NJUST4316(立体几何投影的面积交)

题目：Mission Impossible

题意：在天花板上有3个摄像头，下面有一个凸多面体，问摄像头观测不到的在地面上的面积是多少。

思路：先求出每个摄像头对于凸多面体在xoy平面的投影，然后求凸包，然后利用半平面交来求面积交即可，注意求投影时用到一个结论：

如果在空间有3点共线则满足：(z3-z2)/(z2-z1)=(y3-y2)/(y2-y1)=(x3-x2)/(x2-x1)

代码太乱，有时间再整理。

#include <math.h>
#include <stdio.h>
#include <iostream>
#include <algorithm>

using namespace std;

const int N=105;
const double EPS=1e-8;

typedef double DIY;

DIY Area;

struct Point
{
    DIY x,y;
    Point() {}
    Point(DIY _x,DIY _y):x(_x),y(_y){}
} p[N];

Point MakeVector(Point &P,Point &Q)
{
    return Point(Q.x-P.x,Q.y-P.y);
}

DIY CrossProduct(Point P,Point Q)
{
    return P.x*Q.y-P.y*Q.x;
}

Point MinA;

Point stack1[N],stack2[N],stack3[N];
int top1,top2,top3;

DIY dist(Point A,Point B)
{
    return sqrt((A.x-B.x)*(A.x-B.x)+(A.y-B.y)*(A.y-B.y));
}

DIY cross(Point A,Point B,Point C)
{
    return (B.x-A.x)*(C.y-A.y)-(B.y-A.y)*(C.x-A.x);
}

bool cmp1(Point a,Point b)
{
    DIY k=cross(MinA,a,b);
    if(k>0) return 1;
    if(k<0) return 0;
    return dist(MinA,a)>dist(MinA,b);
}

void Graham1(Point *p,int n,int &top)
{
    int i;
    for(i=0; i<n; i++)
        if(p[i].y<p[0].y||(p[i].y==p[0].y&&p[i].x<p[0].x))
            swap(p[i],p[0]);
    MinA=p[0];
    sort(p+1,p+n,cmp1);
    p[n]=p[0];
    stack1[0]=p[0];
    stack1[1]=p[1];
    stack1[2]=p[2];
    top=2;
    for(i=3; i<=n; i++)
    {
        while(cross(stack1[top-1],stack1[top],p[i])<=0&&top>=2) --top;
        stack1[++top]=p[i];
    }
}

void Graham2(Point *p,int n,int &top)
{
    int i;
    for(i=0; i<n; i++)
        if(p[i].y<p[0].y||(p[i].y==p[0].y&&p[i].x<p[0].x))
            swap(p[i],p[0]);
    MinA=p[0];
    sort(p+1,p+n,cmp1);
    p[n]=p[0];
    stack2[0]=p[0];
    stack2[1]=p[1];
    stack2[2]=p[2];
    top=2;
    for(i=3; i<=n; i++)
    {
        while(cross(stack2[top-1],stack2[top],p[i])<=0&&top>=2) --top;
        stack2[++top]=p[i];
    }
}

void Graham3(Point *p,int n,int &top)
{
    int i;
    for(i=0; i<n; i++)
        if(p[i].y<p[0].y||(p[i].y==p[0].y&&p[i].x<p[0].x))
            swap(p[i],p[0]);
    MinA=p[0];
    sort(p+1,p+n,cmp1);
    p[n]=p[0];
    stack3[0]=p[0];
    stack3[1]=p[1];
    stack3[2]=p[2];
    top=2;
    for(i=3; i<=n; i++)
    {
        while(cross(stack3[top-1],stack3[top],p[i])<=0&&top>=2) --top;
        stack3[++top]=p[i];
    }
}

DIY MultiCross(Point P,Point Q,Point R)
{
    return CrossProduct(MakeVector(Q,P),MakeVector(Q,R));
}

struct halfPlane
{
    Point s,t;
    DIY angle;
    halfPlane(){}
    halfPlane(Point _s,Point _t):s(_s),t(_t){}
    halfPlane(DIY sx,DIY sy,DIY tx,DIY ty):s(sx,sy),t(tx,ty){}
    void GetAngle()
    {
        angle=atan2(t.y-s.y,t.x-s.x);
    }
} hp[N],q[N];

