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randvoorhies/bresenham.js

Created August 3, 2017 16:10
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bresenham.js
/**
* Bresenham Curve Rasterizing Algorithms
* @author Zingl Alois
* @date 17.12.2014
* @version 1.3
* @url http://members.chello.at/easyfilter/bresenham.html
*/
function assert(a) {
if (!a) console.log("Assertion failed in bresenham.js "+a);
return a;
}
function plotLine(x0, y0, x1, y1)
{
var dx = Math.abs(x1-x0), sx = x0<x1 ? 1 : -1;
var dy = -Math.abs(y1-y0), sy = y0<y1 ? 1 : -1;
var err = dx+dy, e2; /* error value e_xy */
for (;;){ /* loop */
setPixel(x0,y0);
if (x0 == x1 && y0 == y1) break;
e2 = 2*err;
if (e2 >= dy) { err += dy; x0 += sx; } /* x step */
if (e2 <= dx) { err += dx; y0 += sy; } /* y step */
}
}
function plotEllipse(xm, ym, a, b)
{
var x = -a, y = 0; /* II. quadrant from bottom left to top right */
var e2, dx = (1+2*x)*b*b; /* error increment */
var dy = x*x, err = dx+dy; /* error of 1.step */
do {
setPixel(xm-x, ym+y); /* I. Quadrant */
setPixel(xm+x, ym+y); /* II. Quadrant */
setPixel(xm+x, ym-y); /* III. Quadrant */
setPixel(xm-x, ym-y); /* IV. Quadrant */
e2 = 2*err;
if (e2 >= dx) { x++; err += dx += 2*b*b; } /* x step */
if (e2 <= dy) { y++; err += dy += 2*a*a; } /* y step */
} while (x <= 0);
while (y++ < b) { /* too early stop for flat ellipses with a=1, */
setPixel(xm, ym+y); /* -> finish tip of ellipse */
setPixel(xm, ym-y);
}
}
function plotCircle(xm, ym, r)
{
var x = -r, y = 0, err = 2-2*r; /* bottom left to top right */
do {
setPixel(xm-x, ym+y); /* I. Quadrant +x +y */
setPixel(xm-y, ym-x); /* II. Quadrant -x +y */
setPixel(xm+x, ym-y); /* III. Quadrant -x -y */
setPixel(xm+y, ym+x); /* IV. Quadrant +x -y */
r = err;
if (r <= y) err += ++y*2+1; /* y step */
if (r > x || err > y) err += ++x*2+1; /* x step */
} while (x < 0);
}
function plotEllipseRect(x0, y0, x1, y1)
{ /* rectangular parameter enclosing the ellipse */
var a = Math.abs(x1-x0), b = Math.abs(y1-y0), b1 = b&1; /* diameter */
var dx = 4*(1.0-a)*b*b, dy = 4*(b1+1)*a*a; /* error increment */
var err = dx+dy+b1*a*a, e2; /* error of 1.step */
if (x0 > x1) { x0 = x1; x1 += a; } /* if called with swapped points */
if (y0 > y1) y0 = y1; /* .. exchange them */
y0 += (b+1)>>1; y1 = y0-b1; /* starting pixel */
a = 8*a*a; b1 = 8*b*b;
do {
setPixel(x1, y0); /* I. Quadrant */
setPixel(x0, y0); /* II. Quadrant */
setPixel(x0, y1); /* III. Quadrant */
setPixel(x1, y1); /* IV. Quadrant */
e2 = 2*err;
if (e2 <= dy) { y0++; y1--; err += dy += a; } /* y step */
if (e2 >= dx || 2*err > dy) { x0++; x1--; err += dx += b1; } /* x */
} while (x0 <= x1);
while (y0-y1 <= b) { /* too early stop of flat ellipses a=1 */
setPixel(x0-1, y0); /* -> finish tip of ellipse */
setPixel(x1+1, y0++);
setPixel(x0-1, y1);
setPixel(x1+1, y1--);
}
}
function plotQuadBezierSeg(x0, y0, x1, y1, x2, y2)
{ /* plot a limited quadratic Bezier segment */
var sx = x2-x1, sy = y2-y1;
