poj 1269 Intersecting Lines（判相交交点与平行）

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

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 10379		Accepted: 4651

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

Source

 

//////////////////////////////////////////////////////////////////////////////////////////////////////

题目大意是给定n对线，判断每一对线是平行还是相交并求出交点

 

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

const double eps = 1e-6;
typedef struct Node
{
    double x,y;
} point;

typedef struct
{
    point a,b;
} line;

bool dy(double x,double y){    return x>eps+y;}//x>y

bool xy(double x,double y){    return x<y-eps;}//x<y

bool dyd(double x,double y){    return x>y-eps;}//x>=y

bool xyd(double x,double y){    return x<y+eps;}//x<=y

bool dd(double x,double y){    return fabs(x-y)<eps;}//x==y

double crossProduct(point a,point b,point c)//ab  ac
{
    return (c.x-a.x)*(b.y-a.y)-(b.x-a.x)*(c.y-a.y);
}

bool parallel(line u,line v)
{
    return dd((u.a.x-u.b.x)*(v.a.y-v.b.y)-(v.a.x-v.b.x)*(u.a.y-u.b.y),0.0);
}

point intersection(line u,line v)
{
    point ans=u.a;
    double t = ((u.a.x-v.a.x)*(v.a.y-v.b.y)-(u.a.y-v.a.y)*(v.a.x-v.b.x))/
    ((u.a.x-u.b.x)*(v.a.y-v.b.y)-(u.a.y-u.b.y)*(v.a.x-v.b.x));
    ans.x+=(u.b.x-u.a.x)*t;
    ans.y+=(u.b.y-u.a.y)*t;
    return ans;
}

int main()
{
    line u,v;
    int n;
    while(scanf("%d",&n)!=EOF)
    {
        printf("INTERSECTING LINES OUTPUT
");
        while(n--)
        {
            scanf("%lf%lf%lf%lf",&u.a.x,&u.a.y,&u.b.x,&u.b.y);
            scanf("%lf%lf%lf%lf",&v.a.x,&v.a.y,&v.b.x,&v.b.y);

            if(parallel(u,v))
            {
                if(dd(crossProduct(u.a,u.b,v.a),0.0))
                    printf("LINE
");
                else
                    printf("NONE
");
            }
            else
            {
                point ans=intersection(u,v);
                printf("POINT %.2lf %.2lf
",ans.x,ans.y);
            }
        }
        printf("END OF OUTPUT
");
    }

}

由于并不是太懂，就拷贝别人的了

原帖：http://blog.csdn.net/zxy_snow/article/details/6341282

/×

先判断两条直线是不是同线，不是的话再判断是否平行，再不是的话就只能是相交的，求出交点。

如何判断是否同线？由叉积的原理知道如果p1，p2，p3共线的话那么(p2-p1)X(p3-p1)=0。因此如果p1，p2，p3共线，p1，p2，p4共线，那么两条直线共线。direction（）求叉积，叉积为0说明共线。

如何判断是否平行？由向量可以判断出两直线是否平行。如果两直线平行，那么向量p1p2、p3p4也是平等的。即((p1.x-p2.x)*(p3.y-p4.y)-(p1.y-p2.y)*(p3.x-p4.x))==0说明向量平等。

如何求出交点？这里也用到叉积的原理。假设交点为p0(x0,y0)。则有：

(p1-p0)X(p2-p0)=0

(p3-p0)X(p2-p0)=0

展开后即是

(y1-y2)x0+(x2-x1)y0+x1y2-x2y1=0

(y3-y4)x0+(x4-x3)y0+x3y4-x4y3=0

将x0,y0作为变量求解二元一次方程组。

假设有二元一次方程组

a1x+b1y+c1=0;

a2x+b2y+c2=0

那么

x=(c1*b2-c2*b1)/(a2*b1-a1*b2);

y=(a2*c1-a1*c2)/(a1*b2-a2*b1);

因为此处两直线不会平行，所以分母不会为0。

×/