```
# Zu Chongzhi's calculation of pi (China, about 480 A.D.),
# approximating the circle from inside by a regular 12288-polygon(!):
phi <- seq(0, 2*pi, len=3*2^12+1)
x <- cos(phi)
y <- sin(phi)
pi_approx <- polyarea(x, y)
print(pi_approx, digits=8) #=> 3.1415925 or 355/113
poly_length(x, y) #=> 6.2831852 where 2*pi is 6.2831853
x1 <- x + 0.5; y1 <- y + 0.5
x2 <- rev(x1); y2 <- rev(y1)
poly_center(x1, y1) #=> 0.5 0.5
poly_center(x2, y2) #=> 0.5 0.5
# A simple example
L1 <- matrix(c(0, 0.5, 1, 1, 2,
0, 1, 1, 0.5, 0), nrow = 2, byrow = TRUE)
L2 <- matrix(c(0.5, 0.75, 1.25, 1.25,
0, 0.75, 0.75, 0 ), nrow = 2, byrow = TRUE)
P <- poly_crossings(L1, L2)
P
## x y
## [1,] 1.00 0.750
## [2,] 1.25 0.375
## Not run:
# # Crossings of Logarithmic and Archimedian spirals
# # Logarithmic spiral
# a <- 1; b <- 0.1
# t <- seq(0, 5*pi, length.out = 200)
# xl <- a*exp(b*t)*cos(t) - 1
# yl <- a*exp(b*t)*sin(t)
# plot(xl, yl, type = "l", lwd = 2, col = "blue",
# xlim = c(-6, 3), ylim = c(-3, 4), xlab = "", ylab = "",
# main = "Intersecting Logarithmic and Archimedian spirals")
# grid()
#
# # Archimedian spiral
# a <- 0; b <- 0.25
# r <- a + b*t
# xa <- r * cos(t)
# ya <- r*sin(t)
# lines(xa, ya, type = "l", lwd = 2, col = "red")
# legend(-6.2, -1.0, c("Logarithmic", "Archimedian"),
# lwd = 2, col = c("blue", "red"), bg = "whitesmoke")
#
# L1 <- rbind(xl, yl)
# L2 <- rbind(xa, ya)
# P <- poly_crossings(L1, L2)
# points(P)
# ## End(Not run)
```

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