var xx = x0-x1, yy = y0-y1, xy; /* relative values for checks */
var dx, dy, err, cur = xx*sy-yy*sx; /* curvature */
assert(xx*sx <= 0 && yy*sy <= 0); /* sign of gradient must not change */
if (sx*sx+sy*sy > xx*xx+yy*yy) { /* begin with shorter 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 */
}
function plotQuadBezier(x0, y0, x1, y1, x2, y2)
{ /* plot any quadratic Bezier curve */
var x = x0-x1, y = y0-y1, t = x0-2*x1+x2, r;
if (x*(x2-x1) > 0) { /* horizontal cut at P4? */
if (y*(y2-y1) > 0) /* vertical cut at P6 too? */
if (Math.abs((y0-2*y1+y2)/t*x) > Math.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 = Math.floor(t+0.5); y = Math.floor(r+0.5);
r = (y1-y0)*(t-x0)/(x1-x0)+y0; /* intersect P3 | P0 P1 */
plotQuadBezierSeg(x0,y0, x,Math.floor(r+0.5), x,y);
r = (y1-y2)*(t-x2)/(x1-x2)+y2; /* intersect P4 | P1 P2 */
x0 = x1 = x; y0 = y; y1 = Math.floor(r+0.5); /* P0 = P4, P1 = P8 */
}
if ((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 = Math.floor(r+0.5); y = Math.floor(t+0.5);
r = (x1-x0)*(t-y0)/(y1-y0)+x0; /* intersect P6 | P0 P1 */
plotQuadBezierSeg(x0,y0, Math.floor(r+0.5),y, x,y);
r = (x1-x2)*(t-y2)/(y1-y2)+x2; /* intersect P7 | P1 P2 */
x0 = x; x1 = Math.floor(r+0.5); y0 = y1 = y; /* P0 = P6, P1 = P7 */
}
plotQuadBezierSeg(x0,y0, x1,y1, x2,y2); /* remaining part */
}
function plotQuadRationalBezierSeg(x0, y0, x1, y1, x2, y2, w)
{ /* plot a limited rational Bezier segment, squared weight */
var sx = x2-x1, sy = y2-y1; /* relative values for checks */
var dx = x0-x2, dy = y0-y2, xx = x0-x1, yy = y0-y1;
var xy = xx*sy+yy*sx, cur = xx*sy-yy*sx, err; /* curvature */
assert(xx*sx <= 0.0 && yy*sy <= 0.0); /* sign of gradient must not change */
if (cur != 0.0 && w > 0.0) { /* no straight line */
if (sx*sx+sy*sy > xx*xx+yy*yy) { /* begin with shorter part */
x2 = x0; x0 -= dx; y2 = y0; y0 -= dy; cur = -cur; /* swap P0 P2 */
}
xx = 2.0*(4.0*w*sx*xx+dx*dx); /* differences 2nd degree */
yy = 2.0*(4.0*w*sy*yy+dy*dy);
sx = x0 < x2 ? 1 : -1; /* x step direction */
sy = y0 < y2 ? 1 : -1; /* y step direction */
xy = -2.0*sx*sy*(2.0*w*xy+dx*dy);
if (cur*sx*sy < 0.0) { /* negated curvature? */
xx = -xx; yy = -yy; xy = -xy; cur = -cur;
}
dx = 4.0*w*(x1-x0)*sy*cur+xx/2.0+xy; /* differences 1st degree */
dy = 4.0*w*(y0-y1)*sx*cur+yy/2.0+xy;
if (w < 0.5 && (dy > xy || dx < xy)) { /* flat ellipse, algorithm fails */
cur = (w+1.0)/2.0; w = Math.sqrt(w); xy = 1.0/(w+1.0);
sx = Math.floor((x0+2.0*w*x1+x2)*xy/2.0+0.5);/*subdivide curve in half */
sy = Math.floor((y0+2.0*w*y1+y2)*xy/2.0+0.5);
dx = Math.floor((w*x1+x0)*xy+0.5); dy = Math.floor((y1*w+y0)*xy+0.5);
plotQuadRationalBezierSeg(x0,y0, dx,dy, sx,sy, cur);/* plot separately */
dx = Math.floor((w*x1+x2)*xy+0.5); dy = Math.floor((y1*w+y2)*xy+0.5);
plotQuadRationalBezierSeg(sx,sy, dx,dy, x2,y2, cur);
return;
}
err = dx+dy-xy; /* error 1.step */
do {
setPixel(x0,y0); /* plot curve */
if (x0 == x2 && y0 == y2) return; /* last pixel -> curve finished */
