【计算机图形学】曲线和曲面

1题

//

// 程序名称：Bezier曲线

// 功    能：绘制Bezier曲线，并采用自行设计输入和交互修改数据点的方式。

// 编译环境：VS2019，EasyX_20220116

// 最后修改：2022-4-28

#include <graphics.h>

#include <conio.h>

#include <iostream>

#include <math.h>

using namespace std;

//特征多边形的特征

int n;

int px[100], py[100];

//初始化特征多边形

void initialize() {

    cout << "please input the number of your points (no more than 100)" << endl;

    cin >> n;

    cout << "please input each coordinate of your points in order" << endl;

    for (int i = 0; i < n; i++) {

         cin >> px[i] >> py[i];

    }

    cout << "successfully set" << endl;

}文章来源地址https://www.toymoban.com/news/detail-456478.html

//递归

int Factorial(int nn) {

    if (nn == 0 || nn == 1) return 1;

    else return nn * Factorial(nn - 1);

}

//绘制bezier曲线

void Bezier() {

    int nn = n - 1;

    //line(px[0], py[0], px[1], py[1]);

    double i = 0, n1, n2;

    n1 = Factorial(nn);

    moveto(px[0], py[0]);

    POINT point;

    //在[0,1]内循环

    for (; i <= 1; i += 0.01) {

         point.x = 0;

         point.y = 0;

         for (int j = 0; j <= nn; j++) {

             n2 = (n1 / (Factorial(j) * Factorial(nn - j))) * pow(i, j) * pow(1 - i, nn - j);

             point.x += n2 * px[j];

             point.y += n2 * py[j];

         }

         lineto(point.x, point.y);

    }

    lineto(px[n - 1], py[n - 1]);

    //描绘用户定义的点

    for (int ii = 0; ii < n; ii++) {

         setfillcolor(GREEN);

         fillcircle(px[ii], py[ii], 3);

    }

}

//主函数

int main() {

    //初始化

    initialize();

    initgraph(640, 480);

    ExMessage m;

    //绘制图像

    while (1) {

         m = getmessage(EX_MOUSE | EX_KEY);

         switch (m.message) {

             //右键修改坐标信息

             case WM_RBUTTONDOWN:

                  closegraph();

                  int arr, newx, newy;

                  //输入数组中的坐标，进行修改

                  cout << "which point do you want to change? please input its array number" << endl;

                  while (1) {

                      cin >> arr;

                      if (arr >= n) {

                          cout << "input error, try again" << endl;

                      }

                      else {

                          break;

                      }

                  }

                  cout << "please input the reset coordinate" << endl;

                  cin >> newx >> newy;

                  px[arr] = newx;

                  py[arr] = newy;

                  cout << "successfully reset, please wait a second to see a new graph" << endl;

                  initgraph(640, 480);

             //左键绘制

             case WM_LBUTTONDOWN:

                  Bezier();

             //退出图形化界面

             case WM_KEYDOWN:

                  if (m.vkcode == VK_ESCAPE) {

                      return 0;

                  }

         }

    }

    _getch();

    closegraph();

    return 0;

}

2题

//

// 程序名称：Bezier曲面

// 功    能：绘制Bezier曲面，双三次类型。

// 编译环境：VS2019，EasyX_20220116

// 最后修改：2022-5-9

#include <graphics.h>

#include <conio.h>

#include <iostream>

#include <cmath>

using namespace std;

#define pi 3.1415926

#define MAX 20

//三维点类

class Point3D {

public:

    double x, y, z;

    Point3D(int x = 0, int y = 0, int z = 0) {

         this->x = x;

         this->y = y;

         this->z = z;

    };

    void operator=(Point3D& a) {

         x = a.x;

         y = a.y;

         z = a.z;

    };

    Point3D operator*(double a) {

         return Point3D(a * x, a * y, a * z);

    };

    Point3D operator+(Point3D a) {

         return Point3D(x + a.x, y + a.y, z + a.z);

    };

    void operator+=(Point3D a) {

         x += a.x;

         y += a.y;

         z += a.z;

    };

};

//求阶乘

long int Factorial(int n) {

    int i, sum = 1;

    if (n == 0) {

         return 1;

    }

    for (i = 2; i <= n; i++) {

         sum *= i;

    }

    return sum;

}

//Bernstein函数

double Bernstein(int i, int n, double t) {

    return (double)Factorial(n) / Factorial(i) / Factorial(n - i) * pow(t, i) * pow(1 - t, n - i);

}

//Bezier曲面求解

void BezierCurve(Point3D p[][MAX], int n, int m, int nd, int md, double sita, double fai) {

    double hu = 1.0 / nd;

    double hv = 1.0 / md;

    double u = 0, v = 0;

    int i, j, k, l;

    sita = sita * pi / 180;

    fai = fai * pi / 180;

    int** pB = new int* [nd + 1];

    for (i = 0; i <= nd; i++) {

         pB[i] = new int[2 * md + 2];