Point IntersectPoint(halfPlane P,halfPlane Q)
{
    DIY a1=CrossProduct(MakeVector(P.s,Q.t),MakeVector(P.s,Q.s));
    DIY a2=CrossProduct(MakeVector(P.t,Q.s),MakeVector(P.t,Q.t));
    return Point((P.s.x*a2+P.t.x*a1)/(a2+a1),(P.s.y*a2+P.t.y*a1)/(a2+a1));
}

bool cmp2(halfPlane P,halfPlane Q)
{
    if(fabs(P.angle-Q.angle)<EPS)
        return MultiCross(P.s,P.t,Q.s)>0;
    return P.angle<Q.angle;
}

bool IsParallel(halfPlane P,halfPlane Q)
{
    return fabs(CrossProduct(MakeVector(P.s,P.t),MakeVector(Q.s,Q.t)))<EPS;
}

void HalfPlaneIntersect(int n,int &m)
{
    sort(hp,hp+n,cmp2);
    int i,l=0,r=1;
    for(m=i=1; i<n; ++i)
        if(hp[i].angle-hp[i-1].angle>EPS) hp[m++]=hp[i];
    n=m; m=0;
    q[0]=hp[0];q[1]=hp[1];
    for(i=2; i<n; i++)
    {
        if(IsParallel(q[r],q[r-1])||IsParallel(q[l],q[l+1])) return;
        while(l<r&&MultiCross(hp[i].s,hp[i].t,IntersectPoint(q[r],q[r-1]))>0) --r;
        while(l<r&&MultiCross(hp[i].s,hp[i].t,IntersectPoint(q[l],q[l+1]))>0) ++l;
        q[++r]=hp[i];
    }
    while(l<r&&MultiCross(q[l].s,q[l].t,IntersectPoint(q[r],q[r-1]))>0) --r;
    while(l<r&&MultiCross(q[r].s,q[r].t,IntersectPoint(q[l],q[l+1]))>0) ++l;
    q[++r]=q[l];
    for(i=l; i<r; ++i)
        p[m++]=IntersectPoint(q[i],q[i+1]);
}

void Solve(Point *p1,Point *p2,Point *p3,int &top1,int &top2,int &top3,int &m)
{
    int i,j;
    Point a,b;
    Point O;
    O.x=O.y=0;
    int num=0;
    for(i=0;i<top1;i++)
    {
        hp[num]=halfPlane(p1[i],p1[(i+1)%top1]);
        hp[num].GetAngle();
        num++;
    }
    for(i=0;i<top2;i++)
    {
        hp[num]=halfPlane(p2[i],p2[(i+1)%top2]);
        hp[num].GetAngle();
        num++;
    }
    for(i=0;i<top3;i++)
    {
        hp[num]=halfPlane(p3[i],p3[(i+1)%top3]);
        hp[num].GetAngle();
        num++;
    }
    HalfPlaneIntersect(num,m);
    Area=0;
    p[m]=p[0];
    for(i=0;i<m;++i)
        Area+=cross(O,p[i],p[i+1]);
    if(Area<0) Area=-Area;
    Area/=2.0;
}

int main()
{
    int n,m,t,i,j,k;
    Point p1[N],p2[N],p3[N];
    top1=top2=top3=0;
    Point tmp[N];
    DIY x[N],y[N],z[N];
    while(cin>>n)
    {
        for(i=0; i<n; i++)
        {
            cin>>x[i]>>y[i]>>z[i];
        }
        for(i=0;i<3;i++)
            cin>>tmp[i].x>>tmp[i].y;
        for(i=0;i<n;i++)
        {
            p1[i].y=y[i]+(0-z[i])*(y[i]-tmp[0].y)/(z[i]-100);
            p1[i].x=x[i]+(0-z[i])*(x[i]-tmp[0].x)/(z[i]-100);
        }
        for(i=0;i<n;i++)
        {
            p2[i].y=y[i]+(0-z[i])*(y[i]-tmp[1].y)/(z[i]-100);
            p2[i].x=x[i]+(0-z[i])*(x[i]-tmp[1].x)/(z[i]-100);
        }
        for(i=0;i<n;i++)
        {
            p3[i].y=y[i]+(0-z[i])*(y[i]-tmp[2].y)/(z[i]-100);
            p3[i].x=x[i]+(0-z[i])*(x[i]-tmp[2].x)/(z[i]-100);
        }
        int k=0;
        Graham1(p1,n,top1);
        Graham2(p2,n,top2);
        Graham3(p3,n,top3);
        Solve(stack1,stack2,stack3,top1,top2,top3,m);
        printf("%.2lf\n",Area);
    }
    return 0;
}