x1 = 2*err > dy; y1 = 2*(err+yy) < -dy;/* save value for test of x step */
if (2*err < dx || y1) { y0 += sy; dy += xy; err += dx += xx; }/* y step */
if (2*err > dx || x1) { x0 += sx; dx += xy; err += dy += yy; }/* x step */
} while (dy <= xy && dx >= xy); /* gradient negates -> algorithm fails */
}
plotLine(x0,y0, x2,y2); /* plot remaining needle to end */
}
function plotQuadRationalBezier(x0, y0, x1, y1, x2, y2, w)
{ /* plot any quadratic rational Bezier curve */
var x = x0-2*x1+x2, y = y0-2*y1+y2;
var xx = x0-x1, yy = y0-y1, ww, t, q;
assert(w >= 0.0);
if (xx*(x2-x1) > 0) { /* horizontal cut at P4? */
if (yy*(y2-y1) > 0) /* vertical cut at P6 too? */
if (Math.abs(xx*y) > Math.abs(yy*x)) { /* which first? */
x0 = x2; x2 = xx+x1; y0 = y2; y2 = yy+y1; /* swap points */
} /* now horizontal cut at P4 comes first */
if (x0 == x2 || w == 1.0) t = (x0-x1)/x;
else { /* non-rational or rational case */
q = Math.sqrt(4.0*w*w*(x0-x1)*(x2-x1)+(x2-x0)*(x2-x0));
if (x1 < x0) q = -q;
t = (2.0*w*(x0-x1)-x0+x2+q)/(2.0*(1.0-w)*(x2-x0)); /* t at P4 */
}
q = 1.0/(2.0*t*(1.0-t)*(w-1.0)+1.0); /* sub-divide at t */
xx = (t*t*(x0-2.0*w*x1+x2)+2.0*t*(w*x1-x0)+x0)*q; /* = P4 */
yy = (t*t*(y0-2.0*w*y1+y2)+2.0*t*(w*y1-y0)+y0)*q;
ww = t*(w-1.0)+1.0; ww *= ww*q; /* squared weight P3 */
w = ((1.0-t)*(w-1.0)+1.0)*Math.sqrt(q); /* weight P8 */
x = Math.floor(xx+0.5); y = Math.floor(yy+0.5); /* P4 */
yy = (xx-x0)*(y1-y0)/(x1-x0)+y0; /* intersect P3 | P0 P1 */
plotQuadRationalBezierSeg(x0,y0, x,Math.floor(yy+0.5), x,y, ww);
yy = (xx-x2)*(y1-y2)/(x1-x2)+y2; /* intersect P4 | P1 P2 */
y1 = Math.floor(yy+0.5); x0 = x1 = x; y0 = y; /* P0 = P4, P1 = P8 */
}
if ((y0-y1)*(y2-y1) > 0) { /* vertical cut at P6? */
if (y0 == y2 || w == 1.0) t = (y0-y1)/(y0-2.0*y1+y2);
else { /* non-rational or rational case */
q = Math.sqrt(4.0*w*w*(y0-y1)*(y2-y1)+(y2-y0)*(y2-y0));
if (y1 < y0) q = -q;
t = (2.0*w*(y0-y1)-y0+y2+q)/(2.0*(1.0-w)*(y2-y0)); /* t at P6 */
}
q = 1.0/(2.0*t*(1.0-t)*(w-1.0)+1.0); /* sub-divide at t */
xx = (t*t*(x0-2.0*w*x1+x2)+2.0*t*(w*x1-x0)+x0)*q; /* = P6 */
yy = (t*t*(y0-2.0*w*y1+y2)+2.0*t*(w*y1-y0)+y0)*q;
ww = t*(w-1.0)+1.0; ww *= ww*q; /* squared weight P5 */
w = ((1.0-t)*(w-1.0)+1.0)*Math.sqrt(q); /* weight P7 */
x = Math.floor(xx+0.5); y = Math.floor(yy+0.5); /* P6 */
xx = (x1-x0)*(yy-y0)/(y1-y0)+x0; /* intersect P6 | P0 P1 */
plotQuadRationalBezierSeg(x0,y0, Math.floor(xx+0.5),y, x,y, ww);
xx = (x1-x2)*(yy-y2)/(y1-y2)+x2; /* intersect P7 | P1 P2 */
x1 = Math.floor(xx+0.5); x0 = x; y0 = y1 = y; /* P0 = P6, P1 = P7 */
}
plotQuadRationalBezierSeg(x0,y0, x1,y1, x2,y2, w*w); /* remaining */
}
function plotRotatedEllipse(x, y, a, b, angle)
{ /* plot ellipse rotated by angle (radian) */
var xd = a*a, yd = b*b;
var s = Math.sin(angle), zd = (xd-yd)*s; /* ellipse rotation */
xd = Math.sqrt(xd-zd*s), yd = Math.sqrt(yd+zd*s); /* surrounding rect */
a = Math.floor(xd+0.5); b = Math.floor(yd+0.5); zd = zd*a*b/(xd*yd);
