在產品零件設計中，許多自由曲面是通過自由曲線來構造的。對于自由曲線的設計，設計人員經常需要大致勾畫出曲線的形狀，用戶希望有一種方法能不再采用一般的代數描述，而采用直觀的具有明顯幾何意義的操作，使得設計的曲線能夠逼近曲線的形狀。采用插值方法，用戶設計的曲線形狀不但受曲線上型值點的約束，而且受到邊界條件影響，用戶不能靈活調整曲線形狀。但在產品設計中，曲線的設計是經過多次修改和調整來完成的，已有的方法完成這樣的功能并不容易。Bézier方法的出現改善了上述設計方法的不足，使用戶能方便地實現曲線形狀的修改。
一、Bézier曲線定義
1. 定義：用基函數表示的Bézier曲線為
Bernstein基函數：
當N=3時，即特征多邊形有四個頂點，由上式可得到三次Bernstein基函數：
因此根據定義可把三次Bézier曲線表示為：
也可根據上式寫成矩陣形式，有點類似二次型：
由上可知，只要給定特征多邊形的四個頂點矢量P0P1P2P3，并得用上式即可構造一條三次Bézier曲線。只要改變t值即可計算曲線上的點。對于曲線設計來說，采用這種特征多邊形頂點的方法比較方便、直觀。
二、編程實現
利用OpenGL的glut庫根據Bézier曲線定義來實現曲線的繪制，源程序如下：
// Bezier Curve Program
#include <gl\glut.h>
#define POINTS_NUM 100
// point structure
typedef struct SPoint{
float x;
float y;
}Point;
Point ctrlPoint[4];
void Initialize(void);
void DrawScene(void);
void myReshape(GLsizei w, GLsizei h);
void PlotBezierCurveDirectly(void);
Point PointOnBezierCurve(int i, int k, float t);
void PlotBezierCurve(void);
void main(int argc, char* argv[]) {
glutInit(&argc, argv); // Initialize GLUT
glutInitDisplayMode(GLUT_SINGLE|GLUT_RGB); // Set display mode
glutInitWindowPosition(50,100); // Set top-left display window position
glutInitWindowSize(400, 300); // set display window width and height
glutCreateWindow("Bézier Curve"); // Create display window
Initialize(); // Execute initialization procedure
glutDisplayFunc(DrawScene); // Send graphics to display window
glutReshapeFunc(myReshape); //
glutMainLoop(); // Display everything and wait
}
/*
*/
void Initialize(void) {
//glClearColor(1.0, 1.0, 1.0, 0.0); // Set Display-window color to white
glMatrixMode(GL_PROJECTION); // Set projection parameters
glLoadIdentity();
gluOrtho2D(0.0, 200, 0.0, 150); //
} // Initialize
/*
*/
void myReshape(GLsizei w, GLsizei h) {
// Reset viewport and projection parameter
glViewport(0, 0, w, h);
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
gluOrtho2D(0, w, 0, h);
glMatrixMode(GL_MODELVIEW);
} // myReshape
/*
*/
void DrawScene(void) {
glClear(GL_COLOR_BUFFER_BIT); // Clear display window
PlotBezierCurveDirectly();
PlotBezierCurve();
glFlush(); // Process all OpenGL routines as quickly possible
} // DrawScene
/*
plot the Bezier Curve according to Definition.
