Problem:
Score : 600 points

Problem Statement

There are N holes in a two-dimensional plane. The coordinates of the i-th hole are (x_i,y_i).

Let R=10^{10^{10^{10}}}. Ringo performs the following operation:

Randomly choose a point from the interior of a circle of radius R centered at the origin, and put Snuke there. Snuke will move to the hole with the smallest Euclidean distance from the point, and fall into that hole. If there are multiple such holes, the hole with the smallest index will be chosen.

For every i (1 \leq i \leq N), find the probability that Snuke falls into the i-th hole.

Here, the operation of randomly choosing a point from the interior of a circle of radius R is defined as follows:

Pick two real numbers x and y independently according to uniform distribution on [-R,R].

If x^2+y^2\leq R^2, the point (x,y) is chosen. Otherwise, repeat picking the real numbers x,y until the condition is met.

Constraints

2 \leq N \leq 100

|x_i|,|y_i| \leq 10^6(1\leq i\leq N)

All given points are pairwise distinct.

All input values are integers.

Input

Input is given from Standard Input in the following format:

N
x_1 y_1
:
x_N y_N

Output

Print N real numbers. The i-th real number must represent the probability that Snuke falls into the i-th hole.

The output will be judged correct when, for all output values, the absolute or relative error is at most 10^{-5}.

Sample Input 1

2
0 0
1 1

Sample Output 1

0.5
0.5

If Ringo put Snuke in the region x+y\leq 1, Snuke will fall into the first hole. The probability of this happening is very close to 0.5.
Otherwise, Snuke will fall into the second hole, the probability of which happening is also very close to 0.5.

Sample Input 2

5
0 0
2 8
4 5
2 6
3 10

Sample Output 2

0.43160120892732328768
0.03480224363653196956
0.13880483535586193855
0.00000000000000000000
0.39479171208028279727

Solution in Kotlin (1.0.0) (judge verdict: Accepted)

data class Double2D(val x: Double, val y: Double) {
  operator fun plus(o: Double2D): Double2D {
    return Double2D(x + o.x, y + o.y)
  }
  operator fun minus(o: Double2D): Double2D {
    return Double2D(x - o.x, y - o.y)
  }
  operator fun times(o: Double): Double2D {
    return Double2D(x * o, y * o)
  }
  fun norm2(): Double {
    return x * x + y * y
  }
  fun norm(): Double {
    return Math.sqrt(x * x + y * y)
  }
}

fun cross(v1: Double2D, v2: Double2D): Double {
  return v1.x * v2.y - v1.y * v2.x
}

fun dot(v1: Double2D, v2: Double2D): Double {
  return v1.x * v2.x + v1.y * v2.y
}

fun ccw(p: Double2D, q: Double2D, r: Double2D): Int {
  val vq = q - p
  val vr = r - p
  if (cross(vq, vr) > 0)   return +1     // counter clockwise
  if (cross(vq, vr) < 0)   return -1     // clockwise
  if (dot(vq, vr) < 0)     return +2     // r-p-q on line
  if (vq.norm2() < vr.norm2()) return -2 // p-q-r on line
  return 0
}

fun compareDouble2D(p1: Double2D, p2: Double2D): Int {
  if (p1.x != p2.x) {
    return java.lang.Double.compare(p1.x, p2.x)
  }
  return java.lang.Double.compare(p1.y, p2.y)
}

fun convexHull(inputPs: List<Double2D>): List<Double2D> {
  val ps = inputPs.sortedWith(java.util.Comparator { p1, p2 -> compareDouble2D(p1, p2) })
  val n = ps.size
  val qs = mutableListOf<Double2D>()
  for (i in 0 until n) {
    while (qs.size >= 2 &&
        // Use "<= 0" here to exclude a point on an edge.
        ccw(qs[qs.size - 2], qs[qs.size - 1], ps[i]) == -1) {
      qs.removeAt(qs.size - 1)
    }
    qs.add(ps[i])
  }
  val lowerHullSize = qs.size
  for (i in n - 2 downTo 0) {
    while (qs.size >= lowerHullSize &&
        // Use "<= 0" here to exclude a point on an edge.
        ccw(qs[qs.size - 2], qs[qs.size - 1], ps[i]) == -1) {
      qs.removeAt(qs.size - 1)
    }
    qs.add(ps[i])
  }
  qs.removeAt(qs.size - 1)
  return qs
}

fun main(args: Array<String>) {
  val n = readLine()!!.toInt()
  // No List(n, {}) constructor for Kotlin 1.0.
  val ps = Array(n, {
    val (x, y) = readLine()!!.split(" ").map(String::toInt)
    Double2D(x.toDouble(), y.toDouble()) })
      .asList()
  val qs = convexHull(ps)
  val m = qs.size
  val args = mutableMapOf<Double2D, Double>()
  for (i in 0 until m) {
    val qm = qs[i]
    val qp = qs[(i + (m - 1)) % m]
    val qn = qs[(i + 1) % m]
    val v1 = qn - qm
    val v2 = qp - qm
    val cos_theta = dot(v1, v2) / v1.norm() / v2.norm()
    val theta = Math.acos(cos_theta)
    args[qs[i]] = Math.PI - theta
  }
  for (p in ps) {
    if (args.containsKey(p)) {
      val arg = args[p]!!
      println(String.format("%.9f", arg / (2.0 * Math.PI)))
    } else {
      println(0.0)
    }
  }
}