plotRotatedEllipseRect(x-a,y-b, x+a,y+b, (4*zd*Math.cos(angle)));
}
function plotRotatedEllipseRect(x0, y0, x1, y1, zd)
{ /* rectangle enclosing the ellipse, integer rotation angle */
var xd = x1-x0, yd = y1-y0, w = xd*yd;
if (zd == 0) return plotEllipseRect(x0,y0, x1,y1); /* looks nicer */
if (w != 0.0) w = (w-zd)/(w+w); /* squared weight of P1 */
assert(w <= 1.0 && w >= 0.0); /* limit angle to |zd|<=xd*yd */
xd = Math.floor(xd*w+0.5); yd = Math.floor(yd*w+0.5); /* snap to int */
plotQuadRationalBezierSeg(x0,y0+yd, x0,y0, x0+xd,y0, 1.0-w);
plotQuadRationalBezierSeg(x0,y0+yd, x0,y1, x1-xd,y1, w);
plotQuadRationalBezierSeg(x1,y1-yd, x1,y1, x1-xd,y1, 1.0-w);
plotQuadRationalBezierSeg(x1,y1-yd, x1,y0, x0+xd,y0, w);
}
function plotCubicBezierSeg(x0, y0, x1, y1, x2, y2, x3, y3)
{ /* plot limited cubic Bezier segment */
var f, fx, fy, leg = 1;
var sx = x0 < x3 ? 1 : -1, sy = y0 < y3 ? 1 : -1; /* step direction */
var xc = -Math.abs(x0+x1-x2-x3), xa = xc-4*sx*(x1-x2), xb = sx*(x0-x1-x2+x3);
var yc = -Math.abs(y0+y1-y2-y3), ya = yc-4*sy*(y1-y2), yb = sy*(y0-y1-y2+y3);
var 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 */
return plotQuadBezierSeg(x0,y0, (3*x1-x0)>>1,(3*y1-y0)>>1, 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 : Math.floor(Math.sqrt(1+1024/x1)); /* calc 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 */
exit:
for (pxy = 0, fx = fy = f; x0 != x3 && y0 != y3; ) {
setPixel(x0,y0); /* plot curve */
do { /* move sub-steps of one pixel */
if (pxy == 0) if (dx > xy || dy < xy) break exit; /* confusing */
if (pxy == 1) if (dx > 0 || dy < 0) break exit; /* 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;
} else if (y1 > 0) break exit;
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 == 0 && dx < 0 && dy > 0) pxy = 1; /* pixel ahead valid */
}
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 */
}
function plotCubicBezier(x0, y0, x1, y1, x2, y2, x3, y3)
{ /* plot any cubic Bezier curve */
var n = 0, i = 0;
var xc = x0+x1-x2-x3, xa = xc-4*(x1-x2);
var xb = x0-x1-x2+x3, xd = xb+4*(x1+x2);
var yc = y0+y1-y2-y3, ya = yc-4*(y1-y2);
var yb = y0-y1-y2+y3, yd = yb+4*(y1+y2);
var fx0 = x0, fx1, fx2, fx3, fy0 = y0, fy1, fy2, fy3;
var t1 = xb*xb-xa*xc, t2, t = new Array(5);
/* sub-divide curve at gradient sign changes */
if (xa == 0) { /* horizontal */
if (Math.abs(xc) < 2*Math.abs(xb)) t[n++] = xc/(2.0*xb); /* one change */
} else if (t1 > 0.0) { /* two changes */
t2 = Math.sqrt(t1);
t1 = (xb-t2)/xa; if (Math.abs(t1) < 1.0) t[n++] = t1;
t1 = (xb+t2)/xa; if (Math.abs(t1) < 1.0) t[n++] = t1;
}
t1 = yb*yb-ya*yc;
if (ya == 0) { /* vertical */
if (Math.abs(yc) < 2*Math.abs(yb)) t[n++] = yc/(2.0*yb); /* one change */
} else if (t1 > 0.0) { /* two changes */
t2 = Math.sqrt(t1);
t1 = (yb-t2)/ya; if (Math.abs(t1) < 1.0) t[n++] = t1;
t1 = (yb+t2)/ya; if (Math.abs(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 = Math.floor(fx3+0.5); y3 = Math.floor(fy3+0.5); /* scale bounds */
