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POJ 1410 Intersection（判斷線段交和點(diǎn)在矩形內(nèi)）

Intersection

Time Limit:?1000MS	?	Memory Limit:?10000K
Total Submissions:?9996	?	Accepted:?2632

Description

You are to write a program that has to decide whether a given line segment intersects a given rectangle.?

An example:?
line: start point: (4,9)?
end point: (11,2)?
rectangle: left-top: (1,5)?
right-bottom: (7,1)?

?
Figure 1: Line segment does not intersect rectangle?

The line is said to intersect the rectangle if the line and the rectangle have at least one point in common. The rectangle consists of four straight lines and the area in between. Although all input values are integer numbers, valid intersection points do not have to lay on the integer grid.?

Input

The input consists of n test cases. The first line of the input file contains the number n. Each following line contains one test case of the format:?
xstart ystart xend yend xleft ytop xright ybottom?

where (xstart, ystart) is the start and (xend, yend) the end point of the line and (xleft, ytop) the top left and (xright, ybottom) the bottom right corner of the rectangle. The eight numbers are separated by a blank. The terms top left and bottom right do not imply any ordering of coordinates.

Output

For each test case in the input file, the output file should contain a line consisting either of the letter "T" if the line segment intersects the rectangle or the letter "F" if the line segment does not intersect the rectangle.

Sample Input

1 4 9 11 2 1 5 7 1

Sample Output

F

Source

Southwestern European Regional Contest 1995

題意：

給了一個(gè)線段和矩形。如果線段和矩形的邊相交，或者線段在矩形內(nèi)。輸出T，否則輸出F

思路：

先判斷矩形4條邊是否與線段相交，然后判斷線段兩個(gè)端點(diǎn)是否在矩形內(nèi)

