【计算几何】-poj1070-Deformed Wheel-附测试数据

题意

给你一个斜坡（从左往右，斜率越来越大），再给你一个多边形，再给你一个重心，然后问你最后这个多边形的重心会在哪里，假设地面摩擦力无穷大，且运动过程中重心必须时刻保持向下。

题解

我们先考虑一下简单的情况就是多边形往左边滚(即逆时针旋转)

设支撑点为q，重心为g，旋转角度为k

那么我们得满足什么条件呢

cos(angle(g-q))<=0

如何确定选择的角度呢？二分就行了，判断条件就是

1.我们把这个多边形围绕q点选择后，所有的点都在斜面的上面

2.cos(angle(g-q)+k)<=0

这样好像就行了

但是要注意一些坑点的就是可能会往右边滚，所以分一些情况就行了

代码

#include <iostream>
#include <vector>
#include <cmath>
#include <algorithm>
#include <cstdio>

using namespace std;
const double eps=1e-10;
const double eps2=1e-13;
const double PI=acos(-1.0);

struct Point{
    double x,y;
    Point(double x=0,double y=0):x(x),y(y){}
};
typedef Point Vector;

Vector operator -(Point a,Point b){
    return Vector(a.x-b.x,a.y-b.y);
}
Vector operator +(Point a,Point b){
    return Vector(a.x+b.x,a.y+b.y);
}
Vector operator *(Vector a,double p){
    return Vector(a.x*p,a.y*p);
}
Vector operator /(Vector a,double p){
    return Vector(a.x/p,a.y/p);
}
bool operator <(const Point& a,const Point& b){
    return a.x<b.x||(a.x==b.x&&a.y<b.y);//在有精度需求，比如使用lower_bound的时候，加上dcmp()
}
int dcmp(double x){
    if(fabs(x)<eps)return 0;
    if(x<0)return -1;
    return 1;
}
bool operator ==(const Point& a,const Point& b){
    return dcmp(a.x-b.x)==0&&dcmp(a.y-b.y)==0;
}
double Dot(Vector a,Vector b){
    return a.x*b.x+a.y*b.y;
}//点积
double Cross(Vector a,Vector b){
    return a.x*b.y-a.y*b.x;
}//叉积
double Length(Vector a){
    return sqrt(Dot(a,a));
}//长度
//返回逆时针旋转90度的单位法向量；
Vector Normal(Vector a){
    double l=Length(a);
    return Vector(-a.y/l,a.x/l);
}
//返回向量夹角，无方向
double Angle(Vector a,Vector b){
    return acos(Dot(a,b)/Length(a)/Length(b));
}
//逆时针旋转向量
Vector Rotate(Vector a,double rad){
    return Vector(a.x*cos(rad)-a.y*sin(rad),a.x*sin(rad)+a.y*cos(rad));
}
//求p+v*t与q+w*t的交点，使用时确保Cross(v,w)不等于0
Point GetlineIntersection(Point p,Vector v,Point q,Vector w){
    Vector u=p-q;
    double t=Cross(w,u)/Cross(v,w);
    return p+v*t;
}
//求p到直线ab的距离
double DistanceToline(Point p,Point a,Point b){
    Vector v1=p-a,v2=b-a;
    return fabs(Cross(v1,v2)/Length(v2));
}
//求p到线段ab的距离
double DistanceToSegment(Point p,Point a,Point b){
    if(a==b)return Length(p-a);
    Vector v1=b-a,v2=p-a,v3=p-b;
    if(dcmp(Dot(v1,v2)<0))return Length(p-a);
    else if(dcmp(Dot(v1,v3))>0)return Length(p-b);
    else return fabs(Cross(v1,v2)/Length(v1));
}
//线段a1a2与线段b1b2规范相交返回真
bool SegmenProperIntersection(Point a1,Point a2,Point b1,Point b2){
    double c1=Cross(a2-a1,b1-a1),c2=Cross(a2-a1,b2-a1);
    double c3=Cross(b2-b1,a1-b1),c4=Cross(b2-b1,a2-b1);
    return dcmp(c1)*dcmp(c2)<0&&dcmp(c3)*dcmp(c4)<0;
}
//点p在线段a1a2上返回真
bool OnSegment(Point p,Point a1,Point a2){
    return dcmp(Cross(a1-p,a2-p))==0&&dcmp(Dot(a1-p,a2-p))<0;
}
//点p在ab上的投影
Point GetLineProjection(Point P,Point A,Point B)
{
    Vector v=B-A;
    return A+v*(Dot(v,P-A)/Dot(v,v));
}
//与 x 轴的夹角，取值范围为 (-π，π]
double angle(Vector v){
    return atan2(v.y,v.x);
}
//求线段a1,a2到线段b1,b2的最短距离
double disSegmenttoSegment(Point a1,Point a2,Point b1,Point b2)
{
    double ans=DistanceToSegment(a1,b1,b2);
    ans=min(ans,DistanceToSegment(a2,b1,b2));
    ans=min(ans,DistanceToSegment(b1,a1,a2));
    ans=min(ans,DistanceToSegment(b2,a1,a2));
    return ans;
}
//返回单位向量；
Vector normal(Vector a){
    double l=Length(a);
    return Vector(a.x/l,a.y/l);
}
int ConvecHull(Point* p,int n,Point* ch)
{
    sort(p,p+n);
    int m=0;
    for(int i=0;i<n;i++){
        while(m>1&&dcmp(Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2])<=0))m--;//注意<=与<的区别
        ch[m++]=p[i];
    }
    int k=m;
    for(int i=n-2;i>=0;i--){
        while(m>k&&dcmp(Cross(ch[m-1]-ch[m-2],p[i]-ch[m-2])<=0))m--;//注意<=与<的区别
        ch[m++]=p[i];
    }
    if(n>1)m--;
    return m;
}