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;
}
}
function plotLineAA(x0, y0, x1, y1)
{ /* draw a black (0) anti-aliased line on white (255) background */
var dx = Math.abs(x1-x0), sx = x0 < x1 ? 1 : -1;
var dy = Math.abs(y1-y0), sy = y0 < y1 ? 1 : -1;
var err = dx-dy, e2, x2; /* error value e_xy */
var ed = dx+dy == 0 ? 1 : Math.sqrt(dx*dx+dy*dy);
for ( ; ; ){ /* pixel loop */
setPixelAA(x0,y0, 255*Math.abs(err-dx+dy)/ed);
e2 = err; x2 = x0;
if (2*e2 >= -dx) { /* x step */
if (x0 == x1) break;
if (e2+dy < ed) setPixelAA(x0,y0+sy, 255*(e2+dy)/ed);
err -= dy; x0 += sx;
}
if (2*e2 <= dy) { /* y step */
if (y0 == y1) break;
if (dx-e2 < ed) setPixelAA(x2+sx,y0, 255*(dx-e2)/ed);
err += dx; y0 += sy;
}
}
}
function plotCircleAA(xm, ym, r)
{ /* draw a black anti-aliased circle on white background */
var x = r, y = 0; /* II. quadrant from bottom left to top right */
var i, x2, e2, err = 2-2*r; /* error of 1.step */
r = 1-err;
for ( ; ; ) {
i = 255*Math.abs(err+2*(x+y)-2)/r; /* get blend value of pixel */
setPixelAA(xm+x, ym-y, i); /* I. Quadrant */
setPixelAA(xm+y, ym+x, i); /* II. Quadrant */
setPixelAA(xm-x, ym+y, i); /* III. Quadrant */
setPixelAA(xm-y, ym-x, i); /* IV. Quadrant */
if (x == 0) break;
e2 = err; x2 = x; /* remember values */
if (err > y) { /* x step */
i = 255*(err+2*x-1)/r; /* outward pixel */
if (i < 255) {
setPixelAA(xm+x, ym-y+1, i);
setPixelAA(xm+y-1, ym+x, i);
setPixelAA(xm-x, ym+y-1, i);
setPixelAA(xm-y+1, ym-x, i);
}
err -= --x*2-1;
}
if (e2 <= x2--) { /* y step */
i = 255*(1-2*y-e2)/r; /* inward pixel */
if (i < 255) {
setPixelAA(xm+x2, ym-y, i);
setPixelAA(xm+y, ym+x2, i);
setPixelAA(xm-x2, ym+y, i);
setPixelAA(xm-y, ym-x2, i);
}
err -= --y*2-1;
}
}
}
function plotEllipseRectAA(x0, y0, x1, y1)
{ /* draw a black anti-aliased rectangular ellipse on white background */
var a = Math.abs(x1-x0), b = Math.abs(y1-y0), b1 = b&1; /* diameter */
var dx = 4*(a-1)*b*b, dy = 4*(b1+1)*a*a; /* error increment */
var f, ed, i, err = b1*a*a-dx+dy; /* error of 1.step */
if (a == 0 || b == 0) return plotLine(x0,y0, x1,y1);
if (x0 > x1) { x0 = x1; x1 += a; } /* if called with swapped points */
if (y0 > y1) y0 = y1; /* .. exchange them */
y0 += (b+1)>>1; y1 = y0-b1; /* starting pixel */
a = 8*a*a; b1 = 8*b*b;
for (;;) { /* approximate ed=Math.sqrt(dx*dx+dy*dy) */
i = Math.min(dx,dy); ed = Math.max(dx,dy);
if (y0 == y1+1 && err > dy && a > b1) ed = 255*4/a; /* x-tip */
else ed = 255/(ed+2*ed*i*i/(4*ed*ed+i*i)); /* approximation */
i = ed*Math.abs(err+dx-dy); /* get intensity value by pixel err */
setPixelAA(x0,y0, i); setPixelAA(x0,y1, i);
setPixelAA(x1,y0, i); setPixelAA(x1,y1, i);
if (f = 2*err+dy >= 0) { /* x step, remember condition */
if (x0 >= x1) break;
i = ed*(err+dx);
if (i < 256) {
setPixelAA(x0,y0+1, i); setPixelAA(x0,y1-1, i);
setPixelAA(x1,y0+1, i); setPixelAA(x1,y1-1, i);
} /* do error increment later since values are still needed */
}
if (2*err <= dx) { /* y step */
