使用MFC的CDC类绘制三维坐标系及球面函数

系列链接

概述

本文使用MFC的CDC类绘制三维坐标系及球面函数。首先计算推导出三维坐标在二维平面显示的坐标变换方程（使用斜二测视图），使用球面的参数方程，然后定义图形缩放比例规模、坐标轴位移，变换坐标系和规模等，最后绘制坐标轴及球面函数。

如果对绘制二维坐标系还不太熟悉可以先看上面系列链接的：使用MFC的CDC类绘制二维坐标系及正余弦函数，本文对二维绘制及绘制函数部分不再赘述。因为二维坐标系的博文已经分模块讲解地比较清楚了，而与三维坐标系的基本思路相同，所以本文大部分直接使用注释讲解。

三维转二维的推导

若图片字体太小请右键图片在新标签页中打开图片

Transform3Dto2D

上图可知，只要使用Transform3Dto2D()函数，即可方便的把三维坐标转化为二维坐标（斜二测视图）。

球面参数方程

在三维空间直角坐标系中，以原点为球心、半径为 r 的球面的方程为 x^2 + y^2 + z^2 = r^2，其参数方程为

SphericalParameterEquation

新建项目

Visual Studio- 新建项目 - MFC应用程序 - 命名为GraphicsExercise3D - 确定 - 下一步 - 应用程序类型选择单个文档 - 完成

GraphicsExercise3DView.h

在GraphicsExercise3DView.h添加以下内容

// 操作
public:
    void SetScale(int scale);
    void SetTransformOrigin(float transformOriginX, float transformOriginY);
    void SetPlotSphere(float radius, float stepPhi, float stepTheta);
    void SetSlantRadian(float slant);

    float TransformScale(float num);
    float TransformOriginX(float x);
    float TransformOriginY(float y);
    float TransformOriginScaleX(float x);
    float TransformOriginScaleY(float y);
    void Transform3Dto2D(float &x, float &y, float z);

private:
    int scale;
    float radius, stepPhi, stepTheta, slant, transformOriginX, transformOriginY;

GraphicsExercise3DView.cpp

引入数学函数库

#include <math.h>

定义π

#ifndef PI
#define PI 3.14159
#endif // !PI

在构造函数初始化

CGraphicsExercise3DView::CGraphicsExercise3DView()
{
    // TODO: 在此处添加构造代码

    // 设置斜二测视图倾斜角度（弧度制）
    SetSlantRadian(PI / 4);

    // 设置规模比例
    SetScale(70);

    // 设置坐标系在x、y方向的位移（不改变规模情况下，即移动像素）
    SetTransformOrigin(300, 350);

    // 设置球面半径radius、取样步长stepPhi、stepTheta
    SetPlotSphere(2.0, 0.01, 0.1);
}

设置初始化参数的Set函数

// 设置规模
void CGraphicsExercise3DView::SetScale(int scale)
{
    this->scale = scale;
}

// 设置坐标系原点在x、y方向的位移（不改变规模情况下，即移动像素）
void CGraphicsExercise3DView::SetTransformOrigin(float transformOriginX, float transformOriginY)
{
    this->transformOriginX = transformOriginX;
    this->transformOriginY = transformOriginY;
}

// 设置球面半径radius、取样步长stepPhi、stepTheta
void CGraphicsExercise3DView::SetPlotSphere(float radius, float stepPhi, float stepTheta)
{
    this->radius = radius;
    this->stepPhi = stepPhi;
    this->stepTheta = stepTheta;
}

// 设置斜二测视图的倾斜角（单位弧度）
void CGraphicsExercise3DView::SetSlantRadian(float slant)
{
    this->slant = slant;
}

坐标及规模变换

// 变换规模
float CGraphicsExercise3DView::TransformScale(float num)
{
    return num * scale;
}

// 坐标系X轴方向位移
float CGraphicsExercise3DView::TransformOriginX(float x)
{
    return x + transformOriginX / scale;
}

// 坐标系y轴方向位移
float CGraphicsExercise3DView::TransformOriginY(float y)
{
    return y - transformOriginY / scale;
}

