POJ 2826 An Easy Problem?! （线段相交）

题意：两块木板可以承装的雨水的最大面积，注意雨水是垂直下落的。



题解：多种情况需要考虑清楚。

#include<cstdio>
#include<cmath>
#define eps 1e-8
#define zero(x) (((x)>0?(x):-(x))<eps)
struct Point{double x,y;};
struct Line{Point a,b;};

//计算cross product (P1-P0)x(P2-P0)
double xmult(Point p1,Point p2,Point p0){
return (p1.x-p0.x)*(p2.y-p0.y)-(p2.x-p0.x)*(p1.y-p0.y);
}
double xmult(double x1,double y1,double x2,double y2,double x0,double y0){
return (x1-x0)*(y2-y0)-(x2-x0)*(y1-y0);
}

//计算dot product (P1-P0).(P2-P0)
double dmult(Point p1,Point p2,Point p0){
return (p1.x-p0.x)*(p2.x-p0.x)+(p1.y-p0.y)*(p2.y-p0.y);
}
double dmult(double x1,double y1,double x2,double y2,double x0,double y0){
return (x1-x0)*(x2-x0)+(y1-y0)*(y2-y0);
}

//两点距离
double distance(Point p1,Point p2){
return sqrt((p1.x-p2.x)*(p1.x-p2.x)+(p1.y-p2.y)*(p1.y-p2.y));
}
double distance(double x1,double y1,double x2,double y2){
return sqrt((x1-x2)*(x1-x2)+(y1-y2)*(y1-y2));
}

//判三点共线
int dots_inline(Point p1,Point p2,Point p3){
return zero(xmult(p1,p2,p3));
}
int dots_inline(double x1,double y1,double x2,double y2,double x3,double y3){
return zero(xmult(x1,y1,x2,y2,x3,y3));
}

//判点是否在线段上,包括端点
int dot_online_in(Point p,Line l){
return zero(xmult(p,l.a,l.b))&&(l.a.x-p.x)*(l.b.x-p.x)<eps&&(l.a.y-p.y)*(l.b.y-p.y)<eps;
}
int dot_online_in(Point p,Point l1,Point l2){
return zero(xmult(p,l1,l2))&&(l1.x-p.x)*(l2.x-p.x)<eps&&(l1.y-p.y)*(l2.y-p.y)<eps;
}
int dot_online_in(double x,double y,double x1,double y1,double x2,double y2){
return zero(xmult(x,y,x1,y1,x2,y2))&&(x1-x)*(x2-x)<eps&&(y1-y)*(y2-y)<eps;
}

//判点是否在线段上,不包括端点
int dot_online_ex(Point p,Line l){
return dot_online_in(p,l)&&(!zero(p.x-l.a.x)||!zero(p.y-l.a.y))&&(!zero(p.x-l.b.x)||!zero(p.y-l.b.y));
}
int dot_online_ex(Point p,Point l1,Point l2){
return dot_online_in(p,l1,l2)&&(!zero(p.x-l1.x)||!zero(p.y-l1.y))&&(!zero(p.x-l2.x)||!zero(p.y-l2.y));
}
int dot_online_ex(double x,double y,double x1,double y1,double x2,double y2){
return dot_online_in(x,y,x1,y1,x2,y2)&&(!zero(x-x1)||!zero(y-y1))&&(!zero(x-x2)||!zero(y-y2));
}

//判两点在线段同侧,点在线段上返回0
int same_side(Point p1,Point p2,Line l){
return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)>eps;
}
int same_side(Point p1,Point p2,Point l1,Point l2){
return xmult(l1,p1,l2)*xmult(l1,p2,l2)>eps;
}

//判两点在线段异侧,点在线段上返回0
int opposite_side(Point p1,Point p2,Line l){
return xmult(l.a,p1,l.b)*xmult(l.a,p2,l.b)<-eps;
}
int opposite_side(Point p1,Point p2,Point l1,Point l2){
return xmult(l1,p1,l2)*xmult(l1,p2,l2)<-eps;
}

//判两直线平行
int parallel(Line u,Line v){
return zero((u.a.x-u.b.x)*(v.a.y-v.b.y)-(v.a.x-v.b.x)*(u.a.y-u.b.y));
}
int parallel(Point u1,Point u2, Point v1,Point v2){
return zero((u1.x-u2.x)*(v1.y-v2.y)-(v1.x-v2.x)*(u1.y-u2.y));
}

//判两直线垂直
int perpendicular(Line u,Line v){
return zero((u.a.x-u.b.x)*(v.a.x-v.b.x)+(u.a.y-u.b.y)*(v.a.y-v.b.y));
}
int perpendicular(Point u1,Point u2,Point v1,Point v2){
return zero((u1.x-u2.x)*(v1.x-v2.x)+(u1.y-u2.y)*(v1.y-v2.y));
}

//判两线段相交,包括端点和部分重合
int intersect_in(Line u,Line v){
if (!dots_inline(u.a,u.b,v.a)||!dots_inline(u.a,u.b,v.b))
return !same_side(u.a,u.b,v)&&!same_side(v.a,v.b,u);
return dot_online_in(u.a,v)||dot_online_in(u.b,v)||dot_online_in(v.a,u)||dot_online_in(v.b,u);
}
int intersect_in(Point u1,Point u2,Point v1,Point v2){
if (!dots_inline(u1,u2,v1)||!dots_inline(u1,u2,v2))
return !same_side(u1,u2,v1,v2)&&!same_side(v1,v2,u1,u2);
return dot_online_in(u1,v1,v2)||dot_online_in(u2,v1,v2)||dot_online_in(v1,u1,u2)||dot_online_in(v2,u1,u2);
}

