Polygon Revolution

Given a convex polygon with $N$ vertices $p_1, \dots, p_N$ and a line $L$ on a 2-dimensional plane. You can generate a 3-dimensional solid of revolution by the revolution of the convex polygon around the axis $L$. Calculate the volume of this solid.

When the axis $L$ is an external line of the convex polygon, the task is much easier, because the following theorem helps. However, note that the axis may intersect the convex polygon.

The second theorem of Pappus: The volume $V$ of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area $A$ of the lamina and the distance traveled by the lamina's geometric centroid $x$.

$V = Ad_2 = 2\pi Ax$

Figure 5: Sample revolutions

The first line of the input is a positive integer $T$, the number of test cases. The first line of each test case is a positive integer $N$ ($2 < N \le 100$), the number of vertices of the convex polygon. Then $N$ lines follow. The $i$-th ($1 \le i \le N$) line contains two real numbers $X_i$ ($0 \le X_i \le 10000$) and $Y_i$ ($0 \le Y_i \le 10000$), the coordinates of the vertices of the convex polygon in clockwise order. Finally, three real numbers $A, B, C$ ($-1000 \le A, B, C \le 1000$) represent the equation of the axis $Ax + By + C = 0$.

The output consists of $T$ lines, one line for each test case. Each line contains one real number, the volume $V$ of the solid of revolution, with an error not greater than $0.1$.