polyarea: Area of a Polygon

Description

Calculates the area and length of a polygon given by the vertices in the vectors x and y.

Usage

polyarea(x, y)
  poly_length(x, y)
  poly_center(x, y)
  poly_crossings(L1, L2)

Arguments

x-coordinates of the vertices defining the polygon

y-coordinates of the vertices

L1, L2

matrices of type 2xn with x- and y-coordinates.

Value

Area or length of the polygon resp. sum of the enclosed areas; or the coordinates of the center of gravity.
poly_crossings returns a matrix with column names x and y representing the crossing points.

Details

polyarea calculates the area of a polygon defined by the vertices with coordinates x and y. Areas to the left of the vertices are positive, those to the right are counted negative.

The computation is based on the Gauss polygon area formula. The polygon automatically be closed, that is the last point need not be / should not be the same as the first.

If some points of self-intersection of the polygon line are not in the vertex set, the calculation will be inexact. The sum of all areas will be returned, parts that are circulated in the mathematically negative sense will be counted as negative in this sum.

If x, y are matrices of the same size, the areas of all polygons defined by corresponding columns are computed.

poly_center calculates the center (of mass) of the figure defined by the polygon. Self-intersections should be avoided in this case. The mathematical orientation of the polygon does not have influence on the center coordinates.

poly_length calculates the length of the polygon

poly_crossings calculates the crossing points of two polygons given as matrices with x- and y-coordinates in the first and second row. Can be used for finding the crossing points of parametrizised curves.

Examples

Run this code

# 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

# 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)