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문제

3차원 공간 위에 N개의 구면거울이 있다.

이때, (0,0,0)에서 (u,v,w)방향으로 레이저를 발사하려고 한다. 레이저는 항상 직선으로 움직인다.

I에서 발사한 레이저가 구면 거울의 Q에서 반사되었을 때, 점 N을 구면 거울의 바깥에 있으면서, 구면 거울의 중심과 Q를 잇는 일직선위의 한 점이라고 하자. 이때, 레이저는 다음과 같은 조건을 만족하는 R방향으로 반사딘다.

(1) R은 I, Q, N으로 만든 평면 위의 한 점이다. ∠IQN = ∠NQR

레이저가 마지막으로 반사되는 점의 위치를 찾는 프로그램을 작성하시오.

입력

첫째 줄에 테스트 케이스의 개수 T가 주어진다. 각 테스트 케이스의 첫째 줄에는 구면 거울의 개수 N이 주어진다. 둘째 줄에는 레이저를 발사한 방향 u, v, w가 공백으로 구분되어 주어진다.

다음 N개의 줄에는 구면 거울을 설명하는 4개의 정수 xi,yi,zi,ri가 주어진다. 구면 거울의 중심은 (xi,yi,zi)이고, 반지름은 ri이다.

1 ≤ N ≤ 100

-100 ≤ u,v,w ≤ 100

-100 ≤ xi,yi,zi ≤ 100

5 ≤ ri ≤ 30

u2 + v2 + w2 > 0

두 구면 거울 사이의 거리는 적어도 0.1이다. (0,0,0)은 모든 구면 거울의 바깥점이고, 모든 구면 거울과 적어도 0.1만큼 떨어져 있다.

레이저는 구면 거울에 적어도 1번, 많아야 5번 반사된다. 또, 반사각(그림에서 θ)은 항상 85도보다 작다.

출력

각각의 테스트 케이스에 대해서, 마지막으로 반사되는 점의 좌표를 공백으로 구분하여 출력한다. 모든 좌표는 소수점 넷째 자리에서 반올림해서 셋째 자리까지 출력한다.

예제 입력 1

2
3
-20 -20 -24
100 100 100 30
10 8 3 5
-70 -70 -84 5
4
0 47 84
-23 41 42 8
45 -10 14 19
-5 28 47 12
-27 68 34 14

