SGU 124 Broken line(計算幾何)

Description:

There is a closed broken line on a plane with sides parallel to coordinate axes, without self-crossings and self-contacts. The broken line consists of K segments. You have to determine, whether a given point with coordinates (X0,Y0) is inside this closed broken line, outside or belongs to the broken line.

Input:

The first line contains integer K (4 Ј K Ј 10000) - the number of broken line segments. Each of the following N lines contains coordinates of the beginning and end points of the segments (4 integer xi1,yi1,xi2,yi2; all numbers in a range from -10000 up to 10000 inclusive). Number separate by a space. The segments are given in random order. Last line contains 2 integers X0 and Y0 - the coordinates of the given point delimited by a space. (Numbers X0, Y0 in a range from -10000 up to 10000 inclusive).

Output:

The first line should contain:

INSIDE - if the point is inside closed broken line,

OUTSIDE - if the point is outside,

BORDER - if the point belongs to broken line.

Sample Input:

4
0 0 0 3
3 3 3 0
0 3 3 3
3 0 0 0
2 2

Sample Output:

INSIDE

題目連結

題意為判斷一點與多邊形的關係（在多邊形內部、外部或者邊界）

首先判斷點是否在多邊形邊界上（這很容易），之後再用射線法進行判斷

射線法為從判斷點引一條水平射線，判斷與多邊形的邊界交點數量，若為偶數則在多邊形外部，若為奇數則在多邊形內部

其中又有一些特殊情況，首先跳過多邊形與 $x$ 軸平行的邊界線，之後對於射線與多邊形頂點相交的情況其在判斷與多邊形邊界線段是否相交時按照一定規則只在其中一頂點相交時才判斷相交，在另一頂點相交時不判斷相交

可根據下圖理解射線法的原理（紅色區域為多邊形，兩條射線為 $ST, UV$ ）

AC程式碼:

#include <bits/stdc++.h>
using namespace std;
typedef double db;
const db inf = 1e20;
const int maxn = 1e4 + 5;
const db eps = 1e-9;

int Sgn(db Key) {return fabs(Key) < eps ? 0 : (Key < 0 ? -1 : 1);}
int Cmp(db Key1, db Key2) {return Sgn(Key1 - Key2);}
struct Point {db X, Y;};
typedef Point Vector;
Vector operator - (Vector Key1, Vector Key2) {return (Vector){Key1.X - Key2.X, Key1.Y - Key2.Y};}
Vector operator + (Vector Key1, Vector Key2) {return (Vector){Key1.X + Key2.X, Key1.Y + Key2.Y};}
db operator * (Vector Key1, Vector Key2) {return Key1.X * Key2.X + Key1.Y * Key2.Y;}
db operator ^ (Vector Key1, Vector Key2) {return Key1.X * Key2.Y - Key1.Y * Key2.X;}
struct Line {Point S, T;};
typedef Line Segment;
typedef Line Ray;
bool IsPointOnSeg(Point Key1, Segment Key2) {
    return Sgn((Key1 - Key2.S) ^ (Key2.T - Key2.S)) == 0 && Sgn((Key1 - Key2.S) * (Key1 - Key2.T)) <= 0;
}
bool IsSegInterSeg(Segment Key1, Segment Key2) {
    return
        max(Key1.S.X, Key1.T.X) >= min(Key2.S.X, Key2.T.X) &&
        max(Key2.S.X, Key2.T.X) >= min(Key1.S.X, Key1.T.X) &&
        max(Key1.S.Y, Key1.T.Y) >= min(Key2.S.Y, Key2.T.Y) &&
        max(Key2.S.Y, Key2.T.Y) >= min(Key1.S.Y, Key1.T.Y) &&
        Sgn((Key2.S - Key1.T) ^ (Key1.S - Key1.T)) * Sgn((Key2.T - Key1.T) ^ (Key1.S - Key1.T)) <= 0 &&
        Sgn((Key1.S - Key2.T) ^ (Key2.S - Key2.T)) * Sgn((Key1.T - Key2.T) ^ (Key2.S - Key2.T)) <= 0;
}

int N;
Segment Segs[maxn];
Point Dot;
Ray Judge;

bool IsPointOnPolygon() {
    for (int i = 1; i <= N; ++i)
        if (IsPointOnSeg(Dot, Segs[i])) return true;
    return false;
}

bool IsPointInPolygon() {
    int Cnt = 0;
    for (int i = 1; i <= N; ++i) {
        if (Cmp(Segs[i].S.Y, Segs[i].T.Y) == 0) continue;
        if (IsSegInterSeg(Judge, Segs[i]) && Cmp(Segs[i].T.Y, Dot.Y)) {
            Cnt++;
        }
    }
    return Cnt & 1;
}

int main(int argc, char *argv[]) {
    scanf("%d", &N);
    for (int i = 1; i <= N; ++i) {
        scanf("%lf%lf%lf%lf", &Segs[i].S.X, &Segs[i].S.Y, &Segs[i].T.X, &Segs[i].T.Y);
        if (Cmp(Segs[i].S.Y, Segs[i].T.Y) > 0) swap(Segs[i].S, Segs[i].T);
    }
    scanf("%lf%lf", &Dot.X, &Dot.Y);
    Judge = (Ray){Dot, (Point){inf, Dot.Y}};
    if (IsPointOnPolygon()) printf("BORDER\n");
    else if (IsPointInPolygon()) printf("INSIDE\n");
    else printf("OUTSIDE\n");
    return 0;
}