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문제

3차원 공간에 다면체가 있다. 그런데 아쉽게도 우리는 그 다면체가 어떻게 생겼는지 알지 못하고 xz평면과 yz평면에 빛에 수직으로 비친 그림자의 모양만 알고 있다. 즉 정사영의 모양만을 알고 있다는 것이다. 밑의 그림처럼 생긴 다면체가 있다면 그 다면체의 정사영은 밑의 Pxz, Pyz가 주어질 것이다.

우리가 해야 할 일은 다음과 같다. xz평면과 yz평면의 정사영의 모양이 주어져 있을 때, 다면체의 부피를 구하여라. 단, 두 개의 정사영 가지고 정확한 모양을 알 수 없기 때문에 나올 수 있는 부피는 여러 가지가 나올 수 있다. 그럴 경우에는 가능한 부피 중 가장 큰 부피를 출력하면 된다.

입력

첫째 줄에 xz평면에 비친 정사영의 꼭짓점의 개수 Nxz (1 ≤ Nxz ≤ 1,000)가 주어진다. 그리고 두 번째 줄부터 Nxz+1번째 줄까지 Pxz의 꼭짓점의 좌표 x,z가 공백을 사이에 두고 주어진다. 그리고 Nxz+2번째 줄에는 yz평면에 비친 정사영의 꼭짓점의 개수 Nyz (1 ≤ Nyz ≤ 1,000)가 주어지고 마찬가지로 Nxz+3번째 줄부터 Nxz+Nyz+2번째 줄까지 Pyz의 좌표 y, z가 공백을 사이에 두고 주어진다.

출력

첫 줄에 다면체 부피의 최댓값을 출력한다. 절대/상대 오차는 10-2까지 허용한다.

