zoukankan      html  css  js  c++  java
  • [LOJ 2070] 「SDOI2016」平凡的骰子

    [LOJ 2070] 「SDOI2016」平凡的骰子

    【题目链接】

    链接

    【题解】

    原题求的是球面面积

    可以理解为首先求多面体重心,然后算球面多边形的面积

    求重心需要将多面体进行四面体剖分,从而计算出每一个四面体的重心和体积,加权平均即为整个多面体的重心

    四面体体积可以用一个点引出的三条向量的积乘 (frac 1 6)

    四面体重心坐标是四个顶点坐标平均数

    根据题目提示,球面多边形面积为三个二面角之和减去 (pi),那么我们需要求二面角

    先求出法向量,然后点积求向量二面角

    【代码】

    // Copyright lzt
    #include<stdio.h>
    #include<cstring>
    #include<cstdlib>
    #include<algorithm>
    #include<vector>
    #include<map>
    #include<set>
    #include<cmath>
    #include<iostream>
    #include<queue>
    #include<string>
    #include<ctime>
    using namespace std;
    typedef long long ll;
    typedef pair<int, int> pii;
    typedef long double ld;
    typedef unsigned long long ull;
    typedef pair<long long, long long> pll;
    #define fi first
    #define se second
    #define pb push_back
    #define mp make_pair
    #define rep(i, j, k)  for (register int i = (int)(j); i <= (int)(k); i++)
    #define rrep(i, j, k) for (register int i = (int)(j); i >= (int)(k); i--)
    #define Debug(...) fprintf(stderr, __VA_ARGS__)
    
    inline ll read() {
      ll x = 0, f = 1;
      char ch = getchar();
      while (ch < '0' || ch > '9') {
      if (ch == '-') f = -1;
      ch = getchar();
      }
      while (ch <= '9' && ch >= '0') {
      x = 10 * x + ch - '0';
      ch = getchar();
      }
      return x * f;
    }
    #define enter putchar('
    ')
    #define space putchar(' ')
    #define MAXN 1000005
    #define mo 999999137
    typedef long long int64;
    typedef double db;
    template <class T>
    void read(T &res) {
        res = 0;
        T f = 1;
        char c = getchar();
        while (c < '0' || c > '9') {
            if (c == '-')
                f = -1;
            c = getchar();
        }
        while (c >= '0' && c <= '9') {
            res = res * 10 + c - '0';
            c = getchar();
        }
        res *= f;
    }
    template <class T>
    void out(T x) {
        if (x < 0) {
            x = -x;
            putchar('-');
        }
        if (x >= 10)
            out(x / 10);
        putchar('0' + x % 10);
    }
    const db PI = acos(-1.0);
    struct Point {
        db x, y, z;
        Point() {}
        Point(db _x, db _y, db _z) {
            x = _x;
            y = _y;
            z = _z;
        }
        friend Point operator+(const Point &a, const Point &b) { return Point(a.x + b.x, a.y + b.y, a.z + b.z); }
        friend Point operator-(const Point &a, const Point &b) { return Point(a.x - b.x, a.y - b.y, a.z - b.z); }
        friend Point operator*(const Point &a, const db &d) { return Point(a.x * d, a.y * d, a.z * d); }
        friend Point operator/(const Point &a, const db &d) { return Point(a.x / d, a.y / d, a.z / d); }
        friend Point operator*(const Point &a, const Point &b) {
            return Point(a.y * b.z - a.z * b.y, -a.x * b.z + a.z * b.x, a.x * b.y - a.y * b.x);
        }
        friend db dot(const Point &a, const Point &b) { return a.x * b.x + a.y * b.y + a.z * b.z; }
        Point operator-=(const Point &b) { return *this = *this - b; }
        Point operator+=(const Point &b) { return *this = *this + b; }
        Point operator/=(const db &d) { return *this = *this / d; }
        Point operator*=(const db &d) { return *this = *this * d; }
        db norm() { return sqrt(x * x + y * y + z * z); }
    } P[55], G;
    vector<Point> S[85];
    int N, F;
    Point GetG(Point p, Point a, Point b, Point c) { return (p + a + b + c) / 4.0; }
    db GetV(Point p, Point a, Point b, Point c) {
        a -= p;
        b -= p;
        c -= p;
        db res = abs(dot(a, b * c));
        res /= 6.0;
        return res;
    }
    Point CalcG() {
        Point t = Point(0.0, 0.0, 0.0);
        db sv = 0.0;
        for (int i = 1; i <= F; ++i) {
            int s = S[i].size();
            for (int j = 0; j <= s - 1; ++j) {
                Point tmp = GetG(P[1], S[i][j], S[i][(j + 1) % s], S[i][(j + 2) % s]);
                db v = GetV(P[1], S[i][j], S[i][(j + 1) % s], S[i][(j + 2) % s]);
                sv += v;
                t += tmp * v;
            }
        }
        t /= sv;
        return t;
    }
    db CalcTangle(Point p, Point x, Point y, Point z) {
        x -= p;
        y -= p;
        z -= p;
        return acos(dot(x * y, x * z) / (x * y).norm() / (x * z).norm());
    }
    void Init() {
        read(N);
        read(F);
        db x, y, z;
        for (int i = 1; i <= N; ++i) {
            scanf("%lf%lf%lf", &x, &y, &z);
            P[i] = Point(x, y, z);
        }
        int k, a;
        for (int i = 1; i <= F; ++i) {
            read(k);
            for (int j = 1; j <= k; ++j) {
                read(a);
                S[i].pb(P[a]);
            }
        }
    }
    void Solve() {
        Point G = CalcG();
        for (int i = 1; i <= F; ++i) {
            int s = S[i].size();
            db x = -(s - 2) * PI;
            for (int j = 0; j < s; ++j) {
                x += CalcTangle(G, S[i][j], S[i][(j + 1) % s], S[i][(j - 1 + s) % s]);
            }
            printf("%.7lf
    ", x / (4 * PI));
        }
    }
    int main() {
        Init();
        Solve();
        return 0;
    }
    
  • 相关阅读:
    [转]进程间通信----pipe和fifo
    [转]udev
    [转]netlink
    [转]进程间通信-----管道
    [转]socket
    [转]armv8 memory system
    [转]内核态和用户态
    [转]dpdk内存管理
    meeting and robert rules
    notion
  • 原文地址:https://www.cnblogs.com/wawawa8/p/10677682.html
Copyright © 2011-2022 走看看