    }

    Point3D ptemp(0, 0, 0);

    for (i = 0; i <= nd; i++, u += hu) {

         v = 0;

         for (j = 0; j <= 2 * md + 1; j += 2, v += hv) {

             ptemp.x = 0;

             ptemp.y = 0;

             ptemp.z = 0;

             for (k = 0; k <= n; k++) {

                  for (l = 0; l <= m; l++) {

                      ptemp += p[k][l] * Bernstein(k, n, u) * Bernstein(l, m, v);

                  }

             }

             pB[i][j] = ptemp.x * cos(sita) - ptemp.y * sin(sita) + 500;

             pB[i][j + 1] = -ptemp.x * sin(sita) * sin(fai) - ptemp.y * cos(sita) * sin(fai) + ptemp.z * cos(fai) + 400;

         }

         drawpoly(md + 1, pB[i]);

    }

    for (i = 0; i < nd + 1; i++) {

         delete[]pB[i];

    }

    delete[]pB;

}

//画控制多边形

void drawControlPoly_3D(Point3D p[][MAX], int n, int m, double sita, double fai, int color) {

    int** p2d = new int* [n + 1];

    int i, j;

    setcolor(color);

    sita = sita * pi / 180;

    fai = fai * pi / 180;

    for (i = 0; i < n + 1; i++) {

         p2d[i] = new int[2 * m + 4];

    }

    char str[80];

    //3D->2D

    for (i = 0; i <= n; i++) {

         for (j = 0; j <= m; j++) {

             p2d[i][2 * j] = p[i][j].x * cos(sita) - p[i][j].y * sin(sita) + 500;

             p2d[i][2 * j + 1] = -p[i][j].x * sin(sita) * sin(fai) - p[i][j].y * cos(sita) * sin(fai) + p[i][j].z * cos(fai) + 400;

         }

         p2d[i][2 * j] = p2d[i][0];

         p2d[i][2 * j + 1] = p2d[i][1];

         drawpoly(m + 2, p2d[i]);

    }

    for (j = 0; j <= 2 * m + 1; j += 2) {

         moveto(p2d[0][j], p2d[0][j + 1]);

         for (i = 1; i <= n; i++) {

             lineto(p2d[i][j], p2d[i][j + 1]);

         }

    }

    for (i = 0; i < n + 1; i++) {

         delete[]p2d[i];

    }

    delete[]p2d;

}

//双几次曲面

void BezierCurve_Shuang(Point3D p[][MAX], int pn, int pm, int nd, int md, double sita, double fai, int shuangjici) {

    int i, j, k, l;

    Point3D point[MAX][MAX];

    for (i = 0; i < pn - shuangjici; i += shuangjici) {

         for (j = 0; j < pm - shuangjici; j += shuangjici) {

             for (k = 0; k <= shuangjici; k++) {

                  for (l = 0; l <= shuangjici; l++) {

                      point[k][l] = p[i + k][j + l];

                  }

             }

             BezierCurve(point, shuangjici, shuangjici, nd, md, sita, fai);

         }

    }

}

//主函数

int main() {

    int n;

    double fai, sita;

    //固定角度了。

    sita = 20;

    fai = 160;

    n = 3;

    //顶点集合

    Point3D p[][MAX] = {

         {Point3D(0,0,0),Point3D(150,150,150),Point3D(250,150,150),Point3D(400,0,0)},

        {Point3D(50,200,0),Point3D(150,250,150),Point3D(250,250,150),Point3D(450,200,0)},

        {Point3D(20,400,0),Point3D(150,350,150),Point3D(250,350,150),Point3D(420,400,0)},

        {Point3D(0,600,0),Point3D(150,550,150),Point3D(250,550,150),Point3D(400,600,0)}

    };

    /*

    cout << "please input sita and fai" << endl;

    cin >> sita >> fai;

    cout << "please input your n" << endl;

    cin >> n;

    */

    //初始化界面

    initgraph(1000, 700);

    setcolor(RED);

    //Bezier曲面

    BezierCurve_Shuang(p, 4, 4, 100, 100, sita, fai, n);

    //画控制多边形

    drawControlPoly_3D(p, 3, 3, sita, fai, GREEN);

    _getch();

    closegraph();

    return 0;

}

2题改进版本

//

// 程序名称：Bezier曲面

// 功    能：绘制Bezier曲面，并采用自行设计输入和交互修改数据点的方式。

// 编译环境：VS2019，EasyX_20220116

// 最后修改：2022-5-13

#include <iostream>

#include <graphics.h>

#include <conio.h>

#include <math.h>

using namespace std;

#define PI 3.1415926

//定义 N * N 的控制点数组

const int N = 8;

float controlPoints[N][N][3] ={

    { {-150, -150, 100}, {-100, -150, 0}, {-50, -150, 0}, {0, -150, 50}, {50, -150, 0}, {100, -150, 0}, {150, -150, -100}, {200, -150, 0} },

    { {-150, -100, 0}, {-100, -100, 50}, {-50, -100, -100}, {0, -100, 0}, {50, -100, 0}, {100, -100, 100}, {150, -100, 0}, {200, -100, -50} },

