POJ 3304 Segments (判断直线与线段相交)

题目链接：POJ 3304

Problem Description

Given n segments in the two dimensional space, write a program, which determines if there exists a line such that after projecting these segments on it, all projected segments have at least one point in common.

Input

Input begins with a number T showing the number of test cases and then, T test cases follow. Each test case begins with a line containing a positive integer n ≤ 100 showing the number of segments. After that, n lines containing four real numbers x1 y1 x2 y2 follow, in which (x1, y1) and (x2, y2) are the coordinates of the two endpoints for one of the segments.

Output

For each test case, your program must output "Yes!", if a line with desired property exists and must output "No!" otherwise. You must assume that two floating point numbers a and b are equal if |a - b| < 10-8.

Sample Input

3
2
1.0 2.0 3.0 4.0
4.0 5.0 6.0 7.0
3
0.0 0.0 0.0 1.0
0.0 1.0 0.0 2.0
1.0 1.0 2.0 1.0
3
0.0 0.0 0.0 1.0
0.0 2.0 0.0 3.0
1.0 1.0 2.0 1.0

Sample Output

Yes!
Yes!
No!

Solution

题意

给定 \(n\) 条线段，判断是否存在一条直线，使得所有线段在这条直线上的投影有公共部分。

题解

枚举 ToLeftTest

如果存在满足条件的一条直线，那么该直线的一条垂线一定与所有线段相交。

如果 \(n \le 2\) 时一定存在。

枚举所有线段的端点中任意两点构造的直线，判断是否所有线段都与之相交。

判断线段与直线相交的方法：判断线段的两个端点是否在直线的同一侧。

注意不要枚举两个相同的端点。

Code

#include <cstdio>
#include <iostream>
#include <cmath>
#include <algorithm>
#include <vector>
using namespace std;
typedef long long ll;
typedef double db;
const db eps = 1e-10;
const db pi = acos(-1.0);
const ll inf = 0x3f3f3f3f3f3f3f3f;
const ll maxn = 1e3 + 10;

inline int dcmp(db x) {
    if(fabs(x) < eps) return 0;
    return x > 0? 1: -1;
}

class Point {
public:
    double x, y;
    Point(double x = 0, double y = 0) : x(x), y(y) {}
    void input() {
        scanf("%lf%lf", &x, &y);
    }
    bool operator<(const Point &a) const {
        return (!dcmp(x - a.x))? dcmp(y - a.y) < 0: x < a.x;
    }
    bool operator==(const Point &a) const {
        return dcmp(x - a.x) == 0 && dcmp(y - a.y) == 0;
    }
    db dis2(const Point a) {
        return pow(x - a.x, 2) + pow(y - a.y, 2);
    }
    db dis(const Point a) {
        return sqrt(dis2(a));
    }

    db dis2() {
        return x * x + y * y;
    }
    db dis() {
        return sqrt(dis2());
    }
    Point operator+(const Point a) {
        return Point(x + a.x, y + a.y);
    }
    Point operator-(const Point a) {
        return Point(x - a.x, y - a.y);
    }
    Point operator*(double p) {
        return Point(x * p, y * p);
    }
    Point operator/(double p) {
        return Point(x / p, y / p);
    }
    db dot(const Point a) {
        return x * a.x + y * a.y;
    }
    db cross(const Point a) {
        return x * a.y - y * a.x;
    }
};
typedef Point Vector;

class Line {
public:
    Point s, e;
    Line() {}
    Line(Point s, Point e) : s(s), e(e) {}
    void input() {
        scanf("%lf%lf%lf%lf", &s.x, &s.y, &e.x, &e.y);
    }
    int toLeftTest(Point p) {
        if((e - s).cross(p - s) > 0) return 1;
        else if((e - s).cross(p - s) < 0) return -1;
        return 0;
    }
};

Line line[maxn];
Point point[maxn];

int main() {
    int T;
    scanf("%d", &T);
    while(T--) {
        int n;
        scanf("%d", &n);
        for(int i = 0; i < n; ++i) {
            line[i].input();
            point[i * 2] = line[i].s;
            point[i * 2 + 1] = line[i].e;
        }
        if(n < 3) {
            printf("Yes!\n");
            continue;
        }
        int fg = 0;
        for(int i = 0; i < 2 * n; ++i) {
            for(int j = i + 1; j < 2 * n; ++j) {
                if(point[i] == point[j]) continue;
                Line l = Line(point[i], point[j]);
                int flag = 1;
                for(int k = 0; k < n; ++k) {
                    if(l.toLeftTest(line[k].s) + l.toLeftTest(line[k].e) && l.toLeftTest(line[k].s) == l.toLeftTest(line[k].e)) {
                        flag = 0;
                        break;
                    }
                }
                if(flag == 1) {
                    printf("Yes!\n");
                    fg = 1;
                    break;
                }
            }
            if(fg == 1) {break;}
        }
        if(!fg) {
            printf("No!\n");
        }
    }
    return 0;
}