# Josua Kugler, Christian Merten

poly1 <- structure(
  list(sides = c(2, 2, 2, 2), angles = c(pi/3, 2*pi/3, pi/3, 2*pi/3)),
  class = c("Quadrilateral", "Polygon"))
poly2 <- structure(
  list(sides = c(3, 4, 3, 4), angles = rep(pi/2, 4)),
  class = c("Rectangle", "Quadrilateral", "Polygon"))
poly3 <- structure(
  list(sides = c(2, 2, 2), angles = rep(pi/3, 3)),
  class = c("Triangle", "Polygon"))
poly4 <- structure(
  list(sides = rep(1, 5), angles = rep(3/5*pi, 5)),
  class = c("Polygon"))

get_corners <- function(x) {
  n <- length(x$sides)
  if (n < 3) return(invisible(NULL))
  out <- matrix(ncol = 2, nrow = length(x$sides))
  direction <- c(1, 0)
  point <- c(0, 0)
  out[1, ] <- point
  for (i in 1:(n-1)) {
    point <- point + direction * x$sides[i]
    out[i+1, ] <- point
    a <- pi - x$angles[i]
    direction <- matrix(c(cos(a), sin(a), -sin(a), cos(a)), ncol=2) %*% direction
  }
  out
}

plot_polygon <- function(x) {
  corners <- get_corners(x)
  plot(range(corners[,1]), range(corners[,2]), type = 'n', axes=FALSE, ann=FALSE, asp=1)
  polygon(corners[,1], corners[,2], lwd = 3)
}

# a)
plot.Polygon <- plot_polygon

# b)
Rectangle <- function(w, h) structure(
    list(sides = c(w, h, w, h), angles = rep(pi/2, 4)),
    class = c("Rectangle", "Quadrilateral", "Polygon"))

Triangle <- function(s1, s2, a1) {
    b <- sqrt(s1^2 + s2^2 - 2*s1*s2*cos(a1))
    a2 <- acos((b^2 + s2^2 - s1^2)/(2 * s2 * b))
    structure(list(sides = c(s1, s2, b),
                   angles = c(a1, a2, 2*pi-a1-a2)),
              class = c("Triangle", "Polygon"))
}

# c)
validate_Triangle <- function(x) {
    stopifnot("no triangle" = "Triangle" %in% class(x))
    tri <- Triangle(x$sides[1], x$sides[2], x$angles[1])
    stopifnot("side and angles do not fit" = identical(c(tri$angles, tri$sides), c(x$angles, x$sides)))
}

# d)
circumference <- function(obj) {
    UseMethod("circumference")
}

circumference.Polygon <- function(p) sum(p$sides)

# e)
area <- function(obj) {
    UseMethod("area")
}

area.Triangle <- function(p) {
    h <- p$sides[1] * sin(p$angles[1])
    p$sides[2]*h*0.5
}

area.Quadrilateral <- function(p)
    1/2 * (p$sides[1]*p$sides[4]*sin(p$angles[4]) + p$sides[2]*p$sides[3]*sin(p$angles[2]))