【计算机图形学】曲线和曲面（Bezier曲线 & Bezier曲面）

模块5 曲线和曲面

一实验目的

编写曲线和曲面的算法

二实验内容

1：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。

实验结果如下图所示：

第一步：输入特征多边形的顶点个数，并按照顺序输入顶点的坐标。

第二步：点击左键生成bezier曲线（白色部分）和多边形顶点（绿色部分）。

第三步：点击右键修改顶点，首先输入需要修改的顶点在数组中的坐标（即顺序个数-1），此处以（100，100）为例，将其修改为（100，300）。

第四步：通过敲击回车键，可观察修改后的bezier曲线和顶点。

第五步：如果需要继续修改顶点坐标，用户可以继续点击右键，重复上述操作。此处不再赘述。如果需要退出图像绘制界面，用户可以通过敲击esc键，即可退出本实验界面。

2：绘制Bezier曲面，双三次类型。

实验结果如下图所示：

2.1：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，改进。

实验结果如下图所示：

修改数据点：

新贝塞尔曲面：

三程序说明

最终的实验代码如下表所示：

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;

}

//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;

}

【计算机图形学】曲线和曲面（Bezier曲线 & Bezier曲面）

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