Bresenham算法的一种变体适用于圆，椭圆和抛物线等二次函数，因此也应适用于二次Bezier曲线。

我打算尝试一种实现，但是随后我在网上找到了一个实现：http
:
//members.chello.at/~easyfilter/bresenham.html。

如果您需要更多细节或其他示例，上述页面上有指向100页PDF的链接，其中详细介绍了该方法：http
:
//members.chello.at/~easyfilter/Bresenham.pdf。

这是来自Alois Zingl网站的代码，用于绘制任何二次Bezier曲线。第一个例程将曲线在水平和垂直梯度变化处细分：

void plotQuadBezier(int x0, int y0, int x1, int y1, int x2, int y2)
{ /* plot any quadratic Bezier curve */
  int x = x0-x1, y = y0-y1;
  double t = x0-2*x1+x2, r;
  if ((long)x*(x2-x1) > 0) { /* horizontal cut at P4? */
    if ((long)y*(y2-y1) > 0) /* vertical cut at P6 too? */
      if (fabs((y0-2*y1+y2)/t*x) > abs(y)) { /* which first? */
        x0 = x2; x2 = x+x1; y0 = y2; y2 = y+y1; /* swap points */
      } /* now horizontal cut at P4 comes first */
    t = (x0-x1)/t;
    r = (1-t)*((1-t)*y0+2.0*t*y1)+t*t*y2; /* By(t=P4) */
    t = (x0*x2-x1*x1)*t/(x0-x1); /* gradient dP4/dx=0 */
    x = floor(t+0.5); y = floor(r+0.5);
    r = (y1-y0)*(t-x0)/(x1-x0)+y0; /* intersect P3 | P0 P1 */
    plotQuadBezierSeg(x0,y0, x,floor(r+0.5), x,y);
    r = (y1-y2)*(t-x2)/(x1-x2)+y2; /* intersect P4 | P1 P2 */
    x0 = x1 = x; y0 = y; y1 = floor(r+0.5); /* P0 = P4, P1 = P8 */
  }
  if ((long)(y0-y1)*(y2-y1) > 0) { /* vertical cut at P6? */
    t = y0-2*y1+y2; t = (y0-y1)/t;
    r = (1-t)*((1-t)*x0+2.0*t*x1)+t*t*x2; /* Bx(t=P6) */
    t = (y0*y2-y1*y1)*t/(y0-y1); /* gradient dP6/dy=0 */
    x = floor(r+0.5); y = floor(t+0.5);
    r = (x1-x0)*(t-y0)/(y1-y0)+x0; /* intersect P6 | P0 P1 */
    plotQuadBezierSeg(x0,y0, floor(r+0.5),y, x,y);
    r = (x1-x2)*(t-y2)/(y1-y2)+x2; /* intersect P7 | P1 P2 */
    x0 = x; x1 = floor(r+0.5); y0 = y1 = y; /* P0 = P6, P1 = P7 */
  }
  plotQuadBezierSeg(x0,y0, x1,y1, x2,y2); /* remaining part */
}

第二个例程实际上绘制了一个Bezier曲线段（一个没有梯度变化的段）：

void plotQuadBezierSeg(int x0, int y0, int x1, int y1, int x2, int y2)
{ /* plot a limited quadratic Bezier segment */
  int sx = x2-x1, sy = y2-y1;
  long xx = x0-x1, yy = y0-y1, xy; /* relative values for checks */
  double dx, dy, err, cur = xx*sy-yy*sx; /* curvature */
  assert(xx*sx <= 0 && yy*sy <= 0); /* sign of gradient must not change */
  if (sx*(long)sx+sy*(long)sy > xx*xx+yy*yy) { /* begin with longer part */
    x2 = x0; x0 = sx+x1; y2 = y0; y0 = sy+y1; cur = -cur; /* swap P0 P2 */
  }
  if (cur != 0) { /* no straight line */
    xx += sx; xx *= sx = x0 < x2 ? 1 : -1; /* x step direction */
    yy += sy; yy *= sy = y0 < y2 ? 1 : -1; /* y step direction */
    xy = 2*xx*yy; xx *= xx; yy *= yy; /* differences 2nd degree */
    if (cur*sx*sy < 0) { /* negated curvature? */
      xx = -xx; yy = -yy; xy = -xy; cur = -cur;
    }
    dx = 4.0*sy*cur*(x1-x0)+xx-xy; /* differences 1st degree */
    dy = 4.0*sx*cur*(y0-y1)+yy-xy;
    xx += xx; yy += yy; err = dx+dy+xy; /* error 1st step */
    do {
      setPixel(x0,y0); /* plot curve */
      if (x0 == x2 && y0 == y2) return; /* last pixel -> curve finished */
      y1 = 2*err < dx; /* save value for test of y step */
      if (2*err > dy) { x0 += sx; dx -= xy; err += dy += yy; } /* x step */
      if ( y1 ) { y0 += sy; dy -= xy; err += dx += xx; } /* y step */
    } while (dy < 0 && dx > 0); /* gradient negates -> algorithm fails */
  }
  plotLine(x0,y0, x2,y2); /* plot remaining part to end */
}