    { {-150, -50, 0}, {-100, -50, 0}, {-50, -50, 50}, {0, -50, 0}, {50, -50, 0}, {100, -50, 0}, {150, -50, -50}, {200, -50, 0} },

    { {-150, 0, -50}, {-100, 0, -100}, {-50, 0, 0}, {0, 0, 150}, {50, 0, 0}, {100, 0, -100}, {150, 0, -50}, {200, 0, -50} },

    { {-150, 50, 0}, {-100, 50, 0}, {-50, 50, -100}, {0, 50, 0}, {50, 50, 50}, {100, 50, 0}, {150, 50, 0}, {200, 50, 50} },

    { {-150, 100, 50}, {-100, 100, 0}, {-50, 100, 0}, {0, 100, -50}, {50, 100, -50}, {100, 100, 50}, {150, 100, 0}, {200, 100, 0} },

    { {-150, 150, 0}, {-100, 150, 100}, {-50, 150, -100}, {0, 150, 0}, {50, 150, 0}, {100, 150, 0}, {150, 150, -150}, {200, 150, 50} },

    { {-150, 200, 0}, {-100, 200, 0}, {-50, 200, 50}, {0, 200, -50}, {50, 200, 0}, {100, 200, -100}, {150, 200, 0}, {200, 200, 50} }

};

//计算贝塞尔基函数的C值

int C(int n, int k) {

    if (k > n) {

        return 0;

    }

    int c = 1;

    for (int i = 0; i < k; i++) {

        c *= (n - i);

        c /= (i + 1);

    }

    return c;

}

//勾勒函数

void display(float x, float y, float z) {

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400), int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300), RED);

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400) + 1, int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300), RED);

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400) - 1, int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300), RED);

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400), int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300) + 1, RED);

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400), int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300) - 1, RED);

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400) + 1, int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300) + 1, RED);

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400) + 1, int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300) - 1, RED);

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400) - 1, int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300) + 1, RED);

    putpixel(int(0.7071 * x - 0.7071 * y + 0.5f + 400) - 1, int(-0.4082 * x - 0.4082 * y + 0.8165 * z + 0.5f + 300) - 1, RED);

}

//Bezier曲面绘制

void Bezier() {

    //绘制控制点

    for (int i = 0; i < N - 1; i++) {

        for (int j = 0; j < N - 1; j++) {

            fillcircle(int(0.7071 * controlPoints[i][j][0] - 0.7071 * controlPoints[i][j][1] + 0.5f + 400), int(-0.4082 * controlPoints[i][j][0] - 0.4082 * controlPoints[i][j][1] + 0.8165 * controlPoints[i][j][2] + 0.5f + 300), 3);

        }

    }

    // 绘制贝塞尔曲面

    const int nPoints = 50;  // 每行/列计算的点数

    float uStep = 1.0f / nPoints;

    float vStep = 1.0f / nPoints;

    for (float u = 0; u < 1; u += uStep) {

        for (float v = 0; v < 1; v += vStep) {

            float x = 0, y = 0, z = 0;

            for (int i = 0; i < N; i++) {

                for (int j = 0; j < N; j++) {

                    // 计算贝塞尔基函数的值

                    float basisU = powf(1 - u, N - 1 - i) * powf(u, i) * float(C(N - 1, i));

                    float basisV = powf(1 - v, N - 1 - j) * powf(v, j) * float(C(N - 1, j));

                    float basis = basisU * basisV;

                    // 使用贝塞尔基函数的值和控制点计算曲面上的点

                    x += basis * controlPoints[i][j][0];

                    y += basis * controlPoints[i][j][1];

                    z += basis * controlPoints[i][j][2];

                }

            }

            // 将曲面上的点连接起来

            display(x, y, z);

        }

    }

}

//主函数

int main(){

    initgraph(800, 600);

    Bezier();

    ExMessage m;

    while (1) {

        m = getmessage(EX_MOUSE | EX_KEY);

        switch (m.message) {

        case WM_RBUTTONDOWN:

            //右键修改坐标信息

            closegraph();

            int row, column, newx, newy, newz;

            //输入数组中的坐标，进行修改

            cout << "which point do you want to change? please input its row & column array number" << endl;

            while (1) {

                cin >> row;

                cin >> column;

                if (row >= N) {

                    cout << "row number input error, try again" << endl;

                }

                else if (column >= N) {

                    cout << "column number input error, try again" << endl;

                }

                else {

                    break;

                }

            }

            cout << "please input the reset coordinate" << endl;

            cin >> newx >> newy >> newz;

            controlPoints[row][column][0] = newx;

            controlPoints[row][column][1] = newy;

            controlPoints[row][column][2] = newz;

            cout << "successfully reset, please wait a second to see a new graph" << endl;

            initgraph(800, 600);

        case WM_LBUTTONDOWN:

            //左键绘制

            Bezier();

        case WM_KEYDOWN:

            //退出图形化界面

            if (m.vkcode == VK_ESCAPE) {

                return 0;

            }

        }

    }

    _getch();

    closegraph();

    return 0;

}