1. 直线参数方程
   如果是在二维坐标系下，已知两个点p1(x1, y1), p2(x2, y2), 很容易求得两点之间的斜率，然后使用y = kx + b计算出k，b值得到直线方程。同理在三维空间下仍然可以使用直线参数方程。
   x = x1 + dx * t,
   y = y1 + dy * t,
   z = z1 + dz * t

   例如：
   已知p1(x1, y1, z1)， p2(x2, y2, z2)个点，可以求得p1p2两个点的方向向量d。
   d = p2 – p1 = ((x2 – x1), (y2 – y1), (z2 – z1))。
   可以求得t = |d| / √d。

2. c++代码求解方程
   a.首先定义点的结构

   struct Points3D
   { 
   	float x;
   	float y;
   	float z;
   };
   

   b . 计算两个点的方向向量

   	void LinearEquation::GetDirection(float * Direction)
   	{ 
   		
   		Direction[0] = m_p2.x - m_p1.x;
   		Direction[1] = m_p2.y - m_p1.y;
   		Direction[2] = m_p2.z - m_p1.z;
   	}
   

   c.然后计算向量模长

   float GetLength(float* Matrix)
   { 
   	float x = pow(Matrix[0], 2);
   	float y = pow(Matrix[1], 2);
   	float z = pow(Matrix[2], 2);
   
   	return sqrt(x + y + z);
   }
   

   d.得到参数t以及dx，dy，dz，主要代码如下

   void GetEquationParameters(float& dx, float& dy, float& dz, float& t)
   { 
   	float Direction[3];
   	GetDirection(Direction); ///得到两点的方向向量
   
   	float L_D = GetLength(Direction); //得到方向向量的模长
   
   	float std_D[3];
   	memcpy(std_D, Direction, sizeof(Direction));
   	GetNormVector(std_D);  //得到单位向量
   
   	dx = std_D[0];
   	dy = std_D[1];
   	dz = std_D[2];
   
   	float L_std_D = GetLength(std_D);
   
   	t = L_D / L_std_D;  //t = |d| / √d;
   }
   

   e. 得到两点之间的所有点，interval代表采点的间隔。

   void GetLinearPoints(float interval, std::vector<Points3D>& points)
   		{ 
   			float dx, dy, dz, t;
   			GetEquationParameters(dx, dy, dz, t);
   		
   			
   			for (int i = 0; i < int(m_NumPoints / interval); i++){ 
   				float x = m_p1.x + dx * t * interval * i;
   				float y = m_p1.y + dy * t * interval * i;
   				float z = m_p1.z + dz * t * interval * i;
   		
   				float dis = sqrt(pow(m_p2.x - x, 2) + pow(m_p2.y - y, 2) + pow(m_p2.z - z, 2));
   				if (dis < 1.0)
   					break;
   		
   				Points3D pt;
   				pt.x = x;
   				pt.y = y;
   				pt.z = z;
   				points.push_back(pt);
   		
   			}
   		
   		}
   

3. 使用VTK进行点绘制显示

   #ifndef INITIAL_OPENGL
   #define INITIAL_OPENGL
   
   #include <vtkAutoInit.h>
   VTK_MODULE_INIT(vtkRenderingOpenGL2)
   VTK_MODULE_INIT(vtkInteractionStyle)
   #endif //不加这5行会报错
   
   
   #include "vtkPoints.h"
   #include "vtkSmartPointer.h"
   #include "vtkActor.h"
   #include "vtkCellArray.h"
   #include "vtkPolyData.h"
   #include "vtkPolyDataMapper.h"
   #include "vtkRenderer.h"
   #include "vtkRenderWindow.h"
   #include "vtkRenderWindowInteractor.h"
   #include "vtkProperty.h"
   
   void VtkDrawPoints(vector<Points3D> pt)
   { 
   	vtkSmartPointer<vtkPoints> points = vtkSmartPointer<vtkPoints>::New();
   	vtkSmartPointer<vtkCellArray> vertices = vtkSmartPointer<vtkCellArray>::New();
   	vtkSmartPointer<vtkPolyData> polydata = vtkSmartPointer<vtkPolyData>::New();
   	vtkSmartPointer<vtkPolyDataMapper> mapper = vtkSmartPointer<vtkPolyDataMapper>::New();
   	vtkSmartPointer<vtkActor> actor = vtkSmartPointer<vtkActor>::New();
   	vtkSmartPointer<vtkRenderer> render = vtkSmartPointer<vtkRenderer>::New();
   	vtkSmartPointer<vtkRenderWindow> window = vtkSmartPointer<vtkRenderWindow>::New();
   	vtkSmartPointer<vtkRenderWindowInteractor> win_render = vtkSmartPointer<vtkRenderWindowInteractor>::New();
   
   	int Num = pt.size();
   
   	for (int i = 0; i < Num; i++) { 
   		vtkIdType id = points->InsertNextPoint(pt[i].x, pt[i].y, pt[i].z);
   		vertices->InsertNextCell(1);
   		vertices->InsertCellPoint(id);
   	}
   
   	
   	polydata->SetPoints(points);
   	polydata->SetVerts(vertices);
   	
   	
   	mapper->SetInputData(polydata);
   	
   	actor->SetMapper(mapper);
   	actor->GetProperty()->SetColor(255, 0, 0);
   	actor->GetProperty()->SetPointSize(2);
   
   	render->AddActor(actor);
   	render->SetBackground(0, 0, 0);
   
   	window->AddRenderer(render);
   	window->SetSize(600, 600);
   
   	win_render->SetRenderWindow(window);
   	win_render->Initialize();
   	win_render->Start();
   }
   

完整代码可以访问此链接c++求空间直线方程并使用vtk进行绘制显示。