三维空间两条直线的最短距离、最近点及C++算法实现

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在双目视觉立体空间重建中，会根据两个相机中的物体图像坐标，求取给定三维坐标系的三维坐标，而可以根据物体

图像坐标、相机内参、给定坐标系的相机外参，求取相机光轴线的方程，从而实现立体重建，内外参、直线方程请执行

搜索学习，本文主要是解决在已知空间两直线求最短距离及最近点。如有疑问或错误，请与评论区指出，谢谢。

一、理论：

已知空间中两线段，如果它们无限变粗，判断是否相交。（主要讨论不在同一平面的情况）

线段AB，线段CD

问题的关键是求出这两条任意直线之间的最短距离，以及在这个距离上的两线最接近点坐标，判断该点是否在线段AB

和线段CD上。

首先将直线方程化为对称式，得到其方向向量n1=（a1,b1,c1),n2=(a2,b2,c2).

再将两向量叉乘得到其公垂向量N=（x,y,z）,在两直线上分别选取点A,B(任意)，得到向量AB，

求向量AB在向量N方向的投影即为两异面直线间的距离了（就是最短距离了）。

最短距离的求法：d=|向量N*向量AB|/|向量N|（上面是两向量的数量积，下面是取模）。

设交点为C,D，带入公垂线N的对称式中，又因为C,D两点分别满足一开始的直线方程，所以得到关于C（或D）的两个

连等方程，分别解出来就好了！

二、方程式

直线1：(平面式)

面e:A_1e*x+B_1e*y+C_1e*z+D_1e=0;

面f:A_1f*x+B_1f*y+C_1f*z+D_1f=0;

 

直线2：（平面式）

面e:A_2e*x+B_2e*y+C_2e*z+D_2e=0;

面f:A_2f*x+B_2f*y+C_2f*z+D_2f=0;

 

直线1的方向向量：（direction vector）

DV_1=(A_1e,B_1e,C_1e)X(A_1f,B_1f,C_1f)

=(B_1e*C_1f-B_1f*C_1e, C_1f*A_1e-A_1e*C_1f,A_1e*B_1f-A_1f*B_1e);

DV_2=(B_2e*C_2f-B_2f*C_2e, C_2f*A_2e-A_2e*C_2f,A_2e*B_2f-A_2f*B_2e);

 

在直线1、2上任取一点R1，R2

领Z_R1=1,Z_R2=1;

直线1带入Z_R 1有：

面e:A_1e*x+B_1e*y+C_1e*Z_R1+D_1e=0;

面f:A_1f*x+B_1f*y+C_1f*Z_R1+D_1f=0;
即：



P_R1=(X_R1,Y_R1,Z_R1);

 

同理：



P_R2=(X_R2,Y_R2,Z_R2);

 

两条直线的公垂线：（public vertical vector）

DV_PVV=(DV_1) X( DV_2);

在两条直线任意选的点的方向向量：

DV_RP=(P_R1-P_R2);

 

任选直线的方向向量与公垂线向量的数量积：即此直线在公垂线上的投影(d=|n*AB|/|N|):



当DIS小于某个给定阈值时，求最近点。

直线1的对称性方程：



令公垂线与直线1交于点m,交线与点n;

有



直线2：



并且把m、n带入公垂线方程，有：



把用公垂线表示的点m和n,分别带入直线1和2的对称性方程,消除点m和n, 有：

直线1:



直线2：



上述两式相减：



 

求出t_1m,t_2n,则m和n



最近点：



#include<opencv2/core/core.hpp>
using namespace cv;
Point3d two_space_linear_collision_get_nearest_point(double A_0e, double B_0e, double C_0e, double D_0e, double A_0f, double B_0f, double C_0f, double D_0f,
double A_1e, double B_1e, double C_1e, double D_1e, double A_1f, double B_1f, double C_1f, double D_1f)
{
//直线0的A面的D,A,B,C
double line0_flatE_d = D_0e;

Mat line0_flatE = (Mat_<double>(1, 3) << A_0e, B_0e, C_0e);

//直线0的B面的D,A,B,C
double line0_flatF_d = D_0f;
Mat line0_flatF = (Mat_<double>(1, 3) << A_0f, B_0f, C_0f);

//直线1的A面的D,A,B,C
double line1_flatE_d = D_1e;
Mat line1_flatE = (Mat_<double>(1, 3) << A_1e, B_1e, C_1e);