AC代碼1：

轉(zhuǎn)角法判斷點(diǎn)在多邊形內(nèi)：實(shí)現(xiàn)簡(jiǎn)單，效率較低

//判斷線段相交和點(diǎn)在矩形內(nèi) #include <iostream> #include <stdio.h> #include <string.h> #include <algorithm> #include <queue> #include <map> #include <vector> #include <set> #include <string> #include <math.h>using namespace std; const double eps = 1e-8; int sgn(double x) {if(fabs(x) < eps)return 0;if(x < 0)return -1;else return 1; } struct Point {double x,y;Point(){}Point(double _x,double _y){x = _x;y = _y;}Point operator -(const Point &b)const{return Point(x - b.x,y - b.y);}//叉積double operator ^(const Point &b)const{return x*b.y - y*b.x;}//點(diǎn)積double operator *(const Point &b)const{return x*b.x + y*b.y;}//繞原點(diǎn)旋轉(zhuǎn)角度B（弧度值），后x,y的變化void transXY(double B){double tx = x,ty = y;x = tx*cos(B) - ty*sin(B);y = tx*sin(B) + ty*cos(B);} }; struct Line {Point s,e;double k;Line(){}Line(Point _s,Point _e){s = _s;e = _e;k = atan2(e.y - s.y,e.x - s.x);}//兩條直線求交點(diǎn)，//第一個(gè)值為0表示直線重合，為1表示平行，為2是相交//只有第一個(gè)值為2時(shí)，交點(diǎn)才有意義pair<int,Point> operator &(const Line &b)const{Point res = s;if(sgn((s-e)^(b.s-b.e)) == 0){if(sgn((s-b.e)^(b.s-b.e)) == 0)return make_pair(0,res);//重合else return make_pair(1,res);//平行}double t = ((s-b.s)^(b.s-b.e))/((s-e)^(b.s-b.e));res.x += (e.x-s.x)*t;res.y += (e.y-s.y)*t;return make_pair(2,res);} }; //兩點(diǎn)間距離 double dist(Point a,Point b) {return sqrt((a-b)*(a-b)); } //判斷線段相交 bool inter(Line l1,Line l2) {returnmax(l1.s.x,l1.e.x) >= min(l2.s.x,l2.e.x) &&max(l2.s.x,l2.e.x) >= min(l1.s.x,l1.e.x) &&max(l1.s.y,l1.e.y) >= min(l2.s.y,l2.e.y) &&max(l2.s.y,l2.e.y) >= min(l1.s.y,l1.e.y) &&sgn((l2.s-l1.s)^(l1.e-l1.s))*sgn((l2.e-l1.s)^(l1.e-l1.s)) <= 0 &&sgn((l1.s-l2.s)^(l2.e-l1.s))*sgn((l1.e-l2.s)^(l2.e-l2.s)) <= 0; } //判斷點(diǎn)在線段上 bool OnSeg(Point p,Line L) {returnsgn((L.s-p)^(L.e-p))==0&&sgn((p.x-L.s.x)*(p.x-L.e.x))<=0&&sgn((p.y-L.s.y)*(p.y-L.e.y))<=0; }//判斷點(diǎn)在凸多邊形內(nèi)，點(diǎn)形成一個(gè)凸包，而且按逆時(shí)針排序 //如果是順時(shí)針，就把<0改成>0 //點(diǎn)的編號(hào)：0~n-1 //返回值：-1點(diǎn)在凸多邊形外，0點(diǎn)在凸多邊形邊上，1點(diǎn)在凸多邊形內(nèi) int inConvexPoly(Point a,Point p[],int n) {for(int i=0;i<n;i++){if(sgn((p[i]-a)^(p[(i+1)%n]-a))<0)return -1;else if(OnSeg(a,Line(p[i],p[(i+1)%n])))return 0;}return 1; }//判斷點(diǎn)在任意多邊形內(nèi) //射線法，poly[]的頂點(diǎn)數(shù)要大于等于3，點(diǎn)的編號(hào)：0~n-1 //返回值 //-1：點(diǎn)在多邊形外 //0：點(diǎn)在多邊形邊界上 //1：點(diǎn)在多邊形內(nèi) int inPoly(Point p,Point poly[],int n) {int cnt=0;Line ray,side;ray.s=p;ray.s.y=p.y;ray.e.x=-100000000000.0;//-INF,注意取值，防止越界for(int i=0;i<n;i++){side.s=poly[i];side.e=poly[(i+1)%n];if(OnSeg(p,side)) return 0;//如果平行軸，則不考慮if(sgn(side.s.y-side.e.y)==0)continue;if(OnSeg(side.s,ray)){if(sgn(side.s.y-side.e.y)>0)cnt++;}else if(OnSeg(side.e,ray)){if(sgn(side.e.y-side.s.y)>0)cnt++;}else if(inter(ray,side))cnt++;}if(cnt%2==1)return 1;elsereturn -1; } int main() {int T;scanf("%d",&T);while(T--){Line line;scanf("%lf%lf%lf%lf",&line.s.x,&line.s.y,&line.e.x,&line.e.y);double x1,y1,x2,y2;scanf("%lf%lf%lf%lf",&x1,&y1,&x2,&y2);if(x1>x2) swap(x1,x2);if(y1>y2) swap(y1,y2);Point p[5];//0~3p[0]=Point(x1,y1);//left_bottomp[1]=Point(x2,y1);//right_bottomp[2]=Point(x2,y2);//right_topp[3]=Point(x1,y2);//left_topLine bottom=Line(p[1],p[0]);if(inter(line,bottom)){printf("T\n");continue;}Line right=Line(p[2],p[1]);if(inter(line,right)){printf("T\n");continue;}Line top=Line(p[3],p[2]);if(inter(line,top)){printf("T\n");continue;}Line left=Line(p[3],p[0]);if(inter(line,left)){printf("T\n");continue;}if(inConvexPoly(line.s,p,4)>=0||inConvexPoly(line.e,p,4)>=0){printf("T\n");continue;}printf("F\n");continue;}return 0; }

