[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('\n')
#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\n", x / (4 * PI));
    }
}
int main() {
    Init();
    Solve();
    return 0;
}