暑假集训#2 div1 J 四点直角 J - Space Invader 四点共面+跨立实验

题意：给你四个点，判断能否先依次通过A,B两点，然后再在某个地方只进行一次直角转弯再一次经过C,D两点；

#include <iostream>
#include <cstdio>
#include <cmath>
#include <cstring>
#include <algorithm>
using namespace std;
typedef long long ll;
const double eps=1e-12;
struct Point{
   double x,y,z;
   void read(){
      scanf("%lf %lf %lf",&x,&y,&z);
   };
};

int dcmp(double a)
{
    if(fabs(a)<eps) return 0;
    return a>0?1:-1;
}

double Dot(Point a,Point b)
{
    return a.x*b.x+a.y*b.y+a.z*b.z;
}

Point cross(Point a,Point b)
{
     return Point{a.y*b.z-a.z*b.y,a.z*b.x-a.x*b.z,a.x*b.y-a.y*b.x};
}
Point operator-(Point a,Point b)
{
    return Point{a.x-b.x,a.y-b.y,a.z-b.z};
}
double dis(Point a,Point b)
{
    return sqrt((a.x-b.x)*(a.x-b.x)+(a.y-b.y)*(a.y-b.y)+(a.z-b.z)*(a.z-b.z));
}
int main()
{
    Point a,b,c,d;
    a.read();b.read();c.read();d.read();
    Point ab=b-a,cd=d-c,ca=a-c,cb=b-c,ad=d-a,ac=c-a;
    if(Dot(ab,cd)!=0||Dot(ab,cross(ad,cd))!=0) {printf("Invalid\n");return 0;}
    double x1=Dot(cross(cd,ca),cross(cd,cb)),
       x2=Dot(cross(ab,ac),cross(ab,ad));
    if(dcmp(x1)>=0&&dcmp(x2)>=0)
    {
        if(dis(b,c)>=dis(a,c)||dis(c,b)>=dis(d,b)) printf("Invalid\n");
        else printf("Valid\n");
    }
    else printf("Invalid\n");
    return 0;
}

分析：这道题涉及到了几个知识点：

1.判断四点是否共面

if(Dot(ab,cross(ad,cd))!=0)

代码如上，假设平面有A,B,C,D四点，那么先求出a,c,d三点确定的平面的一个法向量方向的向量，

然后再判断向量ab是否垂直该向量。

2.因为即使ab与cd垂直且共面，也不一定就满足条件，因为可能是ab与cd规范相交（不在端点处

相交），那么这个时候就需要进行跨立实验判断是否两条线段都只在对方的同一侧了

3.即使二满足了，也不一定是正确的，因为题目要求的是一次经过A,B,C,D四点，那么还要判断下

四个点的相对位置关系，不过因为已经确定了垂直关系，那么这个时候只需要比较下点之间的距离

就可以确定了

J - Space Invader

Time Limit:1000MS     Memory Limit:65536KB     64bit IO Format:%I64d & %I64u

Submit Status Practice URAL 2099

Description

Winter in Yekaterinburg is the longest time of the year. And everyone spends long winter evenings in his own way. For example, Eugene creates computer games in the evenings. His current game has a very simple game play — user controls a spaceship flying in 3D space. To test the mechanics of the movement of his spaceship Eugene wants to solve the following problem: can his spaceship, moving in a straight line, fly through the points A and B, then make a turn at 90°, and then fly through the points C and D, continuing to move in a straight line. The points A, B, C, D should be visited by the spaceship in just that order, the turning point may coincide with the point B or the point C. The spaceship should be considered as a material point.

Input

There are four lines, each containing integers x_i, y_i, z_i that are the coordinates of the points A, B, C, D respectively(−10 ⁶ ≤ x_i, y_i, z_i ≤ 10 ⁶) . All points are pairwise different.

Output

If the spaceship can fly through the data points, output “Valid”, otherwise output “Invalid”.

Sample Input

input	output
-2 0 0 -1 0 0 0 1 0 0 2 0	Valid