*/
void PlotBezierCurveDirectly(void) {
// Basis functions
#define B0(t) ((1.0 - t) * (1.0 - t) * (1.0 - t))
#define B1(t) (3.0 * t * (1.0 - t) * (1.0 - t))
#define B2(t) (3.0 * t * t * (1.0 - t))
#define B3(t) (t * t * t)
// initialize control points
Point controlPoint[4];
Point plotPoint[POINTS_NUM];
controlPoint[0].x = 100.0;
controlPoint[0].y = 100.0;
controlPoint[1].x = 300.0;
controlPoint[1].y = 200.0;
controlPoint[2].x = 400.0;
controlPoint[2].y = 200.0;
controlPoint[3].x = 600.0;
controlPoint[3].y = 100.0;
// calculate the points on the Bezier curve
glColor3f(1.0, 0.0, 0.0f); // set line segment geometry color to red
glBegin(GL_LINE_STRIP);
for (int i = 0; i < POINTS_NUM; ++i) {
float t = 1.0 * i / POINTS_NUM;
plotPoint[i].x = (B0(t) * controlPoint[0].x) +
(B1(t) * controlPoint[1].x) +
(B2(t) * controlPoint[2].x) +
(B3(t) * controlPoint[3].x);
plotPoint[i].y = (B0(t) * controlPoint[0].y) +
(B1(t) * controlPoint[1].y) +
(B2(t) * controlPoint[2].y) +
(B3(t) * controlPoint[3].y);
glVertex2f(plotPoint[i].x, plotPoint[i].y);
}
glEnd();
// draw control points
glColor3f(1.0, 1.0, 0.0f);
glPointSize(6.0);
glBegin(GL_POINTS);
glVertex2f(controlPoint[0].x, controlPoint[0].y);
glVertex2f(controlPoint[1].x, controlPoint[1].y);
glVertex2f(controlPoint[2].x, controlPoint[2].y);
glVertex2f(controlPoint[3].x, controlPoint[3].y);
glEnd();
// draw control polygon
glColor3f(1.0, 0.0, 1.0f);
glBegin(GL_LINE_STRIP);
glVertex2f(controlPoint[0].x, controlPoint[0].y);
glVertex2f(controlPoint[1].x, controlPoint[1].y);
glVertex2f(controlPoint[2].x, controlPoint[2].y);
glVertex2f(controlPoint[3].x, controlPoint[3].y);
glEnd();
} // PlotBezierCurveDirectly
/*
*/
Point PointOnBezierCurve(int i, int k, float t) {
if (k == 0) {
return ctrlPoint[i];
}
else {
Point point;
Point iPrev = PointOnBezierCurve(i, k-1, t);
Point iNext = PointOnBezierCurve(i+1, k-1, t);
point.x = (1.0 - t) * iPrev.x + t * iNext.x;
point.y = (1.0 - t) * iPrev.y + t * iNext.y;
return point;
}
} // PointOnBezierCurve
/*
Use recursive algorithm to plot Bezier Curve.
*/
void PlotBezierCurve(void) {
// initialize control points
ctrlPoint[0].x = 100.0;
ctrlPoint[0].y = 200.0;
ctrlPoint[1].x = 300.0;
ctrlPoint[1].y = 300.0;
ctrlPoint[2].x = 400.0;
ctrlPoint[2].y = 500.0;
ctrlPoint[3].x = 600.0;
ctrlPoint[3].y = 300.0;
Point plotPoint[POINTS_NUM];
// calculate the points on the Bezier curve
glColor3f(1.0, 0.0, 0.0f); // set line segment geometry color to red
glBegin(GL_LINE_STRIP);
for (int i = 0; i < POINTS_NUM; ++i) {
float t = 1.0 * i / POINTS_NUM;
plotPoint[i] = PointOnBezierCurve(0,3,t);
glVertex2f(plotPoint[i].x, plotPoint[i].y);
}
glEnd();
// draw control points
glColor3f(1.0, 1.0, 1.0f);
glPointSize(6.0);
glBegin(GL_POINTS);
glVertex2f(ctrlPoint[0].x, ctrlPoint[0].y);
glVertex2f(ctrlPoint[1].x, ctrlPoint[1].y);
glVertex2f(ctrlPoint[2].x, ctrlPoint[2].y);
glVertex2f(ctrlPoint[3].x, ctrlPoint[3].y);
glEnd();
// draw control polygon
glColor3f(0.0, 0.0, 1.0f);
glBegin(GL_LINE_STRIP);
glVertex2f(ctrlPoint[0].x, ctrlPoint[0].y);
glVertex2f(ctrlPoint[1].x, ctrlPoint[1].y);
glVertex2f(ctrlPoint[2].x, ctrlPoint[2].y);
glVertex2f(ctrlPoint[3].x, ctrlPoint[3].y);
glEnd();
} // PlotBezierCurve
程序運行結果如圖1所示：
圖1 根據定義實現的Bézier曲線
可修改特征多邊形頂點變量controlPoint來看看相應的Bézier曲線的變化。
三、Bézier曲線的遞推算法
計算Bézier曲線上的點，可用Bézier曲線方程直接計算，但使用de Casteljau提出的遞推算法則要簡單得多。
Bézier曲線的Bernstein基函數的遞推性質
