三维坐标应用，三维坐标中点到直线的距离公式

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。

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);}}

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。