[POJ][1228][Grandpa's Estate]

题目：http://poj.org/problem?id=1228

题意：给定一些点，求能否确定一个凸包。如果在这些点中求出的凸包的每一条边都至少有三个点（包括顶点），则这个凸包是确定的。

　　  注意一条直线或点数n<5时应输出NO。

#include <stdio.h>
#include <math.h>
#include <string.h>
#include <iostream>
#include <algorithm>

using namespace std;
const double eps = 1e-8;
const double pi = acos(-1.0);

typedef struct
{
    double x, y;
}cpoint;

double sqr(double x) {return x*x;}

double dcmp(double x)
{
    if(x < -eps) return -1; else return x > eps;
}

double cross(cpoint p0, cpoint p1, cpoint p2)
{
    return (p1.x - p0.x) * (p2.y - p0.y) - (p2.x - p0.x) * (p1.y - p0.y);
}

double dot(cpoint p0, cpoint p1, cpoint p2)
{
    return (p1.x - p0.x) * (p2.x - p0.x) + (p1.y - p0.y) * (p2.y - p0.y);
}

double dis(cpoint p1, cpoint p2)
{
    return sqrt(sqr(p1.x - p2.x) + sqr(p1.y - p2.y));
}

double dissqr(cpoint p1, cpoint p2)
{
    return sqr(p1.x - p2.x) + sqr(p1.y - p2.y);
}

int PointEqual(const cpoint p1, const cpoint p2)
{
    return dcmp(p1.x - p2.x) == 0 && dcmp(p1.y - p2.y) == 0;
}

int PointOnSegment(cpoint p0, cpoint p1, cpoint p2)
{
    return dcmp(cross(p0, p1, p2)) == 0 && dcmp(dot(p0, p1, p2)) <= 0;
}

cpoint bp;

int PolarCmp(const cpoint &p1, const cpoint &p2)
{
    int u = dcmp(cross(bp, p1, p2));
    return u > 0 || (u == 0 && dcmp(dissqr(bp, p1)-dissqr(bp, p2)) < 0);
}

int PointInPolygon(cpoint cp, cpoint p[], int n)
{
    int i, k, d1, d2, wn = 0;
    double sum = 0;
    p[n] = p[0];
    for (int i = 0; i < n; i++)
    {
        if (PointOnSegment(cp, p[i], p[i+1])) return 2;
        k = dcmp(cross(p[i], p[i+1], cp));
        d1 = dcmp(p[i+0].y - cp.y);
        d2 = dcmp(p[i+1].y - cp.y);
        if (k > 0 && d1 <= 0 && d2 > 0) wn++;
        if (k < 0 && d2 <= 0 && d1 > 0) wn--;
    }
    return wn != 0;
}

void graham(cpoint pin[], int n, cpoint ch[], int &m)
{
    int i, j, k, u, v;
    memcpy(ch, pin, n*sizeof(cpoint));
    for (i = k = 0; i < n; i++)
    {
        u = dcmp(ch[i].x - ch[k].x);
        v = dcmp(ch[i].y - ch[k].y);
        if (v < 0 || (v == 0 && u < 0)) k = i;
    }
    bp = ch[k];
    sort(ch, ch + n, PolarCmp);
    n = unique(ch, ch + n, PointEqual) - ch;
    if (n <= 1) {m = n; return ;}
    if (dcmp(cross(ch[0], ch[1], ch[n-1])) == 0)
    {
        m = 2; ch[1] = ch[n-1]; return ;
    }
    ch[n++] = ch[0];
    for (i = 1, j = 2; j < n; j++)
    {
        while (i > 0 && dcmp(cross(ch[i-1], ch[i], ch[j])) <= 0) i--;
        ch[++i] = ch[j];
    }
    m = i;
}

double PolygononArea(cpoint p[], int n)
{
    if (n < 3) return 0;
    double s = p[0].y * (p[n-1].x - p[1].x);
    for (int i = 1; i<n; i++)
    {
        s += p[i].y * (p[i-1].x - p[(i+1) % n].x);
    }
    return fabs(s / 2.0);
}

double LengthOfPolygon(cpoint p[], int m)
{
    double length = 0;
    for (int i=0; i<m; i++)
    {
        length += dis(p[i], p[(i+1)%m]);
    }
    return length;
}

cpoint kingdom[1005];
cpoint polygon_kingdom[1005];
int m, v[1005], segment[1005];

int main()
{
    //freopen("D:/a.txt", "r", stdin);
    int ncase, n;
    scanf("%d", &ncase);
    for (int k=0; k<ncase; k++)
    {
        memset(v, 0, sizeof(v));
        memset(segment, 0, sizeof(segment));
        scanf("%d", &n);
        for (int i=0; i<n; i++)
        {
            scanf("%lf%lf", &kingdom[i].x, &kingdom[i].y);
        }
        graham(kingdom, n, polygon_kingdom, m);
        int stop = 0;
        for (int i=0; i<n; i++)
        {
            if (PointInPolygon(kingdom[i], polygon_kingdom, m) == 1)
            {
                printf("NO\n");
                stop = 1;
                break;
            }
        }
        if (stop) continue;
        for (int i=0; i<m; i++)
        {
            for (int j=0; j<n; j++)
            {
                if (v[j]) continue;
                if (PointEqual(kingdom[j], polygon_kingdom[i])) {v[j]=1;continue;}
                if (PointEqual(kingdom[j], polygon_kingdom[(i+1)%m])) {v[j]=1;continue;}
                if (PointOnSegment(kingdom[j], polygon_kingdom[(i+1)%m], polygon_kingdom[i])){segment[i]=1; continue;}
            }
        }
        int sum = 0;
        for (int i=0; i<m; i++)
        {
            sum += segment[i];
        }
        if (n < 6){printf("NO\n"); continue;}
        if (m <= 2){printf("NO\n"); continue;}
        if (m == sum) printf("YES\n");
        else printf("NO\n");
    }
    return 0;
}