![clip_image002[4]](http://m.shnenglu.com/images/cppblog_com/eryar/WindowsLiveWriter/BzierCurve_130E4/clip_image002%5B4%5D_thumb.gif "clip_image002[4]")
由于組合數的遞推性：
![clip_image004[4]](http://m.shnenglu.com/images/cppblog_com/eryar/WindowsLiveWriter/BzierCurve_130E4/clip_image004%5B4%5D_thumb.gif "clip_image004[4]")
因此有
![clip_image006[4]](http://m.shnenglu.com/images/cppblog_com/eryar/WindowsLiveWriter/BzierCurve_130E4/clip_image006%5B4%5D_thumb.gif "clip_image006[4]")
為了保證公式可計算，我們約定：在以上公式中，凡當指標超出范圍，以至記號不具有意義時，都應理解為零，如：
![clip_image008[4]](http://m.shnenglu.com/images/cppblog_com/eryar/WindowsLiveWriter/BzierCurve_130E4/clip_image008%5B4%5D_thumb.gif "clip_image008[4]")
根據Bézier曲線定義可知：
![clip_image010[4]](http://m.shnenglu.com/images/cppblog_com/eryar/WindowsLiveWriter/BzierCurve_130E4/clip_image010%5B4%5D_thumb.gif "clip_image010[4]")
這說明一條n次Bézier曲線可表示為分別由前后n個控制頂點決定的兩條n-1次Bézier曲線的線性組合。由此可得Bézier曲線上某一點遞歸求值的de Casteljau算法：
![clip_image012[4]](http://m.shnenglu.com/images/cppblog_com/eryar/WindowsLiveWriter/BzierCurve_130E4/clip_image012%5B4%5D_thumb.gif "clip_image012[4]")
其中：
——
， 是定義Bézier曲線的控制點；
——
，即是曲線P(t)上具有參數t的點；
用這一遞推公式在給定參數下，求Bézier曲線上一點P(t)非常有效。de Casteljau算法穩定可靠，直觀簡便，可以編寫出十分簡捷的程序，是計算Bézier曲線的基本算法和標準算法。
四、de Casteljau算法的程序實現
根據遞推公式可寫出Bézier曲線的遞歸算法，如下：
/*
*/
Point PointOnBezierCurve(int i, int k, float t) {
if (k == 0) {
return ctrlPoint[i];
}
else {
Point point;
Point iPrev = PointOnBezierCurve(i, k-1, t);
Point iNext = PointOnBezierCurve(i+1, k-1, t);
point.x = (1.0 - t) * iPrev.x + t * iNext.x;
point.y = (1.0 - t) * iPrev.y + t * iNext.y;
return point;
}
} // PointOnBezierCurve
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其中，為了保存數據的方便，控制頂點數組變量ctrlPoint為全局變量。
根據遞推算法實現Bézier曲線與根據定義實現的程序效果如下圖2所示：
圖2 圖中上面部分為用遞推算法實現，下面部分為用定義實現。
源程序如下：
// Bezier Curve Program
// File : BezierCurve.cpp
#include <gl\glut.h>
#define POINTS_NUM 100
// point structure
typedef struct SPoint{
float x;
float y;
}Point;
Point ctrlPoint[4]; // control point for recursive algorithm Bezier Curve
void Initialize(void);
void DrawScene(void);
void myReshape(GLsizei w, GLsizei h);
void PlotBezierCurveDirectly(void);
Point PointOnBezierCurve(int i, int k, float t);
void PlotBezierCurve(void);
void main(int argc, char* argv[]) {
glutInit(&argc, argv); // Initialize GLUT
glutInitDisplayMode(GLUT_SINGLE|GLUT_RGB); // Set display mode
glutInitWindowPosition(50,100); // Set top-left display window position
glutInitWindowSize(400, 300); // set display window width and height
glutCreateWindow("Bézier Curve"); // Create display window
Initialize(); // Execute initialization procedure
glutDisplayFunc(DrawScene); // Send graphics to display window
glutReshapeFunc(myReshape); //
glutMainLoop(); // Display everything and wait
}
/*
*/
void Initialize(void) {
//glClearColor(1.0, 1.0, 1.0, 0.0); // Set Display-window color to white
glMatrixMode(GL_PROJECTION); // Set projection parameters
glLoadIdentity();
gluOrtho2D(0.0, 200, 0.0, 150); //
} // Initialize
/*
*/
void myReshape(GLsizei w, GLsizei h) {
// Reset viewport and projection parameter
glViewport(0, 0, w, h);
glMatrixMode(GL_PROJECTION);
glLoadIdentity();
gluOrtho2D(0, w, 0, h);
glMatrixMode(GL_MODELVIEW);
} // myReshape
/*
*/
void DrawScene(void) {
glClear(GL_COLOR_BUFFER_BIT); // Clear display window
PlotBezierCurveDirectly();
PlotBezierCurve();
glFlush(); // Process all OpenGL routines as quickly possible
} // DrawScene
/*
plot the Bezier Curve according to Definition.