//直线1的B面的D,A,B,C
double line1_flatF_d = D_1f;
Mat line1_flatF = (Mat_<double>(1, 3) << A_1f, B_1f, C_1f);

//两条直线0，1的direction vector（raws = 1, cols = 3）
Mat direction_line0 = line0_flatE.cross(line0_flatF);

Mat direction_line1 = line1_flatE.cross(line1_flatF);

//line0的randpoint
Point3d randPoint0(0.0, 0.0, 0.0);
randPoint0.z = 800;
Mat randPoint0_in = (Mat_<double>(2, 2) << line0_flatE.at<double>(0, 0), line0_flatE.at<double>(0, 1),
line0_flatF.at<double>(0, 0), line0_flatF.at<double>(0, 1));

Mat randPoint0_out = randPoint0_in.inv()*(Mat_<double>(2, 1) << -line0_flatE_d - randPoint0.z*line0_flatE.at<double>(0, 2),
-line0_flatF_d - randPoint0.z*line0_flatF.at<double>(0, 2));
randPoint0.x = randPoint0_out.at<double>(0, 0);
randPoint0.y = randPoint0_out.at<double>(1, 0);

//line1上randpoint
Point3d randPoint1(0.0, 0.0, 0.0);
randPoint1.z = 800;
Mat randPoint1_in = (Mat_<double>(2, 2) << line1_flatE.at<double>(0, 0), line1_flatE.at<double>(0, 1),
line1_flatF.at<double>(0, 0), line1_flatF.at<double>(0, 1));

Mat randPoint1_out = randPoint1_in.inv()*(Mat_<double>(2, 1) << -line1_flatE_d - randPoint1.z*line1_flatE.at<double>(0, 2),
-line1_flatF_d - randPoint1.z*line1_flatF.at<double>(0, 2));
randPoint1.x = randPoint1_out.at<double>(0, 0);
randPoint1.y = randPoint1_out.at<double>(1, 0);

//Public vertical vector(rows=1,cols=3)
Mat publicVerticalRay = direction_line0.cross(direction_line1);

//任选点的方向向量
Mat randRay = (Mat_<double>(3, 1) << randPoint0.x - randPoint1.x, randPoint0.y - randPoint1.y, randPoint0.z - randPoint1.z);

Mat cross_rand_mult = publicVerticalRay*randRay;

//公垂线的模
double public_vertical_vector_length = sqrtf(publicVerticalRay.at<double>(0, 0)*publicVerticalRay.at<double>(0, 0)
+ publicVerticalRay.at<double>(0, 1)*publicVerticalRay.at<double>(0, 1)
+ publicVerticalRay.at<double>(0, 2)*publicVerticalRay.at<double>(0, 2));

//两直线的最小距离d=|公向量*任选向量|/|公向量|
double minDistance = std::abs(cross_rand_mult.at<double>(0, 0)) / public_vertical_vector_length;

Point3d point_3d(0.0,0.0,0.0);

if (minDistance<50)
{

Mat two_line_t_in = (Mat_<double>(3, 3) << publicVerticalRay.at<double>(0, 0), -direction_line0.at<double>(0, 0), direction_line1.at<double>(0, 0),
publicVerticalRay.at<double>(0, 1), -direction_line0.at<double>(0, 1), direction_line1.at<double>(0, 1),
publicVerticalRay.at<double>(0, 2), -direction_line0.at<double>(0, 2), direction_line1.at<double>(0, 2));

Mat two_line_t = two_line_t_in.inv()*randRay;

Point3d node0, node1;

node0.x = two_line_t.at<double>(1, 0)*direction_line0.at<double>(0, 0) + randPoint0.x;
node0.y = two_line_t.at<double>(1, 0)*direction_line0.at<double>(0, 1) + randPoint0.y;
node0.z = two_line_t.at<double>(1, 0)*direction_line0.at<double>(0, 2) + randPoint0.z;

node1.x = two_line_t.at<double>(2, 0)*direction_line1.at<double>(0, 0) + randPoint1.x;
node1.y = two_line_t.at<double>(2, 0)*direction_line1.at<double>(0, 1) + randPoint1.y;
node1.z = two_line_t.at<double>(2, 0)*direction_line1.at<double>(0, 2) + randPoint1.z;

point_3d = (node0 + node1) / 2;

}
else
{
point_3d = Point3d(0.0,0.0,0.0);
}

return point_3d;

}

Have a good day~

注：编写比较匆忙，如有偏差请谅解，未经许可请勿转载