060606  2.939394
POINT(centerPol(x,y),centerPol(x,y),"brown")

# Line and LIne (unlike line, which tracks segments as segm but thin) dot the
# segments (with short/long dashes).

# If the polygon were made of a sheet, the centroid would coincide with the barycenter.
# But centroid coordinates are not the average of the coordinates of the vertices. See
# above in the center.
BF = 3; HF = 3; x = c(1,4,6,1); y = c(0,1,6,4); PLANE(0,6,0,6)
POINT(x,y,"seagreen"); u=mean(x); v=mean(y); POINT(u,v,"brown"); u;v
#  3  2.75

# In the case of isolated masses m1,m2, … located at the points x1,y1, x2,y2, …, if I
# put m = c(m1,m2,…), x = c(x1,x2,…), y = c(y1,y2,…), the barycenter coordinates are:
# sum(m*x)/sum(m); sum(m*y)/sum(m)    But (equivalently) I can compute the weighted mean:
# Wmean(x, m); Wmean(y, m)
# See above on the right, where the blue objects have double mass:
BF = 3; HF = 3; x = c(1,4,6,1); y = c(0,1,6,4); PLANE(0,6,0,6)
POINT(x[1:2],y[1:2],"blue"); Point(x[3:4],y[3:4],"seagreen");
m=c(2,2,1,1); u=Wmean(x,m); v=Wmean(y,m)
POINT(u,v,"brown"); u; v
#  2.833333  2In the figure on the right, y of centroid is 10; we find x:
15·30·20 = 7*pi*5^2 + x*(30*20-pi*5^2)
x = (15*600-7*pi*5^2)/(600-pi*5^2) = 16.20492 # The centroid of Italy.
# I can upload coordinates (separated by ",") from an external file.
# For example, if I copy the coordinates that are in the penisola file and then paste
# them, after I have typed:
# I = c(                   and finally  )  in order to get:
#    I = c(
#    8,46,
#    7.87,45.9,
#    ...
#    8,46
#    )
# Here's what I obtain with these other commands:
X = xTab(I); Y = yTab(I)
BF = 4; HF = 4; Plane(min(X),max(X), min(Y),max(Y)); polyline(X,Y, "blue")
C = centerPol(X,Y); C; POINT(C,C,"red")
# 12.18996  43.48932
Plane(min(X),max(X), min(Y),max(Y)); polyC(X,Y, "green"); polyl(X,Y, "red")
POINT(C,C,"red")                                # It is "Cittŕ di Castello" (see)  