*/
void PlotBezierCurveDirectly(void) {
// Basis functions
#define B0(t) ((1.0 - t) * (1.0 - t) * (1.0 - t))
#define B1(t) (3.0 * t * (1.0 - t) * (1.0 - t))
#define B2(t) (3.0 * t * t * (1.0 - t))
#define B3(t) (t * t * t)
// initialize control points
Point controlPoint[4];
Point plotPoint[POINTS_NUM];
controlPoint[0].x = 100.0;
controlPoint[0].y = 100.0;
controlPoint[1].x = 300.0;
controlPoint[1].y = 200.0;
controlPoint[2].x = 400.0;
controlPoint[2].y = 200.0;
controlPoint[3].x = 600.0;
controlPoint[3].y = 100.0;
// calculate the points on the Bezier curve
glColor3f(1.0, 0.0, 0.0f); // set line segment geometry color to red
glBegin(GL_LINE_STRIP);
for (int i = 0; i < POINTS_NUM; ++i) {
float t = 1.0 * i / POINTS_NUM;
plotPoint[i].x = (B0(t) * controlPoint[0].x) +
(B1(t) * controlPoint[1].x) +
(B2(t) * controlPoint[2].x) +
(B3(t) * controlPoint[3].x);
plotPoint[i].y = (B0(t) * controlPoint[0].y) +
(B1(t) * controlPoint[1].y) +
(B2(t) * controlPoint[2].y) +
(B3(t) * controlPoint[3].y);
glVertex2f(plotPoint[i].x, plotPoint[i].y);
}
glEnd();
// draw control points
glColor3f(1.0, 1.0, 0.0f);
glPointSize(6.0);
glBegin(GL_POINTS);
glVertex2f(controlPoint[0].x, controlPoint[0].y);
glVertex2f(controlPoint[1].x, controlPoint[1].y);
glVertex2f(controlPoint[2].x, controlPoint[2].y);
glVertex2f(controlPoint[3].x, controlPoint[3].y);
glEnd();
// draw control polygon
glColor3f(1.0, 0.0, 1.0f);
glBegin(GL_LINE_STRIP);
glVertex2f(controlPoint[0].x, controlPoint[0].y);
glVertex2f(controlPoint[1].x, controlPoint[1].y);
glVertex2f(controlPoint[2].x, controlPoint[2].y);
glVertex2f(controlPoint[3].x, controlPoint[3].y);
glEnd();
} // PlotBezierCurveDirectly
/*
*/
Point PointOnBezierCurve(int i, int k, float t) {
if (k == 0) {
return ctrlPoint[i];
}
else {
Point point;
Point iPrev = PointOnBezierCurve(i, k-1, t);
Point iNext = PointOnBezierCurve(i+1, k-1, t);
point.x = (1.0 - t) * iPrev.x + t * iNext.x;
point.y = (1.0 - t) * iPrev.y + t * iNext.y;
return point;
}
} // PointOnBezierCurve
/*
Use recursive algorithm to plot Bezier Curve.
*/
void PlotBezierCurve(void) {
// initialize control points
ctrlPoint[0].x = 100.0;
ctrlPoint[0].y = 300.0;
ctrlPoint[1].x = 300.0;
ctrlPoint[1].y = 400.0;
ctrlPoint[2].x = 400.0;
ctrlPoint[2].y = 400.0;
ctrlPoint[3].x = 600.0;
ctrlPoint[3].y = 300.0;
Point plotPoint[POINTS_NUM];
// calculate the points on the Bezier curve
glColor3f(1.0, 0.0, 0.0f); // set line segment geometry color to red
glBegin(GL_LINE_STRIP);
for (int i = 0; i < POINTS_NUM; ++i) {
float t = 1.0 * i / POINTS_NUM;
plotPoint[i] = PointOnBezierCurve(0,3,t);
glVertex2f(plotPoint[i].x, plotPoint[i].y);
}
glEnd();
// draw control points
glColor3f(1.0, 1.0, 1.0f);
glPointSize(6.0);
glBegin(GL_POINTS);
glVertex2f(ctrlPoint[0].x, ctrlPoint[0].y);
glVertex2f(ctrlPoint[1].x, ctrlPoint[1].y);
glVertex2f(ctrlPoint[2].x, ctrlPoint[2].y);
glVertex2f(ctrlPoint[3].x, ctrlPoint[3].y);
glEnd();
// draw control polygon
glColor3f(0.0, 0.0, 1.0f);
glBegin(GL_LINE_STRIP);
glVertex2f(ctrlPoint[0].x, ctrlPoint[0].y);
glVertex2f(ctrlPoint[1].x, ctrlPoint[1].y);
glVertex2f(ctrlPoint[2].x, ctrlPoint[2].y);
glVertex2f(ctrlPoint[3].x, ctrlPoint[3].y);
glEnd();
} // PlotBezierCurve
因此，根據遞歸算法繪制Bézier曲線的程序簡單直觀，便于把數學公式與程序對應。
圖3 修改控制頂點后效果圖