AC代碼2：

射線法判斷點(diǎn)是否在多邊形內(nèi)，貌似要快一點(diǎn)：0 ms

//判斷線段相交和點(diǎn)在矩形內(nèi) #include <iostream> #include <stdio.h> #include <string.h> #include <algorithm> #include <queue> #include <map> #include <vector> #include <set> #include <string> #include <math.h>using namespace std; const double eps = 1e-8; int sgn(double x) {if(fabs(x) < eps)return 0;if(x < 0)return -1;else return 1; } struct Point {double x,y;Point(){}Point(double _x,double _y){x = _x;y = _y;}Point operator -(const Point &b)const{return Point(x - b.x,y - b.y);}//叉積double operator ^(const Point &b)const{return x*b.y - y*b.x;}//點(diǎn)積double operator *(const Point &b)const{return x*b.x + y*b.y;}//繞原點(diǎn)旋轉(zhuǎn)角度B（弧度值），后x,y的變化void transXY(double B){double tx = x,ty = y;x = tx*cos(B) - ty*sin(B);y = tx*sin(B) + ty*cos(B);} }; struct Line {Point s,e;double k;Line(){}Line(Point _s,Point _e){s = _s;e = _e;k = atan2(e.y - s.y,e.x - s.x);}//兩條直線求交點(diǎn)，//第一個(gè)值為0表示直線重合，為1表示平行，為2是相交//只有第一個(gè)值為2時(shí)，交點(diǎn)才有意義pair<int,Point> operator &(const Line &b)const{Point res = s;if(sgn((s-e)^(b.s-b.e)) == 0){if(sgn((s-b.e)^(b.s-b.e)) == 0)return make_pair(0,res);//重合else return make_pair(1,res);//平行}double t = ((s-b.s)^(b.s-b.e))/((s-e)^(b.s-b.e));res.x += (e.x-s.x)*t;res.y += (e.y-s.y)*t;return make_pair(2,res);} }; //兩點(diǎn)間距離 double dist(Point a,Point b) {return sqrt((a-b)*(a-b)); } //判斷線段相交 bool inter(Line l1,Line l2) {returnmax(l1.s.x,l1.e.x) >= min(l2.s.x,l2.e.x) &&max(l2.s.x,l2.e.x) >= min(l1.s.x,l1.e.x) &&max(l1.s.y,l1.e.y) >= min(l2.s.y,l2.e.y) &&max(l2.s.y,l2.e.y) >= min(l1.s.y,l1.e.y) &&sgn((l2.s-l1.s)^(l1.e-l1.s))*sgn((l2.e-l1.s)^(l1.e-l1.s)) <= 0 &&sgn((l1.s-l2.s)^(l2.e-l1.s))*sgn((l1.e-l2.s)^(l2.e-l2.s)) <= 0; } //判斷點(diǎn)在線段上 bool OnSeg(Point p,Line L) {returnsgn((L.s-p)^(L.e-p))==0&&sgn((p.x-L.s.x)*(p.x-L.e.x))<=0&&sgn((p.y-L.s.y)*(p.y-L.e.y))<=0; }//判斷點(diǎn)在凸多邊形內(nèi)，點(diǎn)形成一個(gè)凸包，而且按逆時(shí)針排序 //如果是順時(shí)針，就把<0改成>0 //點(diǎn)的編號(hào)：0~n-1 //返回值：-1點(diǎn)在凸多邊形外，0點(diǎn)在凸多邊形邊上，1點(diǎn)在凸多邊形內(nèi) int inConvexPoly(Point a,Point p[],int n) {for(int i=0;i<n;i++){if(sgn((p[i]-a)^(p[(i+1)%n]-a))<0)return -1;else if(OnSeg(a,Line(p[i],p[(i+1)%n])))return 0;}return 1; }//判斷點(diǎn)在任意多邊形內(nèi) //射線法，poly[]的頂點(diǎn)數(shù)要大于等于3，點(diǎn)的編號(hào)：0~n-1 //返回值 //-1：點(diǎn)在多邊形外 //0：點(diǎn)在多邊形邊界上 //1：點(diǎn)在多邊形內(nèi) int inPoly(Point p,Point poly[],int n) {int cnt=0;Line ray,side;ray.s=p;ray.s.y=p.y;ray.e.x=-100000000000.0;//-INF,注意取值，防止越界for(int i=0;i<n;i++){side.s=poly[i];side.e=poly[(i+1)%n];if(OnSeg(p,side)) return 0;//如果平行軸，則不考慮if(sgn(side.s.y-side.e.y)==0)continue;if(OnSeg(side.s,ray)){if(sgn(side.s.y-side.e.y)>0)cnt++;}else if(OnSeg(side.e,ray)){if(sgn(side.e.y-side.s.y)>0)cnt++;}else if(inter(ray,side))cnt++;}if(cnt%2==1)return 1;elsereturn -1; } int main() {int T;scanf("%d",&T);while(T--){Line line;scanf("%lf%lf%lf%lf",&line.s.x,&line.s.y,&line.e.x,&line.e.y);double x1,y1,x2,y2;scanf("%lf%lf%lf%lf",&x1,&y1,&x2,&y2);if(x1>x2) swap(x1,x2);if(y1>y2) swap(y1,y2);Point p[5];//0~3p[0]=Point(x1,y1);//left_bottomp[1]=Point(x2,y1);//right_bottomp[2]=Point(x2,y2);//right_topp[3]=Point(x1,y2);//left_topLine bottom=Line(p[1],p[0]);if(inter(line,bottom)){printf("T\n");continue;}Line right=Line(p[2],p[1]);if(inter(line,right)){printf("T\n");continue;}Line top=Line(p[3],p[2]);if(inter(line,top)){printf("T\n");continue;}Line left=Line(p[3],p[0]);if(inter(line,left)){printf("T\n");continue;}if(inPoly(line.s,p,4)>=0||inPoly(line.e,p,4)>=0){printf("T\n");continue;}printf("F\n");continue;}return 0; }

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