// 变换坐标系X和规模
float CGraphicsExercise3DView::TransformOriginScaleX(float x)
{
    return TransformScale(TransformOriginX(x));
}

// 变换坐标系Y和规模
float CGraphicsExercise3DView::TransformOriginScaleY(float y)
{
    return -TransformScale(TransformOriginY(y));
}

三维坐标转化为二维坐标

// 使用斜二测视图，把三维坐标点转化为二维平面上的点
void CGraphicsExercise3DView::Transform3Dto2D(float &x, float &y, float z)
{
    x = x - (z * cos(slant)) / 2;
    y = y - (z * sin(slant)) / 2;
}

绘制坐标轴及函数图形

// CGraphicsExercise2View 绘制

void CGraphicsExercise3DView::OnDraw(CDC* pDC)
{
    CGraphicsExercise3DDoc* pDoc = GetDocument();
    ASSERT_VALID(pDoc);
    if (!pDoc)
        return;

    // TODO: 在此处为本机数据添加绘制代码

    float x, y, z;

    // -------------------- 绘制坐标系 -------------------------

    // 坐标x轴
    pDC->MoveTo(TransformOriginScaleX(0), TransformOriginScaleY(0));
    pDC->LineTo(TransformOriginScaleX(radius + 2), TransformOriginScaleY(0));

    // 坐标y轴
    pDC->MoveTo(TransformOriginScaleX(0), TransformOriginScaleY(0));
    pDC->LineTo(TransformOriginScaleX(0), TransformOriginScaleY(radius + 2));

    // 坐标z轴
    x = 0, y = 0;
    Transform3Dto2D(x, y, radius + 5);
    pDC->MoveTo(TransformOriginScaleX(0), TransformOriginScaleY(0));
    pDC->LineTo(TransformOriginScaleX(x), TransformOriginScaleY(y));

    // 坐标x轴的箭头
    pDC->MoveTo(TransformOriginScaleX(radius + 1.8), TransformOriginScaleY(0.2));
    pDC->LineTo(TransformOriginScaleX(radius + 2), TransformOriginScaleY(0));
    pDC->LineTo(TransformOriginScaleX(radius + 1.8), TransformOriginScaleY(-0.2));

    // 坐标y轴的箭头
    pDC->MoveTo(TransformOriginScaleX(-0.2), TransformOriginScaleY(radius + 1.8));
    pDC->LineTo(TransformOriginScaleX(0), TransformOriginScaleY(radius + 2));
    pDC->LineTo(TransformOriginScaleX(0.2), TransformOriginScaleY(radius + 1.8));

    // 坐标z轴的箭头
    x = 0, y = 0.2;
    Transform3Dto2D(x, y, radius + 5 - 0.2);
    pDC->MoveTo(TransformOriginScaleX(x), TransformOriginScaleY(y));
    x = 0, y = 0;
    Transform3Dto2D(x, y, radius + 5);
    pDC->LineTo(TransformOriginScaleX(x), TransformOriginScaleY(y));
    x = 0.2, y = 0;
    Transform3Dto2D(x, y, radius + 5 - 0.2);
    pDC->LineTo(TransformOriginScaleX(x), TransformOriginScaleY(y));

    // -------------------- 绘制刻度线 -------------------------

    // 绘制x轴刻度线
    for (float scaleX = 0.2; scaleX < radius + 1; scaleX += 0.2)
    {
        pDC->MoveTo((int)TransformOriginScaleX(scaleX), (int)TransformOriginScaleY(0));
        pDC->LineTo((int)TransformOriginScaleX(scaleX), (int)TransformOriginScaleY(0.1));
    }

    // 绘制y轴刻度线
    for (float scaleY = 0.2; scaleY <= radius + 1; scaleY += 0.2)
    {
        pDC->MoveTo((int)TransformOriginScaleX(0), (int)TransformOriginScaleY(scaleY));
        pDC->LineTo((int)TransformOriginScaleX(0.1), (int)TransformOriginScaleY(scaleY));
    }