예제 입력 1

5
1 0
1 -1
-1 0
-1 1
0 1
5
-1 1
0 1
1 1
1 -1
-1 -1

예제 출력 1

5.00
[{"problem_id":"1853","problem_lang":"0","title":"\uc815\uc0ac\uc601","description":"<p>3\ucc28\uc6d0 \uacf5\uac04\uc5d0 \ub2e4\uba74\uccb4\uac00 \uc788\ub2e4. \uadf8\ub7f0\ub370 \uc544\uc27d\uac8c\ub3c4 \uc6b0\ub9ac\ub294 \uadf8 \ub2e4\uba74\uccb4\uac00 \uc5b4\ub5bb\uac8c \uc0dd\uacbc\ub294\uc9c0 \uc54c\uc9c0 \ubabb\ud558\uace0 xz\ud3c9\uba74\uacfc yz\ud3c9\uba74\uc5d0 \ube5b\uc5d0 \uc218\uc9c1\uc73c\ub85c \ube44\uce5c \uadf8\ub9bc\uc790\uc758 \ubaa8\uc591\ub9cc \uc54c\uace0 \uc788\ub2e4. \uc989 \uc815\uc0ac\uc601\uc758 \ubaa8\uc591\ub9cc\uc744 \uc54c\uace0 \uc788\ub2e4\ub294 \uac83\uc774\ub2e4. \ubc11\uc758 \uadf8\ub9bc\ucc98\ub7fc \uc0dd\uae34 \ub2e4\uba74\uccb4\uac00 \uc788\ub2e4\uba74 \uadf8 \ub2e4\uba74\uccb4\uc758 \uc815\uc0ac\uc601\uc740 \ubc11\uc758 P<sub>xz<\/sub>, Pyz\uac00 \uc8fc\uc5b4\uc9c8 \uac83\uc774\ub2e4.<\/p>\r\n\r\n<p style=\"text-align: center;\"><img alt=\"\" src=\"\/JudgeOnline\/upload\/201006\/Screen shot 2010-06-11 at 8_49_10 PM.png\" style=\"height:181px; width:351px\" \/><\/p>\r\n\r\n<p>\uc6b0\ub9ac\uac00 \ud574\uc57c \ud560 \uc77c\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. xz\ud3c9\uba74\uacfc yz\ud3c9\uba74\uc758 \uc815\uc0ac\uc601\uc758 \ubaa8\uc591\uc774 \uc8fc\uc5b4\uc838 \uc788\uc744 \ub54c, \ub2e4\uba74\uccb4\uc758 \ubd80\ud53c\ub97c \uad6c\ud558\uc5ec\ub77c. \ub2e8, \ub450 \uac1c\uc758 \uc815\uc0ac\uc601 \uac00\uc9c0\uace0 \uc815\ud655\ud55c \ubaa8\uc591\uc744 \uc54c \uc218 \uc5c6\uae30 \ub54c\ubb38\uc5d0 \ub098\uc62c \uc218 \uc788\ub294 \ubd80\ud53c\ub294 \uc5ec\ub7ec \uac00\uc9c0\uac00 \ub098\uc62c \uc218 \uc788\ub2e4. \uadf8\ub7f4 \uacbd\uc6b0\uc5d0\ub294 \uac00\ub2a5\ud55c \ubd80\ud53c \uc911 \uac00\uc7a5 \ud070 \ubd80\ud53c\ub97c \ucd9c\ub825\ud558\uba74 \ub41c\ub2e4.<\/p>\r\n","input":"<p>\uccab\uc9f8 \uc904\uc5d0 xz\ud3c9\uba74\uc5d0 \ube44\uce5c \uc815\uc0ac\uc601\uc758 \uaf2d\uc9d3\uc810\uc758 \uac1c\uc218 N<sub>xz<\/sub>&nbsp;(1 &le; N<sub>xz<\/sub> &le; 1,000)\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. \uadf8\ub9ac\uace0 \ub450 \ubc88\uc9f8 \uc904\ubd80\ud130 N<sub>xz<\/sub>+1\ubc88\uc9f8 \uc904\uae4c\uc9c0 P<sub>xz<\/sub>\uc758 \uaf2d\uc9d3\uc810\uc758 \uc88c\ud45c x,z\uac00 \uacf5\ubc31\uc744 \uc0ac\uc774\uc5d0 \ub450\uace0 \uc8fc\uc5b4\uc9c4\ub2e4. \uadf8\ub9ac\uace0 N<sub>xz<\/sub>+2\ubc88\uc9f8 \uc904\uc5d0\ub294 yz\ud3c9\uba74\uc5d0 \ube44\uce5c \uc815\uc0ac\uc601\uc758 \uaf2d\uc9d3\uc810\uc758 \uac1c\uc218 N<sub>yz<\/sub> (1 &le; N<sub>yz<\/sub> &le; 1,000)\uac00 \uc8fc\uc5b4\uc9c0\uace0 \ub9c8\ucc2c\uac00\uc9c0\ub85c N<sub>xz<\/sub>+3\ubc88\uc9f8 \uc904\ubd80\ud130 N<sub>xz<\/sub>+N<sub>yz<\/sub>+2\ubc88\uc9f8 \uc904\uae4c\uc9c0 P<sub>yz<\/sub>\uc758 \uc88c\ud45c y, z\uac00 \uacf5\ubc31\uc744 \uc0ac\uc774\uc5d0 \ub450\uace0 \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uccab \uc904\uc5d0 \ub2e4\uba74\uccb4 \ubd80\ud53c\uc758 \ucd5c\ub313\uac12\uc744 \ucd9c\ub825\ud55c\ub2e4. \uc808\ub300\/\uc0c1\ub300 \uc624\ucc28\ub294 10<sup>-2<\/sup>\uae4c\uc9c0 \ud5c8\uc6a9\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"1853","problem_lang":"1","title":"Geodes","description":"<p>In the mountains of Geodesia, geodes are found. These &#39;hollow&#39; stones contain cavities with crystal formations around them. The beautifully colored crystals can be sold for a high price to people both inside and outside of Geodesia. The problem with geodes is that one cannot see from the outside of a given rock whether it is a geode or not. Out of every thousand rocks found in Geodesia only some are geodes.&nbsp;<\/p>\r\n\r\n<p>In order to find out whether a given rock is in fact a geode, one can try to consider its density&nbsp;<\/p>\r\n\r\n<p>\\[\\rho =\\frac{m}{V}\\]<\/p>\r\n\r\n<p>where m represents its mass and V its volume. Since a geode contains empty space, a rock contaning one is expected to have a lower density than other rocks. If the density of a rock is too high, it would be a waste of time and effort to further investigate it.