Intersecting Lines(叉积，方程)

Posted 2021-01-13 fighting-sh

Intersecting Lines

http://poj.org/problem?id=1269

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 18897		Accepted: 8043

Description

We all know that a pair of distinct points on a plane defines a line and that a pair of lines on a plane will intersect in one of three ways: 1) no intersection because they are parallel, 2) intersect in a line because they are on top of one another (i.e. they are the same line), 3) intersect in a point. In this problem you will use your algebraic knowledge to create a program that determines how and where two lines intersect. 
Your program will repeatedly read in four points that define two lines in the x-y plane and determine how and where the lines intersect. All numbers required by this problem will be reasonable, say between -1000 and 1000. 

Input

The first line contains an integer N between 1 and 10 describing how many pairs of lines are represented. The next N lines will each contain eight integers. These integers represent the coordinates of four points on the plane in the order x1y1x2y2x3y3x4y4. Thus each of these input lines represents two lines on the plane: the line through (x1,y1) and (x2,y2) and the line through (x3,y3) and (x4,y4). The point (x1,y1) is always distinct from (x2,y2). Likewise with (x3,y3) and (x4,y4).

Output

There should be N+2 lines of output. The first line of output should read INTERSECTING LINES OUTPUT. There will then be one line of output for each pair of planar lines represented by a line of input, describing how the lines intersect: none, line, or point. If the intersection is a point then your program should output the x and y coordinates of the point, correct to two decimal places. The final line of output should read "END OF OUTPUT".

Sample Input

5
0 0 4 4 0 4 4 0
5 0 7 6 1 0 2 3
5 0 7 6 3 -6 4 -3
2 0 2 27 1 5 18 5
0 3 4 0 1 2 2 5

Sample Output

INTERSECTING LINES OUTPUT
POINT 2.00 2.00
NONE
LINE
POINT 2.00 5.00
POINT 1.07 2.20
END OF OUTPUT

先判断是否平行，平行的话再判断是否共线，否则把向量转换成方程，计算交点

直线的一般式方程AX+BY+C=0中，A B C分别等于：

A = Y2 - Y1

B = X1 - X2

C = X2*Y1 - X1*Y2

 1 #include<iostream>
 2 #include<cstdio>
 3 #include<cstring>
 4 #include<cmath>
 5 #include<string>
 6 #include<algorithm>
 7 #include<queue>
 8 #include<vector>
 9 #define esp 0.00000001
10 using namespace std;
11 
12 struct Vector{
13     double x,y;
14 };
15 
16 struct Line{
17     Vector s,e;
18 }line[35];
19 
20 double Cross(Vector a,Vector b,Vector c){
21     return (b.x-a.x)*(c.y-a.y)-(b.y-a.y)*(c.x-a.x);
22 }
23 
24 
25 int main(){
26     int n;
27     cin>>n;
28     Vector a,b,c,d;
29     double tmp;
30     cout<<"INTERSECTING LINES OUTPUT"<<endl;
31     for(int i=1;i<=n;i++){
32         cin>>a.x>>a.y>>b.x>>b.y>>c.x>>c.y>>d.x>>d.y;
33         tmp=(a.x-b.x)*(c.y-d.y)-(a.y-b.y)*(c.x-d.x);
34         if(fabs(tmp)<esp&&fabs(Cross(a,b,d))<esp){
35             cout<<"LINE"<<endl;
36         }
37         else if(fabs(tmp)<esp){
38             cout<<"NONE"<<endl;
39         }
40         else{
41             double a1=a.y-b.y,b1=b.x-a.x,c1=a.x*b.y-b.x*a.y;//c是叉积
42             double a2=c.y-d.y,b2=d.x-c.x,c2=c.x*d.y-d.x*c.y;
43             double x=(c2*b1-c1*b2)/(b2*a1-b1*a2);
44             double y=(a2*c1-a1*c2)/(b2*a1-b1*a2);
45             printf("POINT %.2f %.2f
",x,y);
46         }
47     }
48     cout<<"END OF OUTPUT"<<endl;
49 }

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