Rotate Sum 3

Time limit1sMemory limit1024 MB

Summary

Given a convex polygon, find the volume swept when it is repeatedly rotated in 3D about its symmetry axes, which move with it.

Level: Hard9 of 10
Topics: Geometry, Math, Implementation
Solved: No attempts yet

Problem

Grammy loves geometry. Today, she takes out her precious convex polygon and plays with it.

Grammy thinks that symmetry is a fun characteristic for a convex polygon, so she draws out all the axes of symmetry on the polygon.

NIO is a naughty boy. He repeats the following operation several times. In each operation, he chooses an axis of symmetry as the rotation axis and rotates the polygon in three-dimensional space around that axis by an arbitrary angle. The axes of symmetry rotate together with the polygon.

Grammy wants to know the total volume swept by the convex polygon during NIO's operations. Please help her.

Input

The first line contains an integer $n$ ( $3 \leq n \leq 10^5$ ), the number of vertices of the convex polygon.

Each of the next $n$ lines contains two integers $x_i$ and $y_i$ ( $-10^9 \leq x_i, y_i \leq 10^9$ ), the coordinates of the $i$ -th vertex. The vertices are given in counterclockwise order. No three vertices lie on the same line.

Output

Output a real number, the volume of the swept region. The answer is correct if the absolute or relative error is less than $10^{-6}$ .

Examples2

Example 1
Input
```
3
0 -1
1 0
0 1
```
Expected output
```
1.047197551197
```
Example 2
Input
```
3
1 1
4 5
1 4
```
Expected output
```
0
```