21990번 - Fiber Shape 스페셜 저지다국어

Imagine a board with $n$ pins put into it, the $i$-th pin is located at $(x_i, y_i)$. For simplicity, we will restrict the problem to the case where the pins are placed in vertices of a convex polygon.

Then, take a non-stretchable string of length $l$, and put it around all the pins. Place a pencil inside the string and draw a curve around the pins, trying to pull the string in every possible direction. The picture below shows an example of a string tied around the pins and pulled by a pencil (a point $P$).

Your task is to find an area inside this curve. Formally, for a given convex polygon $S$ and a length $l$ let's define a fiber shape $F(S, l)$ as a set of points $t$ such that the perimeter of the convex hull of $S \cup \{t\}$ does not exceed $l$. Find an area of $F(S, l)$.

The first line contains two integers $n$ and $l$ ($3 \le n \le 10^4$; $1 \le l \le 8 \cdot 10^5$) --- the number of vertices of the polygon $S$ and the length of the string. Next $n$ lines contain integers $x_i$ and $y_i$ ($-10^5 \le x_i, y_i \le 10^5$) --- coordinates of polygon's vertices in counterclockwise order. All internal angles of the polygon are strictly less than $\pi$. The length $l$ exceeds the perimeter of the polygon by at least $10^{-3}$.

Output a single floating-point number --- the area of the fiber shape $F(S, l)$. Your answer will be considered correct if its absolute or relative error doesn't exceed $10^{-6}$. 

The following pictures illustrate the example tests.