//判两线段相交,不包括端点和部分重合
int intersect_ex(Line u,Line v){
return opposite_side(u.a,u.b,v)&&opposite_side(v.a,v.b,u);
}
int intersect_ex(Point u1,Point u2,Point v1,Point v2){
return opposite_side(u1,u2,v1,v2)&&opposite_side(v1,v2,u1,u2);
}

//计算两直线交点,注意事先判断直线是否平行!
//线段交点请另外判线段相交(同时还是要判断是否平行!)
Point intersection(Line u,Line v){
Point ret=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));
ret.x+=(u.b.x-u.a.x)*t;
ret.y+=(u.b.y-u.a.y)*t;
return ret;
}
Point intersection(Point u1,Point u2,Point v1,Point v2){
Point ret=u1;
double t=((u1.x-v1.x)*(v1.y-v2.y)-(u1.y-v1.y)*(v1.x-v2.x))
/((u1.x-u2.x)*(v1.y-v2.y)-(u1.y-u2.y)*(v1.x-v2.x));
ret.x+=(u2.x-u1.x)*t;
ret.y+=(u2.y-u1.y)*t;
return ret;
}

//点到直线上的最近点
Point ptoline(Point p,Line l){
Point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
return intersection(p,t,l.a,l.b);
}
Point ptoline(Point p,Point l1,Point l2){
Point t=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
return intersection(p,t,l1,l2);
}

//点到直线距离
double disptoline(Point p,Line l){
return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
double disptoline(Point p,Point l1,Point l2){
return fabs(xmult(p,l1,l2))/distance(l1,l2);
}
double disptoline(double x,double y,double x1,double y1,double x2,double y2){
return fabs(xmult(x,y,x1,y1,x2,y2))/distance(x1,y1,x2,y2);
}

//点到线段上的最近点
Point ptoseg(Point p,Line l){
Point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
return distance(p,l.a)<distance(p,l.b)?l.a:l.b;
return intersection(p,t,l.a,l.b);
}
Point ptoseg(Point p,Point l1,Point l2){
Point t=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
if (xmult(l1,t,p)*xmult(l2,t,p)>eps)
return distance(p,l1)<distance(p,l2)?l1:l2;
return intersection(p,t,l1,l2);
}

//点到线段距离
double disptoseg(Point p,Line l){
Point t=p;
t.x+=l.a.y-l.b.y,t.y+=l.b.x-l.a.x;
if (xmult(l.a,t,p)*xmult(l.b,t,p)>eps)
return distance(p,l.a)<distance(p,l.b)?distance(p,l.a):distance(p,l.b);
return fabs(xmult(p,l.a,l.b))/distance(l.a,l.b);
}
double disptoseg(Point p,Point l1,Point l2){
Point t=p;
t.x+=l1.y-l2.y,t.y+=l2.x-l1.x;
if (xmult(l1,t,p)*xmult(l2,t,p)>eps)
return distance(p,l1)<distance(p,l2)?distance(p,l1):distance(p,l2);
return fabs(xmult(p,l1,l2))/distance(l1,l2);
}

//矢量V以P为顶点逆时针旋转angle并放大scale倍
Point rotate(Point v,Point p,double angle,double scale){
Point ret=p;
v.x-=p.x,v.y-=p.y;
p.x=scale*cos(angle);
p.y=scale*sin(angle);
ret.x+=v.x*p.x-v.y*p.y;
ret.y+=v.x*p.y+v.y*p.x;
return ret;
}

int main()
{
int t;
Point top1, bot1, top2, bot2, inter, tmp;
scanf("%d",&t);
while(t--)
{
scanf("%lf %lf %lf %lf",&top1.x, &top1.y, &bot1.x, &bot1.y);
scanf("%lf %lf %lf %lf",&top2.x, &top2.y, &bot2.x, &bot2.y);

if ( top1.y == bot1.y ) { printf("0.00\n"); continue; }
else if ( top1.y < bot1.y ) { tmp = top1; top1 = bot1; bot1 = tmp; }

if ( top2.y == bot2.y ) { printf("0.00\n"); continue; }
else if ( top2.y < bot2.y ) { tmp = top2; top2 = bot2; bot2 = tmp; }

if ( top1.x == top2.x && top1.y == top2.y ) { printf("0.00\n"); continue;}
if ( parallel(bot1,top1,bot2,top2) ) { printf("0.00\n"); continue; }
if ( !intersect_in(bot1,top1,bot2,top2)) { printf("0.00\n");continue;}

inter = intersection(bot1,top1,bot2,top2);
if ( top1.y < top2.y ) { tmp = top1; top1 = top2; top2 = tmp; }

double mark = xmult(top2,top1,inter);
if ( mark > eps && top1.x >= top2.x ) { printf("0.00\n"); continue;}
if ( mark < -eps && top1.x <= top2.x ) { printf("0.00\n"); continue;}
if ( zero(mark) ) { printf("0.00\n"); continue;}

tmp.y = top2.y;
if ( top1.x == inter.x ) tmp.x = inter.x;
else tmp.x = (tmp.y-inter.y)*(top1.x-inter.x) / (top1.y-inter.y) + inter.x;
double res = fabs(xmult(top2,tmp,inter)/2);
printf("%.2lf\n",res);
}
return 0;
}