int n,m;
Point p[20],ch[20],g,s;
Point seg[1000],k;
Point r[1000];
Point q,nq;
int pos,np,rnum,cnt;
double getangle()
{
    double l=0,rf=PI,mf;
    Point ro[20],rg;
    if(cos(angle(g-q))>=0)return 0;
    while((rf-l)>eps2){
        mf=(l+rf)/2;
        for(int i=0;i<m;i++){
            ro[i]=Rotate(ch[i]-q,mf)+q;
        }
        if(cos(angle(g-q)+mf)>=0){
            rf=mf;
            continue;
        }
        bool flag=false;
        for(int i=0;i<m;i++){
            if(ch[i]==q)continue;
            for(int j=0;j<cnt;j++){
                if(Cross(r[j+1]-r[j],ro[i]-r[j])<0){
                    rf=mf;
                    flag=true;
                    break;
                }
            }
            if(flag)break;
        }
        if(!flag){
            l=mf;
        }
    }
    return mf;
}
double getangle2()
{
    double l=0,rf=-PI,mf;
    Point ro[20],rg;
    if(cos(angle(g-q))<=0)return 0;
    while((l-rf)>eps2){
        mf=(l+rf)/2;
        for(int i=0;i<m;i++){
            ro[i]=Rotate(ch[i]-q,mf)+q;
        }
        if(cos(angle(g-q)+mf)<=0){
            rf=mf;
            continue;
        }
        bool flag=false;
        for(int i=0;i<m;i++){
            if(ch[i]==q)continue;
            for(int j=0;j<cnt;j++){
                if(Cross(r[j+1]-r[j],ro[i]-r[j])<0){
                    rf=mf;
                    flag=true;
                    break;
                }
            }
            if(flag)break;
        }
        if(!flag){
            l=mf;
        }
    }
    return mf;
}
int main()
{
    int t;
    cin>>t;
    while(t--){
        cin>>n;
        for(int i=0;i<n;i++)cin>>p[i].x>>p[i].y;
        cin>>g.x>>g.y;
        s=g;
        cnt=0;
        while(cin>>seg[cnt].x>>seg[cnt].y){
            cnt++;
            if(dcmp(seg[cnt-1].y)==0)break;
        }
        cin>>k.x>>k.y;
        m=ConvecHull(p,n,ch);
        r[cnt]=k;
        for(int i=0;i<cnt;i++){
            Vector v=normal(Vector(1,seg[i].y));
            v=v*seg[i].x;
            r[cnt-i-1]=r[cnt-i]-v;
        }
        q.x=999999999;
        for(int i=0;i<m;i++){
            bool flag=false;
            for(int j=0;j<cnt;j++){
                if(dcmp(DistanceToSegment(ch[i],r[j],r[j+1]))==0){
                    if(ch[i].x<q.x)q=ch[i];
                }
            }
        }
        double ag1=getangle();//左边
        double ag2=getangle2();//右边
        bool f;
        if(fabs(ag1)>fabs(ag2)){
            f=true;
        }
        else f=false;
        while(true){
            if(f)q.x=999999999;
            else q.x=-999999999;
            for(int i=0;i<m;i++){
                bool flag=false;
                for(int j=0;j<cnt;j++){
                    if(dcmp(DistanceToSegment(ch[i],r[j],r[j+1]))==0){
                        if(f&&ch[i].x<q.x)q=ch[i];
                        if(!f&&ch[i].x>q.x)q=ch[i];
                    }
                }
            }
            double ag;
            if(f)ag=getangle();
            else ag=getangle2();
            Point o=q;
            for(int i=0;i<m;i++){
                ch[i]=Rotate(ch[i]-o,ag)+o;
            }
            g=Rotate(g-o,ag)+o;
            if(dcmp(ag)==0)break;

        }
        printf("%.3f %.3f\n",g.x,g.y);
    }
    return 0;
}
/*
2
4
40 30
30 37
24 30
30 26
27 29
30 1
100 0
40 30
4
0 0
10 10
0 20
-10 10
10 10
100 0
50 0

1
4
0 0
10 10
0 20
-10 10
10 10
100 0
50 0
*/