MFC绘制二次均匀B样条曲线

B样条具有比Bezier更好的局部修改性与凸包性，引入了节点与控制点一起控制曲线的生成，下面以二次均匀B样条的绘制为例，进行代码实现。

参考《计算几何算法与实现》–孔令德

新建MFC单文档程序，添加类Quadratic_BSpline，用来绘制曲线；

#pragma once
#include"P2.h"

class Quadratic_BSpline
{
    
public:
	Quadratic_BSpline(void);
	~Quadratic_BSpline(void);
public:
	//计算基函数
	 double BasicFunctionValue(double t, int i, int k);

	 //初始化控制点，节点矢量
	 void InitPoint();

	 //绘制样条线
	 void DrawCurve(CDC*pDC);

	 //绘制控制点
	 void DrawControlPoint(CDC*pDC);

private:
	double knot[9];	//定义节点矢量数组
	int n;          //控制点个数减1
	int k;          //次数
	CP2 P[6];       //控制点坐标
};

#include "StdAfx.h"
#include "Quadratic_BSpline.h"

#define ROUND(d) int(d+0.5)//四舍五入宏定义

Quadratic_BSpline::Quadratic_BSpline(void)
{
    
	InitPoint();
}


Quadratic_BSpline::~Quadratic_BSpline(void)
{
    
}

/*----------------------------------------------------
input   : t, i, k分别为参数值，支撑区间左端节点的下标，曲线次数
function: 计算第i个k次B样条基函数值
-----------------------------------------------------*/

double Quadratic_BSpline::BasicFunctionValue(double t, int i, int k)
{
    
	//次数为0时的基函数值
	if(k==0)
	{
    
		if(t>=knot[i]&&t<knot[i+1])
			return 1.0;
		else 
			return 0.0;
	}

	//次数不为0时的基函数计算
	double value1, value2, value;
	if(k>0)
	{
    
		//节点矢量位于定义域之外
		if(t<knot[i]||t>knot[i+k+1])
			return 0.0;

		//节点矢量位于定义域之外
		else
		{
    
			double coff1, coff2;    //系数
			double denominator=0.0; //分母

			//递推公式第一项
			denominator=knot[i+k]-knot[i];
			if(denominator==0.0)
				coff1=0.0;                  //重节点分母为0
			else
				coff1=(t-knot[i])/denominator;

			//递推公式第二项
			denominator=knot[i+k+1]-knot[i+1];
			if(denominator==0.0)
				coff2=0.0;                  //重节点分母为0
			else
				coff2=(knot[i+k+1]-t)/denominator;

			//递推公式第一、二项的值
			value1=coff1*BasicFunctionValue(t,i,k-1);
			value2=coff2*BasicFunctionValue(t,i+1,k-1);

			value=value1+value2;
		}
	}
	return value;
}

/*----------------------------------------------------
input   : 
function: 初始化节点与控制点
-----------------------------------------------------*/
void Quadratic_BSpline::InitPoint()
{
    
	n=5, k=2;

	//节点初始化
	knot[0]=-0.5, knot[1]=-0.25,knot[2]=0.0;
	knot[3]=0.25, knot[4]=0.5,  knot[5]=0.75;
	knot[6]=1.0,  knot[7]=1.25, knot[8]= 1.5;

	//控制点初始化
	P[0].x=-460,  P[0].y=-49;
	P[1].x=-355,  P[1].y=204;
	P[2].x= -63,  P[2].y=241;
	P[3].x= 66,   P[3].y=-117;
	P[4].x= 264,  P[4].y=-101;
	P[5].x= 400,  P[5].y=208;
}

/*----------------------------------------------------
input   : 
function: 绘制曲线
-----------------------------------------------------*/
void Quadratic_BSpline::DrawCurve(CDC*pDC)
{
    
	//绘制控制点
	DrawControlPoint(pDC);

	//创建并选取画笔
	CPen NewPen, *pOldPen;
	NewPen.CreatePen(PS_SOLID,2,RGB(0,0,255));
	pOldPen=pDC->SelectObject(&NewPen);

	//等距取参数点计算值
	double tStep=0.01;
	for(double t=0.0; t<=1.0; t+=tStep)
	{
    
		CP2 p(0,0);//离散点
		for(int i=0; i<=n; i++)
		{
    
			double BValue=BasicFunctionValue(t, i, k);
			p+=P[i]*BValue;
		}
		if(t==0)
			pDC->MoveTo(ROUND(p.x), ROUND(p.y));
		else
			pDC->LineTo(ROUND(p.x), ROUND(p.y));
	}

	//还原画笔，删除创建的画笔
	pDC->SelectObject(pOldPen);	
	NewPen.DeleteObject();      
}

/*----------------------------------------------------
input   : 
function: 绘制控制点
-----------------------------------------------------*/
void Quadratic_BSpline::DrawControlPoint(CDC*pDC)
{
    
	//初始化画笔
	CPen NewPen,*pOldPen;
	NewPen.CreatePen(PS_SOLID,3,RGB(0,0,0));
	pOldPen=pDC->SelectObject(&NewPen);

	//初始化画刷
	CBrush NewBrush,*pOldBrush;
	NewBrush.CreateSolidBrush(RGB(0,0,0));
	pOldBrush=pDC->SelectObject(&NewBrush);

	//加粗控制点
	for(int i=0;i<=n;i++)
	{
    
		if(0==i)
		{
    
			pDC->MoveTo(ROUND(P[i].x),ROUND(P[i].y));
			pDC->Ellipse(ROUND(P[i].x)-5,ROUND(P[i].y)-5,ROUND(P[i].x)+5,ROUND(P[i].y)+5);
		}
		else
		{
    
			pDC->LineTo(ROUND(P[i].x),ROUND(P[i].y));
			pDC->Ellipse(ROUND(P[i].x)-5,ROUND(P[i].y)-5,ROUND(P[i].x)+5,ROUND(P[i].y)+5);
		}
	}

	//还原画笔画刷
	pDC->SelectObject(pOldBrush);
	pDC->SelectObject(pOldPen);
	NewPen.DeleteObject(); 
	NewBrush.DeleteObject(); 
}