原題鏈接 Crane Time Limit: 2000MS Memory Limit: 65536K Total Submissions: 6253 Accepted: 1673 Special Judge Description

ACM has bought a new crane (crane – je?áb) . The crane consists of n segments of various lengths, connected by flexible joints. The end of the i-th segment is joined to the beginning of the i + 1-th one, for 1 ≤ i < n. The beginning of the first segment is fixed at point with coordinates (0, 0) and its end at point with coordinates (0, w), where w is the length of the first segment. All of the segments lie always in one plane, and the joints allow arbitrary rotation in that plane. After series of unpleasant accidents, it was decided that software that controls the crane must contain a piece of code that constantly checks the position of the end of crane, and stops the crane if a collision should happen.

Your task is to write a part of this software that determines the position of the end of the n-th segment after each command. The state of the crane is determined by the angles between consecutive segments. Initially, all of the angles are straight, i.e., 180o. The Operator issues commands that change the angle in exactly one joint. Input

The input consists of several instances, separated by single empty lines.

The first line of each instance consists of two integers 1 ≤ n ≤10 000 and c 0 separated by a single space – the number of segments of the crane and the number of commands. The second line consists of n integers l1,…, ln (1 li 100) separated by single spaces. The length of the i-th segment of the crane is li. The following c lines specify the commands of the operator. Each line describing the command consists of two integers s and a (1 ≤ s < n, 0 ≤ a ≤ 359) separated by a single space – the order to change the angle between the s-th and the s + 1-th segment to a degrees (the angle is measured counterclockwise from the s-th to the s + 1-th segment). Output

The output for each instance consists of c lines. The i-th of the lines consists of two rational numbers x and y separated by a single space – the coordinates of the end of the n-th segment after the i-th command, rounded to two digits after the decimal point.

The outputs for each two consecutive instances must be separated by a single empty line. Sample Input

2 1 10 5 1 90

3 2 5 5 5 1 270 2 90 Sample Output

5.00 10.00

-10.00 5.00 -5.00 10.00 Source

CTU Open 2005 題意：n個棍子原來排成一列豎直向上，然后修改某處棍子與棍子之間的角度，每次修改輸出最后一根棍子的尾端的坐標 思路：每次修改都會影響到該棍子和該棍子之后所有棍子的位置，求最后一個棍子的位置，這就有些類似于線段樹更新區間和這種問題，所以問題就是找到區間和的計算方法（即計算棍子坐標的快速的方法）。而我們注意到這里線段樹的一個區間可以表示成一個向量，每一個單獨的線段其實也可以表示成一個向量，而向量V0(x1,y1)轉過角μ成V1(x2,y2)后坐標的計算有 然后需要一個數組記錄線段與線段之間的夾角，而對于不同的向量還要記錄其左子向量與右子向量的夾角并向上更新