该站点上还提供了抗锯齿代码。

来自Zingl站点的三次贝塞尔曲线的相应函数为

void plotCubicBezier(int x0, int y0, int x1, int y1,
  int x2, int y2, int x3, int y3)
{ /* plot any cubic Bezier curve */
  int n = 0, i = 0;
  long xc = x0+x1-x2-x3, xa = xc-4*(x1-x2);
  long xb = x0-x1-x2+x3, xd = xb+4*(x1+x2);
  long yc = y0+y1-y2-y3, ya = yc-4*(y1-y2);
  long yb = y0-y1-y2+y3, yd = yb+4*(y1+y2);
  float fx0 = x0, fx1, fx2, fx3, fy0 = y0, fy1, fy2, fy3;
  double t1 = xb*xb-xa*xc, t2, t[5];
  /* sub-divide curve at gradient sign changes */
  if (xa == 0) { /* horizontal */
    if (abs(xc) < 2*abs(xb)) t[n++] = xc/(2.0*xb); /* one change */
  } else if (t1 > 0.0) { /* two changes */
    t2 = sqrt(t1);
    t1 = (xb-t2)/xa; if (fabs(t1) < 1.0) t[n++] = t1;
    t1 = (xb+t2)/xa; if (fabs(t1) < 1.0) t[n++] = t1;
  }
  t1 = yb*yb-ya*yc;
  if (ya == 0) { /* vertical */
    if (abs(yc) < 2*abs(yb)) t[n++] = yc/(2.0*yb); /* one change */
  } else if (t1 > 0.0) { /* two changes */
    t2 = sqrt(t1);
    t1 = (yb-t2)/ya; if (fabs(t1) < 1.0) t[n++] = t1;
    t1 = (yb+t2)/ya; if (fabs(t1) < 1.0) t[n++] = t1;
  }
  for (i = 1; i < n; i++) /* bubble sort of 4 points */
    if ((t1 = t[i-1]) > t[i]) { t[i-1] = t[i]; t[i] = t1; i = 0; }
    t1 = -1.0; t[n] = 1.0; /* begin / end point */
    for (i = 0; i <= n; i++) { /* plot each segment separately */
    t2 = t[i]; /* sub-divide at t[i-1], t[i] */
    fx1 = (t1*(t1*xb-2*xc)-t2*(t1*(t1*xa-2*xb)+xc)+xd)/8-fx0;
    fy1 = (t1*(t1*yb-2*yc)-t2*(t1*(t1*ya-2*yb)+yc)+yd)/8-fy0;
    fx2 = (t2*(t2*xb-2*xc)-t1*(t2*(t2*xa-2*xb)+xc)+xd)/8-fx0;
    fy2 = (t2*(t2*yb-2*yc)-t1*(t2*(t2*ya-2*yb)+yc)+yd)/8-fy0;
    fx0 -= fx3 = (t2*(t2*(3*xb-t2*xa)-3*xc)+xd)/8;
    fy0 -= fy3 = (t2*(t2*(3*yb-t2*ya)-3*yc)+yd)/8;
    x3 = floor(fx3+0.5); y3 = floor(fy3+0.5); /* scale bounds to int */
    if (fx0 != 0.0) { fx1 *= fx0 = (x0-x3)/fx0; fx2 *= fx0; }
    if (fy0 != 0.0) { fy1 *= fy0 = (y0-y3)/fy0; fy2 *= fy0; }
    if (x0 != x3 || y0 != y3) /* segment t1 - t2 */
      plotCubicBezierSeg(x0,y0, x0+fx1,y0+fy1, x0+fx2,y0+fy2, x3,y3);
    x0 = x3; y0 = y3; fx0 = fx3; fy0 = fy3; t1 = t2;
  }
}

和

void plotCubicBezierSeg(int x0, int y0, float x1, float y1,
  float x2, float y2, int x3, int y3)
{ /* plot limited cubic Bezier segment */
  int f, fx, fy, leg = 1;
  int sx = x0 < x3 ? 1 : -1, sy = y0 < y3 ? 1 : -1; /* step direction */
  float xc = -fabs(x0+x1-x2-x3), xa = xc-4*sx*(x1-x2), xb = sx*(x0-x1-x2+x3);
  float yc = -fabs(y0+y1-y2-y3), ya = yc-4*sy*(y1-y2), yb = sy*(y0-y1-y2+y3);
  double ab, ac, bc, cb, xx, xy, yy, dx, dy, ex, *pxy, EP = 0.01;

