algorithm - 逐像素贝塞尔曲线

Question

我通过谷歌找到的二次/三次贝塞尔曲线代码主要通过将线分割为一系列点并用直线连接它们来工作。光栅化发生在直线算法中，而不是在贝塞尔算法中。 Bresenham 等算法逐个像素地栅格化一条线，并且可以进行优化(参见 Po-Han Lin's solution)。

什么是二次贝塞尔曲线算法，它像直线算法一样逐像素工作，而不是通过绘制一系列点？

Answer 1

Bresenham 算法的一个变体适用于二次函数，例如圆、椭圆和抛物线，因此它也应该适用于二次贝塞尔曲线。

我打算尝试一个实现，但后来我在网上找到了一个:http://members.chello.at/~easyfilter/bresenham.html .

如果您需要更多详细信息或其他示例，上面提到的页面有一个 100 页 PDF 的链接，详细说明了该方法:http://members.chello.at/~easyfilter/Bresenham.pdf .

以下代码来自 Alois Zingl 的网站，用于绘制任何二次贝塞尔曲线。第一个例程在水平和垂直梯度变化处分割曲线:

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段贝塞尔曲线(没有梯度变化):

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 */
}

网站上还提供了抗锯齿代码。

Zingl 网站上的三次贝塞尔曲线对应的函数是

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 指出的那样，网站上的三次贝塞尔曲线的解决方案并不完整；特别是，抗锯齿三次贝塞尔曲线存在问题，并且对有理三次贝塞尔曲线的讨论不完整。