Find new house 四边形费马点-优快云博客

本文介绍了一种解决费马点问题的方法，通过编程实现寻找平面上一点到已知四点距离之和最短的位置。该问题在平面几何中有广泛应用。

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3rd 吉林省赛 E find new house 费马点

Edward had four friends living in X city, he was the only person among them who
live in Y city, five of them were very good friends, and he hoped that he would be
close to them. Now his wish can come true. His work is transferred to the X city, and
he also had the enough money to buy a house in the new city. But his friends hope
that the sum of distance from Edward¡¯s house to other four houses is the shortest
among all possible, so his friend would spend less time going to his house and
enjoying themselves. Can you help them?
Input
Each case consists of a line consists eight double numbers: x1, y1, x2, y2, x3, y3, x4,
y4, and xi, yi are the x- and y-coordinates of the i-th friend.
Output
For each case, output a single line containing the minimum sum of the distances from
his new house to other four friends. Keep two digits after the decimal point.
Sample Input
0 1 0 -1 1 0 -1 0
Sample Output
4.00
   一个平面几何题：求一点使得到平面中已知四点的距离之和最小。（所谓费马点）
   当四边形为凸四边形时，容易证明对角线交点即为费马点；当为凹四边形时，费马点为凹点。
   理论是这样，编程时也不用判断凹凸。代码见下

#include<stdio.h>
#include<cmath>
double _min(double a,double b){return a>b?b:a;}
struct Point
{
       double x,y;
};
int dblcmp(double d)
{
    if(fabs(d)<1e-8) return 0;
    return d>0?1:-1;
}
double det(double x1,double y1,double x2,double y2)
{
       return x1*y2-x2*y1;
}
double cross(Point a,Point b,Point c)
{
       return det(b.x-a.x,b.y-a.y,c.x-a.x,c.y-a.y);
}
int Segcross(Point a,Point b,Point c,Point d)
{
    return (dblcmp(cross(a,c,d))^dblcmp(cross(b,c,d)))==-2 &&
           (dblcmp(cross(c,a,b))^dblcmp(cross(d,a,b)))==-2;
}
double Seglen(Point a,Point b)
{
       return sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y));
}
int main()
{
    Point f1,f2,f3,f4;
    while(scanf("%lf%lf%lf%lf%lf%lf%lf%lf",&f1.x,&f1.y,&f2.x,&f2.y,&f3.x,&f3.y,&f4.x,&f4.y)!=EOF)
    {
           double ans=1<<30;
           if(Segcross(f1,f2,f3,f4))
           {
                 ans=_min(ans,Seglen(f1,f2)+Seglen(f3,f4));
           }
           if(Segcross(f1,f3,f2,f4))
           {
                 ans=_min(ans,Seglen(f1,f3)+Seglen(f2,f4));
           }
           if(Segcross(f1,f4,f2,f3))
           {
                 ans=_min(ans,Seglen(f1,f4)+Seglen(f2,f3));
           }
           ans=_min(ans,Seglen(f1,f2)+Seglen(f1,f3)+Seglen(f1,f4));
           ans=_min(ans,Seglen(f2,f1)+Seglen(f2,f3)+Seglen(f2,f4));
           ans=_min(ans,Seglen(f3,f1)+Seglen(f3,f2)+Seglen(f3,f4));
           ans=_min(ans,Seglen(f4,f1)+Seglen(f4,f2)+Seglen(f3,f4));
           printf("%.2lf/n",ans);
    }
}
     一句话：好好学数学。