    // 绘制z轴刻度线
    for (float x = 0, y = 0, scaleZ = 0.2; scaleZ <= radius + 4; scaleZ += 0.2, x = 0, y = 0)
    {
        Transform3Dto2D(x, y, scaleZ);
        pDC->MoveTo((int)TransformOriginScaleX(x), (int)TransformOriginScaleY(y));
        pDC->LineTo((int)TransformOriginScaleX(x + 0.1), (int)TransformOriginScaleY(y));
    }


    // -------------------- 绘制文字 -------------------------

    // 绘制x轴的x
    pDC->TextOutW(TransformOriginScaleX(radius + 1.6), TransformOriginScaleY(-0.2), CString("x"));
    // 绘制y轴的y
    pDC->TextOutW(TransformOriginScaleX(-0.2), TransformOriginScaleY(radius + 1.6), CString("y"));
    // 绘制z轴的z
    x = 0.2, y = 0;
    Transform3Dto2D(x, y, radius + 5 - 0.4);
    pDC->TextOutW(TransformOriginScaleX(x), TransformOriginScaleY(y), CString("z"));

    CString s;
    // 绘制x轴刻度文字
    for (float ScaleTextX = 0.4; ScaleTextX < radius + 1; ScaleTextX += 0.4)
    {
        s.Format(_T("%.1f"), ScaleTextX);
        pDC->TextOutW(TransformOriginScaleX(ScaleTextX - 0.1), TransformOriginScaleY(-0.1), s);
    }

    // 绘制y轴刻度文字
    for (float ScaleTextY = 0.4; ScaleTextY <= radius + 1; ScaleTextY += 0.4)
    {
        s.Format(_T("%.1f"), ScaleTextY);
        pDC->TextOutW(TransformOriginScaleX(-0.4), TransformOriginScaleY(ScaleTextY + 0.1), s);
    }

    // 绘制z轴刻度文字
    for (float ScaleTextZ = 0.6; ScaleTextZ <= radius + 4; ScaleTextZ += 0.6)
    {
        s.Format(_T("%.1f"), ScaleTextZ);
        x = 0, y = 0;
        Transform3Dto2D(x, y, ScaleTextZ);
        pDC->TextOutW(TransformOriginScaleX(x + 0.15), TransformOriginScaleY(y + 0.12), s);
    }

    // 绘制函数图的Title
    x = 0, y = 0;
    Transform3Dto2D(x, y, radius + 5);
    pDC->TextOutW(TransformOriginScaleX(x + 3), TransformOriginScaleY(y), CString("x^2 + y^2 + z^2 = r^2"));

    // -------------------- 绘制函数 -------------------------

    // 球面
    float phi, theta;
    for (phi = 0; phi < 2 * PI; phi += stepPhi)
    {
        for (theta = 0; theta < PI; theta += stepTheta)
        {
            x = radius * sin(phi) * cos(theta);
            y = radius * sin(phi) * sin(theta);
            z = radius * cos(phi);

            Transform3Dto2D(x, y, z);

            srand(z);
            pDC->SetPixel(TransformOriginScaleX(x), TransformOriginScaleY(y), RGB(rand() % 255, rand() % 255, rand() % 255));
        }
    }


    //// 三棱锥（测试用）
    //x = 1, y = 0, z = 0;
    //Transform3Dto2D(x, y, z);
    //pDC->MoveTo((int)TransformOriginScaleX(x), (int)TransformOriginScaleY(y));

    //x = 0, y = 1, z = 0;
    //Transform3Dto2D(x, y, z);
    //pDC->LineTo((int)TransformOriginScaleX(x), (int)TransformOriginScaleY(y));

    //x = 0, y = 0, z = 1;
    //Transform3Dto2D(x, y, z);
    //pDC->LineTo((int)TransformOriginScaleX(x), (int)TransformOriginScaleY(y));

    //x = 1, y = 0, z = 0;
    //Transform3Dto2D(x, y, z);
    //pDC->LineTo((int)TransformOriginScaleX(x), (int)TransformOriginScaleY(y));

}

效果图

GraphicsExercise3DCapture