【计算机图形学】基本图形元素:圆的生成算法
08年9月入学,12年7月毕业,结束了我在软件学院愉快丰富的大学生活。此系列是对四年专业课程学习的回顾,索引参见:http://blog.csdn.net/xiaowei_cqu/article/details/7747205
圆的特征
圆被定义为到给定中心位置(xc,yc)距离为r的点集。圆心位于原点的圆有四条对称轴x=0,y=0, x=y和x=-y。若已知圆弧上一点(x,y),可以得到其关于四条对称轴的其它7个点,这种性质称为八分对称性。因此,只要扫描转换八分之一圆弧,就可以求出整个圆弧的象素集。
显示圆弧上的八个对称点的算法:
void CirclePoints(int x,int y,int color){ Putpixel(x,y,color); Putpixel(y,x,color); Putpixel(-x,y,color); Putpixel(y,-x,color); Putpixel(x,-y,color); Putpixel(-y,x,color); Putpixel(-x,-y,color); Putpixel(-y,-x,color);}
果我们构造函数 F(x,y)=x2+y2-R2,则对于圆上的点有F(x,y)=0,对于圆外的点有F(x,y)>0,对于圆内的点F(x,y)<0 。与中点画线法一样,构造判别式:
d=F(M)=F(xp+1,yp-0.5)=(xp+1)2+(yp-0.5)2-R2
若 d<0,则应取P1为下一象素,而且再下一象素的判别式为:
d=F(xp+2,yp-0.5)=(xp+2)2+(yp-0.5)2-R2=d+2xp+3
若d≥0,则应取P2为下一象素,而且下一象素的判别式为
d=F(xp+2,yp-1.5)=(xp+2)2+(yp-1.5)2-R2=d+2(xp-yp)+5
我们这里讨论的第一个象素是(0,R),判别式d的初始值为:
d0=F(1,R-0.5)=1.25-R
void PaintArea::drawCircleMiddle(QPainter &painter,const QPoint ¢er, int r){ int x,y,deltax,deltay,d; x=0;y=r; deltax=3;deltay=2-3-3;d=1-r; while(x<y) { if(d<0) { d+=deltax; deltax+=2; x++; } else { d+=(deltax+deltay); deltax+=2;deltay+=2; x++;y++; } painter.drawPoint(center.x()+x,center.y()+y); painter.drawPoint(center.x()+x,center.y()-y); painter.drawPoint(center.x()-x,center.y()+y); painter.drawPoint(center.x()-x,center.y()-y); painter.drawPoint(center.x()+y,center.y()+x); painter.drawPoint(center.x()+y,center.y()-x); painter.drawPoint(center.x()-y,center.y()+x); painter.drawPoint(center.x()-y,center.y()-x); }}
思想参见直线的Bresenham画法 【计算机图形学】基本图形元素:直线的生成算法
void PaintArea::drawCircleBresenham(QPainter &painter,const QPoint ¢er, int r){ int x,y,delta,delta1,delta2,direction; x=0;y=r; delta=2*(1-r); while(y>=0) { painter.drawPoint(x,y); if(delta<0) { delta1=2*(delta+y)-1; if(delta1<=0)direction=1; else direction=2; } else if(delta>0) { delta2=2*(delta-x)-1; if(delta2<=0)direction=2; else direction=3; } else direction=2; switch(direction) {case 1: x++;delta+=2*x+1; break; case 2: x++; y--; delta+=2*(x-y+1); break; case 3: y--;delta+=(-2*y+1); break; } }}
基本同圆弧算法,只是方程变得复杂F(x,y)=(bx)^2+(ay)^2-(ab)^2.
对称性:4分对称,画第一象限
分段依据:斜率为一点
上段圆弧:
下段圆弧:
【椭圆中点算法流程图】
【算法代码】
void PaintArea::drawEllipseMiddle(QPainter &painter,int xCenter,int yCenter, int Rx, int Ry){ int Rx2=Rx*Rx; int Ry2=Ry*Ry; int twoRx2=2*Rx2; int twoRy2=2*Ry2; int p,x=0,y=Ry,px=0,py=twoRx2*y; void ellipsePlotPoints(QPainter&,int,int,int,int); ellipsePlotPoints(painter,xCenter,yCenter,x,y); //Region1 p=round(Ry-(Rx2*Ry)+(0.25*Rx2)); while(px<py){ x++; px+=twoRy2; if(p<0) p+=Ry2+px; else{ y--; py-=twoRx2; p+=Ry2+px-py; } ellipsePlotPoints(painter,xCenter,yCenter,x,y); } //Region2 p=round(Ry2*(x+0.5)*(x+0.5)+Rx2*(y-1)*(y-1)-Rx2*Ry2); while(y>0){ y--; py-=twoRx2; if(p>0) p+=Rx2-py; else{ x++; px+=twoRy2; p+=Rx2-py+px; } ellipsePlotPoints(painter,xCenter,yCenter,x,y); }}void ellipsePlotPoints(QPainter &painter,int xCenter,int yCenter,int x,int y){ painter.drawPoint(xCenter+x,yCenter+y); painter.drawPoint(xCenter-x,yCenter+y); painter.drawPoint(xCenter+x,yCenter-y); painter.drawPoint(xCenter-x,yCenter-y);}
这个绘图软件是用QT写的,我会另外写一篇介绍编程结构,待续~
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