&nbsp;<\/p>\r\n\r\n<p>But how could one determine the density of each of the enormous number of rocks found in the mountains of Geodesia? Minig crafts collect rocks, which they put on mobile conveyer belts. Weighing the rocks automatically at a fast rate is no problem, but how to quickly determine their volume? Measuring e.g. the volume displacement of a liquid(like H2O), where each rock has to be put in and taken out individually is a time consuming process.&nbsp;<\/p>\r\n\r\n<p>So why not try to estimate the volume just by looking at the rock? Along the conveyer belt, two cameras are placed, perpendicular to each other. Of every rock that passes, they each take a picture. These pictures are sent to a computer for processing. The computer calculates from these pictures an upper bound for the volume of the rock, Vmax. A lower bound for the density of the rock is then given by&nbsp;<\/p>\r\n\r\n<p>\\[\\rho_{min} =\\frac{m}{V_{max}} \\le \\frac{m}{V}=\\rho.\\]<\/p>\r\n\r\n<p>This lower bound can then be used to reject rocks that are certainly too heavy to be geodes.&nbsp;<\/p>\r\n\r\n<p>The computer uses a relatively simple trick for estimating the maximal volume. It considers the picures taken as silhouettes, as parallel projections of a rock onto a planes. Since the two cameras are placed perpendicular to each other, the projection planes are also perpendicular to each other.&nbsp;<\/p>\r\n\r\n<p>In other words, choose a coordinate system in three dimensional space as follows: z runs along the vertical direction and x and y run along the perpendicular horizontal directions in which the two cameras are placed. Let S &isin; R<sup>3<\/sup> represent a rock. Then the cameras yield the two parallel projections&nbsp;<\/p>\r\n\r\n<p>\\[P_{xz} =\\left\\{ (x,z) : (x,y,z) \\in S ~for ~some~y \\in \\mathbb{R} &nbsp;\\right\\} &nbsp;~and&nbsp;<br \/>\r\nP_{yz} =\\left\\{ (y,z) : (x,y,z) \\in S ~for ~some~x \\in \\mathbb{R} &nbsp;\\right\\} \\]<\/p>\r\n\r\n<p>on the xz and yz plane respectively. See Figure 1 for an example.&nbsp;<\/p>\r\n\r\n<p>Now a curious property of the rocks found in Geodesia makes the measurement of the volume easier: they are all convex1. Furthermore, the projections can be considered as polygons. So first each of the two pictures taken of a rock is converted into a convex polygon by an image recognition program. You are to write a program that will do the second step: compute from these two polygons Vmax: the maximum volume of any rock having exactly these two polygon-shaped projections.<\/p>\r\n","input":"<p>The first line of input contains a single number: the number of test cases to follow. Each test case has the following format:&nbsp;<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/geodes(1).png\" style=\"height:238px; width:346px\" \/><\/p>\r\n\r\n<ul>\r\n\t<li>One line with the integer n<sub>xz<\/sub> the number of vertices describing the projection on the xz plane.<\/li>\r\n\t<li>n<sub>xz<\/sub> lines, each with two integers x and z, separated by a single space: the coordinates of one vertex of the xz projection.<\/li>\r\n\t<li>One line with the integer n<sub>yz<\/sub>, the number of vertices describing the projection on the yz plane.<\/li>\r\n\t<li>n<sub>yz<\/sub> lines, each with two integers y and z, seperated by a single space: the coordinates of one vertex of the yz projection.<\/li>\r\n<\/ul>\r\n\r\n<p>Both projections are convex polygons, and the minimum and maximum z coordinates are the same values. The vertices of the polygons are given in clockwise order. Consecutive vertices of the polygons are different, but there may be three or more consecutive vertices on one line. The numbers in the input satisfy 3 &le; n<sub>xz<\/sub>, n<sub>yz<\/sub> &le; 1000, and |x|, |y|, |z| &le; 500.&nbsp;<\/p>\r\n","output":"<p>For every test case in the input, the output should contain a single number on a single line: V<sub>max<\/sub>, the maximum volume of any object having the projections given in the input, rounded in the usual way to two decimals behind the decimal point, A round-off error of 0.01 is permitted in your answer.&nbsp;<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]