예제 출력 1

79.094 79.094 94.913
-21.865 54.977 34.176
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(0,0,0)\uc740 \ubaa8\ub4e0 \uad6c\uba74 \uac70\uc6b8\uc758 \ubc14\uae65\uc810\uc774\uace0, \ubaa8\ub4e0 \uad6c\uba74 \uac70\uc6b8\uacfc \uc801\uc5b4\ub3c4 0.1\ub9cc\ud07c \ub5a8\uc5b4\uc838 \uc788\ub2e4.<\/p>\r\n\r\n<p>\ub808\uc774\uc800\ub294 \uad6c\uba74 \uac70\uc6b8\uc5d0 \uc801\uc5b4\ub3c4 1\ubc88, \ub9ce\uc544\uc57c 5\ubc88 \ubc18\uc0ac\ub41c\ub2e4. \ub610, \ubc18\uc0ac\uac01(\uadf8\ub9bc\uc5d0\uc11c &theta;)\uc740 \ud56d\uc0c1 85\ub3c4\ubcf4\ub2e4 \uc791\ub2e4.<\/p>\r\n","output":"<p>\uac01\uac01\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574\uc11c, \ub9c8\uc9c0\ub9c9\uc73c\ub85c \ubc18\uc0ac\ub418\ub294 \uc810\uc758 \uc88c\ud45c\ub97c \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ud558\uc5ec \ucd9c\ub825\ud55c\ub2e4. \ubaa8\ub4e0 \uc88c\ud45c\ub294 \uc18c\uc218\uc810 \ub137\uc9f8 \uc790\ub9ac\uc5d0\uc11c \ubc18\uc62c\ub9bc\ud574\uc11c \uc14b\uc9f8 \uc790\ub9ac\uae4c\uc9c0 \ucd9c\ub825\ud55c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3889","problem_lang":"1","title":"Spherical Mirrors","description":"<blockquote>A long time ago in a galaxy, far, far away, there were N spheres with various radii. Spheres were mirrors, that is, they had reflective surfaces . . . .<\/blockquote>\r\n\r\n<p>You are standing at the origin of the galaxy (0, 0, 0), and emit a laser ray to the direction (u, v, w). The ray travels in a straight line.<\/p>\r\n\r\n<p>When the laser ray from I hits the surface of a sphere at Q, let N be a point outside of the sphere on the line connecting the sphere center and Q. The re\ufb02ected ray goes to the direction towards R that satis\ufb01es the following conditions: (1) R is on the plane formed by the three points I, Q and N, (2) &ang;IQN = &ang;NQR, as shown in Figure 6.<\/p>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/laser.png\" style=\"height:346px; opacity:0.9; width:374px\" \/><\/p>\r\n\r\n<p>Figure 6: Laser ray re\ufb02ection<\/p>\r\n\r\n<p>After it is reflected several times, finally it goes beyond our observation. Your mission is to write a program that identifies the last reflection point.<\/p>\r\n\r\n<p>&nbsp;<\/p>\r\n","input":"<p>The input consists of multiple datasets, each in the following format.<\/p>\r\n\r\n<pre>\r\n N\r\n u  v  w\r\nx<sub>1<\/sub> y<sub>1<\/sub> z<sub>1<\/sub> r<sub>1<\/sub>\r\n...\r\nx<sub>N<\/sub> y<sub>N<\/sub> z<sub>N<\/sub> r<sub>N<\/sub><\/pre>\r\n\r\n<p>The first line of a dataset contains a positive integer N which is the number of spheres. The next line contains three integers u, v and w separated by single spaces, where (u, v, w) is the direction of the laser ray initially emitted from the origin.<\/p>\r\n\r\n<p>Each of the following N lines contains four integers separated by single spaces. The i-th line corresponds to the i-th sphere, and the numbers represent the center position (xi, yi, zi) and the radius ri.<\/p>\r\n\r\n<p>N, u, v, w, xi, yi, zi and ri satisfy the following conditions.<\/p>\r\n\r\n<pre>\r\n1 &le; N &le; 100\r\n-100 &le; u,v,w &le; 100\r\n-100 &le; x<sub>i<\/sub>,y<sub>i<\/sub>,z<sub>i<\/sub>&nbsp;&le; 100\r\n5 &le; r<sub>i<\/sub>&nbsp;&le; 30\r\nu<sup>2<\/sup>&nbsp;+ v<sup>2<\/sup>&nbsp;+ w<sup>2<\/sup>&nbsp;&gt; 0<\/pre>\r\n\r\n<p>You can assume that the distance between the surfaces of any two spheres is no less than 0.1. You can also assume that the origin (0; 0; 0) is located outside of any sphere, and is at least 0.1 distant from the surface of any sphere.<\/p>\r\n\r\n<p>The ray is known to be reflected by the sphere surfaces at least once, and at most five times. You can assume that the angle between the ray and the line connecting the sphere center and the reflection point, which is known as the angle of reflection (i.e. &theta; in Figure 6), is less than 85 degrees for each point of reflection.<\/p>\r\n\r\n<p>The last dataset is followed by a line containing a single zero.<\/p>\r\n","output":"<p>For each dataset in the input, you should print the x-, y- and z-coordinates of the last reflection point separated by single spaces in a line. No output line should contain extra characters.<\/p>\r\n\r\n<p>No coordinate values in the output should have an error greater than 0.01.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]

출처

ACM-ICPC > Regionals > Asia > Japan > Asia Regional Contest 2008 in Aizu E번

  • 문제를 번역한 사람: baekjoon
  • 문제의 오타를 찾은 사람: kipa00