简单几何(半平面交+二分) LA 3890 Most Distant Point from the Sea

 

题目传送门

题意：凸多边形的小岛在海里，问岛上的点到海最远的距离。

分析：训练指南P279，二分答案，然后整个多边形往内部收缩，如果半平面交非空，那么这些点构成半平面，存在满足的点。

 

/************************************************
* Author        :Running_Time
* Created Time  :2015/11/10 星期二 14:16:17
* File Name     :LA_3890.cpp
 ************************************************/

#include <bits/stdc++.h>
using namespace std;

#define lson l, mid, rt << 1
#define rson mid + 1, r, rt << 1 | 1
typedef long long ll;
const int N = 1e5 + 10;
const int INF = 0x3f3f3f3f;
const int MOD = 1e9 + 7;
const double EPS = 1e-10;
const double PI = acos (-1.0);
int dcmp(double x)  {       //三态函数，减少精度问题
    if (fabs (x) < EPS) return 0;
    else    return x < 0 ? -1 : 1;
}
struct Point    {       //点的定义
    double x, y;
    Point () {}
    Point (double x, double y) : x (x), y (y) {}
    Point operator + (const Point &r) const {       //向量加法
        return Point (x + r.x, y + r.y);
    }
    Point operator - (const Point &r) const {       //向量减法
        return Point (x - r.x, y - r.y);
    }
    Point operator * (double p) const {       //向量乘以标量
        return Point (x * p, y * p);
    }
    Point operator / (double p) const {       //向量除以标量
        return Point (x / p, y / p);
    }
    bool operator < (const Point &r) const {       //点的坐标排序
        return x < r.x || (x == r.x && y < r.y);
    }
    bool operator == (const Point &r) const {       //判断同一个点
        return dcmp (x - r.x) == 0 && dcmp (y - r.y) == 0;
    }
};
typedef Point Vector;       //向量的定义
Point read_point(void)   {      //点的读入
    double x, y;    scanf ("%lf%lf", &x, &y);
    return Point (x, y);
}
double dot(Vector A, Vector B)  {       //向量点积
    return A.x * B.x + A.y * B.y;
}
double cross(Vector A, Vector B)    {       //向量叉积
    return A.x * B.y - A.y * B.x;
}
double length(Vector A) {       //向量长度，点积
    return sqrt (dot (A, A));
}
double polar_angle(Vector A)  {     //向量极角
    return atan2 (A.y, A.x);
}
Vector nomal(Vector A)  {       //向量的单位法向量
    double len = length (A);
    return Vector (-A.y / len, A.x / len);
}
struct Line {
    Point p;
    Vector v;
    double ang;
    Line () {}
    Line (const Point &p, const Vector &v) : p (p), v (v)   {
        ang = polar_angle (v);
    }
    bool operator < (const Line &r) const {
        return ang < r.ang;
    }
    Point point(double a)   {
        return p + v * a;
    }
};

bool point_on_left(Point p, Line L) {
    return cross (L.v, p - L.p) > 0;
}
Point line_line_inter(Point p, Vector V, Point q, Vector W)    {        //两直线交点，参数方程
    Vector U = p - q;
    double t = cross (W, U) / cross (V, W);
    return p + V * t;
}

vector<Point> half_plane_inter(vector<Line> L)    {
    sort (L.begin (), L.end ());
    int first, last, n = L.size ();
    Point *p = new Point[n];
    Line *q = new Line[n];
    q[first=last=0] = L[0];
    for (int i=1; i<n; ++i) {
        while (first < last && !point_on_left (p[last-1], L[i]))    last--;
        while (first < last && !point_on_left (p[first], L[i]))  first++;
        q[++last] = L[i];
        if (dcmp (cross (q[last].v, q[last-1].v)) == 0) {
            last--;
            if (point_on_left (L[i].p, q[last]))    q[last] = L[i];
        }
        if (first < last)   p[last-1] = line_line_inter (q[last-1].p, q[last-1].v, q[last].p, q[last].v);
    }
    while (first < last && !point_on_left (p[last-1], q[first]))    last--;
    vector<Point> ps;
    if (last - first <= 1)  return ps;
    p[last] = line_line_inter (q[last].p, q[last].v, q[first].p, q[first].v);
    for (int i=first; i<=last; ++i) ps.push_back (p[i]);
    return ps;
}

Point ps[110];
Vector V[110], V2[110];

int main(void)    {
    int n;
    while (scanf ("%d", &n) == 1)   {
        if (!n) break;
        for (int i=0; i<n; ++i) ps[i] = read_point ();
        for (int i=0; i<n; ++i) {
            V[i] = ps[(i+1)%n] - ps[i];
            V2[i] = nomal (V[i]);
        }
        double l = 0, r = 20000;
        while (r - l > EPS)    {
            double mid = l + (r - l) / 2;
            vector<Line> L;
            for (int i=0; i<n; ++i) {
                L.push_back (Line (ps[i] + V2[i] * mid, V[i]));
            }
            vector<Point> qs = half_plane_inter (L);
            int sz = qs.size ();
            if (sz == 0)    r = mid;
            else    l = mid;
        }
        printf ("%.6f\n", l);
    }

   //cout << "Time elapsed: " << 1.0 * clock() / CLOCKS_PER_SEC << " s.\n";

    return 0;
}