source("http://macosa.dima.unige.it/r.R") # If I have not already loaded the library
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*S** *07** Centroid (and Barycenter)**
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# 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)[1],centerPol(x,y)[2],"brown")
# **Line** and **LIne** (unlike **line**, which tracks segments as** segm **but thin) dot the
# segments (with short/long dasches).
# 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 2
# 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[1],C[2],"red")
**#** 12.18996 43.48932
Plane(min(X),max(X), min(Y),max(Y)); polyC(X,Y, "green"); polyl(X,Y, "red")
POINT(C[1],C[2],"red") # It is "Cittą di Castello" (see)
# See here for uploading data like those of "penisola" saved as a** table** (**read.table**),
# also (using **dec**=",") if they are written in Italian notation.
* *
# See here to determine the **area** of the **surface of Italy**.
**Other examples of use**