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G - Intersection of Polygons

Score : $600$ points

Problem Statement

The vertices of a convex $N$-gon $P$ in an $xy$-plane are given as $(x_1, y_1), (x_2, y_2), \ldots, (x_N, y_N)$ in the counterclockwise order. (Here, the positive direction of the $x$-axis is right, and the positive direction of the $y$-axis is up.)

Based on this polygon $P$, we consider $M$ convex $N$-gons $P_1, P_2, \ldots, P_M$.
For $i = 1, 2, \ldots, M$, the polygon $P_i$ is obtained by shifting $P$ in the positive direction of the $x$-axis by $u_i$ and in the positive direction of the $y$-axis by $v_i$. In other words, $P_i$ is a convex $N$-gon whose vertices are $(x_1+u_i, y_1+v_i), (x_2+u_i, y_2+v_i), \ldots, (x_N+u_i, y_N+v_i)$.

For each of $Q$ points $(a_1, b_1), (a_2, b_2), \ldots, (a_Q, b_Q)$, determine if "the point is contained in all of the $M$ polygons $P_1, P_2, \ldots, P_M$."

Here, we regard a point is also contained in a polygon if the point is on the polygon's boundary.

Constraints

• $3 \leq N \leq 50$
• $1 \leq M \leq 2 \times 10^5$
• $1 \leq Q \leq 2 \times 10^5$
• $-10^8 \leq x_i, y_i \leq 10^8$
• $-10^8 \leq u_i, v_i \leq 10^8$
• $-10^8 \leq a_i, b_i \leq 10^8$
• All values in input are integers.
• $(x_1, y_1), (x_2, y_2), \ldots, (x_N, y_N)$ forms a convex $N$-gon in the counterclockwise order.
• Each interior angle of the polygon $P$ is less than $180$ degrees.

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Input

Input is given from Standard Input in the following format:

$N$

$x_1$ $y_1$

$x_2$ $y_2$

$\vdots$

$x_N$ $y_N$

$M$

$u_1$ $v_1$

$u_2$ $v_2$

$\vdots$

$u_M$ $v_M$

$Q$

$a_1$ $b_1$

$a_2$ $b_2$

$\vdots$

$a_Q$ $b_Q$

Output

Print $Q$ lines. For $i = 1, 2, \ldots, Q$, the $i$-th line should contain Yes if point $(a_i, b_i)$ is contained in all of the $M$ polygons $P_1, P_2, \ldots, P_M$; it should contain No otherwise.

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5
-2 -3
0 -2
1 0
0 2
-2 1
2
0 1
1 0
6
0 0
1 0
0 1
1 1
-1 -1
-1 -2

Yes
No
Yes
Yes
Yes
No

Polygon $P$ is a pentagon ($5$-gon) whose vertices are $(-2, -3), (0, -2), (1, 0), (0, 2), (-2, 1)$.

• Polygon $P_1$ is a pentagon ($5$-gon) obtained by shifting $P$ in the positive direction of the $x$-axis by $0$ and in the positive direction of the $y$-axis by $1$, so its vertices are $(-2, -2), (0, -1), (1, 1), (0, 3), (-2, 2)$.
• Polygon $P_2$ is a pentagon ($5$-gon) obtained by shifting $P$ in the positive direction of the $x$-axis by $1$ and in the positive direction of the $y$-axis by $0$, so its vertices are $(-1, -3), (1, -2), (2, 0), (1, 2), (-1, 1)$.

Thus, the following $6$ lines should be printed.

• The $1$-st line should be Yes because $(a_1, b_1) = (0, 0)$ is contained in both $P_1$ and $P_2$.
• The $2$-nd line should be No because $(a_2, b_2) = (1, 0)$ is contained in $P_2$ but not in $P_1$.
• The $3$-rd line should be Yes because $(a_3, b_3) = (0, 1)$ is contained in both $P_1$ and $P_2$.
• The $4$-th line should be Yes because $(a_4, b_4) = (1, 1)$ is contained in both $P_1$ and $P_2$.
• The $5$-th line should be Yes because $(a_5, b_5) = (-1, -1)$ is contained in both $P_1$ and $P_2$.
• The $6$-th line should be No because $(a_6, b_6) = (-1, -2)$ is contained in $P_2$ but not in $P_1$.

Note that a point on the boundary of a polygon is also considered to be contained in the polygon.

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10
45 100
-60 98
-95 62
-95 28
-78 -41
-54 -92
-8 -99
87 -94
98 23
87 91
5
-57 -40
-21 -67
25 39
-30 25
39 -20
16
4 5
-34 -8
-63 53
78 84
19 -16
64 9
-13 7
13 53
-20 4
2 -7
3 18
-12 10
-69 -93
2 9
27 64
-92 -100

Yes
Yes
No
No
Yes
No
Yes
No
Yes
Yes
Yes
Yes
No
Yes
No
No