  /* check for curve restrains */
  /* slope P0-P1 == P2-P3 and (P0-P3 == P1-P2 or no slope change) */
  assert((x1-x0)*(x2-x3) < EP && ((x3-x0)*(x1-x2) < EP || xb*xb < xa*xc+EP));
  assert((y1-y0)*(y2-y3) < EP && ((y3-y0)*(y1-y2) < EP || yb*yb < ya*yc+EP));
  if (xa == 0 && ya == 0) { /* quadratic Bezier */
    sx = floor((3*x1-x0+1)/2); sy = floor((3*y1-y0+1)/2); /* new midpoint */
    return plotQuadBezierSeg(x0,y0, sx,sy, x3,y3);
  }
  x1 = (x1-x0)*(x1-x0)+(y1-y0)*(y1-y0)+1; /* line lengths */
  x2 = (x2-x3)*(x2-x3)+(y2-y3)*(y2-y3)+1;
  do { /* loop over both ends */
    ab = xa*yb-xb*ya; ac = xa*yc-xc*ya; bc = xb*yc-xc*yb;
    ex = ab*(ab+ac-3*bc)+ac*ac; /* P0 part of self-intersection loop? */
    f = ex > 0 ? 1 : sqrt(1+1024/x1); /* calculate resolution */
    ab *= f; ac *= f; bc *= f; ex *= f*f; /* increase resolution */
    xy = 9*(ab+ac+bc)/8; cb = 8*(xa-ya);/* init differences of 1st degree */
    dx = 27*(8*ab*(yb*yb-ya*yc)+ex*(ya+2*yb+yc))/64-ya*ya*(xy-ya);
    dy = 27*(8*ab*(xb*xb-xa*xc)-ex*(xa+2*xb+xc))/64-xa*xa*(xy+xa);
    /* init differences of 2nd degree */
    xx = 3*(3*ab*(3*yb*yb-ya*ya-2*ya*yc)-ya*(3*ac*(ya+yb)+ya*cb))/4;
    yy = 3*(3*ab*(3*xb*xb-xa*xa-2*xa*xc)-xa*(3*ac*(xa+xb)+xa*cb))/4;
    xy = xa*ya*(6*ab+6*ac-3*bc+cb); ac = ya*ya; cb = xa*xa;
    xy = 3*(xy+9*f*(cb*yb*yc-xb*xc*ac)-18*xb*yb*ab)/8;
    if (ex < 0) { /* negate values if inside self-intersection loop */
      dx = -dx; dy = -dy; xx = -xx; yy = -yy; xy = -xy; ac = -ac; cb = -cb;
    } /* init differences of 3rd degree */
    ab = 6*ya*ac; ac = -6*xa*ac; bc = 6*ya*cb; cb = -6*xa*cb;
    dx += xy; ex = dx+dy; dy += xy; /* error of 1st step */
    for (pxy = &xy, fx = fy = f; x0 != x3 && y0 != y3; ) {
      setPixel(x0,y0); /* plot curve */
      do { /* move sub-steps of one pixel */
        if (dx > *pxy || dy < *pxy) goto exit; /* confusing values */
        y1 = 2*ex-dy; /* save value for test of y step */
        if (2*ex >= dx) { /* x sub-step */
          fx--; ex += dx += xx; dy += xy += ac; yy += bc; xx += ab;
        }
        if (y1 <= 0) { /* y sub-step */
          fy--; ex += dy += yy; dx += xy += bc; xx += ac; yy += cb;
        }
      } while (fx > 0 && fy > 0); /* pixel complete? */
      if (2*fx <= f) { x0 += sx; fx += f; } /* x step */
      if (2*fy <= f) { y0 += sy; fy += f; } /* y step */
      if (pxy == &xy && dx < 0 && dy > 0) pxy = &EP;/* pixel ahead valid */
    }
    exit: xx = x0; x0 = x3; x3 = xx; sx = -sx; xb = -xb; /* swap legs */
    yy = y0; y0 = y3; y3 = yy; sy = -sy; yb = -yb; x1 = x2;
  } while (leg--); /* try other end */
  plotLine(x0,y0, x3,y3); /* remaining part in case of cusp or crunode */
}

正如Mike’Pomax’Kamermans所指出的那样，该站​​点上三次Bezier曲线的解决方案并不完整。特别是三次贝塞尔曲线的抗锯齿问题，对有理三次贝塞尔曲线的讨论还不完整。