i = ed*(dy-err);
if (i < 256) {
setPixelAA(x0+1,y0, i); setPixelAA(x1-1,y0, i);
setPixelAA(x0+1,y1, i); setPixelAA(x1-1,y1, i);
}
y0++; y1--; err += dy += a;
}
if (f) { x0++; x1--; err -= dx -= b1; } /* x error increment */
}
if (--x0 == x1++) /* too early stop of flat ellipses */
while (y0-y1 < b) {
i = 255*4*Math.abs(err+dx)/b1; /* -> finish tip of ellipse */
setPixelAA(x0,++y0, i); setPixelAA(x1,y0, i);
setPixelAA(x0,--y1, i); setPixelAA(x1,y1, i);
err += dy += a;
}
}
function plotQuadBezierSegAA(x0, y0, x1, y1, x2, y2)
{ /* draw an limited anti-aliased quadratic Bezier segment */
var sx = x2-x1, sy = y2-y1;
var xx = x0-x1, yy = y0-y1, xy; /* relative values for checks */
var dx, dy, err, ed, cur = xx*sy-yy*sx; /* curvature */
assert(xx*sx <= 0 && yy*sy <= 0); /* sign of gradient must not change */
if (sx*sx+sy*sy > xx*xx+yy*yy) { /* begin with shorter 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*(x1-x0)*cur+xx-xy; /* differences 1st degree */
dy = 4.0*sx*(y0-y1)*cur+yy-xy;
xx += xx; yy += yy; err = dx+dy+xy; /* error 1st step */
do {
cur = Math.min(dx+xy,-xy-dy);
ed = Math.max(dx+xy,-xy-dy); /* approximate error distance */
ed += 2*ed*cur*cur/(4*ed*ed+cur*cur);
setPixelAA(x0,y0, 255*Math.abs(err-dx-dy-xy)/ed); /* plot curve */
if (x0 == x2 || y0 == y2) break; /* last pixel -> curve finished */
x1 = x0; cur = dx-err; y1 = 2*err+dy < 0;
if (2*err+dx > 0) { /* x step */
if (err-dy < ed) setPixelAA(x0,y0+sy, 255*Math.abs(err-dy)/ed);
x0 += sx; dx -= xy; err += dy += yy;
}
if (y1) { /* y step */
if (cur < ed) setPixelAA(x1+sx,y0, 255*Math.abs(cur)/ed);
y0 += sy; dy -= xy; err += dx += xx;
}
} while (dy < dx); /* gradient negates -> close curves */
}
plotLineAA(x0,y0, x2,y2); /* plot remaining needle to end */
}
function plotQuadBezierAA(x0, y0, x1, y1, x2, y2)
{ /* plot any anti-aliased quadratic Bezier curve */
var x = x0-x1, y = y0-y1, t = x0-2*x1+x2, r;
if (x*(x2-x1) > 0) { /* horizontal cut at P4? */
if (y*(y2-y1) > 0) /* vertical cut at P6 too? */
if (Math.abs((y0-2*y1+y2)/t*x) > Math.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 = Math.floor(t+0.5); y = Math.floor(r+0.5);
r = (y1-y0)*(t-x0)/(x1-x0)+y0; /* intersect P3 | P0 P1 */
plotQuadBezierSegAA(x0,y0, x,Math.floor(r+0.5), x,y);
r = (y1-y2)*(t-x2)/(x1-x2)+y2; /* intersect P4 | P1 P2 */
x0 = x1 = x; y0 = y; y1 = Math.floor(r+0.5); /* P0 = P4, P1 = P8 */
}
if ((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 = Math.floor(r+0.5); y = Math.floor(t+0.5);
r = (x1-x0)*(t-y0)/(y1-y0)+x0; /* intersect P6 | P0 P1 */
plotQuadBezierSegAA(x0,y0, Math.floor(r+0.5),y, x,y);
r = (x1-x2)*(t-y2)/(y1-y2)+x2; /* intersect P7 | P1 P2 */
x0 = x; x1 = Math.floor(r+0.5); y0 = y1 = y; /* P0 = P6, P1 = P7 */
}
plotQuadBezierSegAA(x0,y0, x1,y1, x2,y2); /* remaining part */
}
function plotQuadRationalBezierSegAA(x0, y0, x1, y1, x2, y2, w)
{ /* draw an anti-aliased rational quadratic Bezier segment, squared weight */
var sx = x2-x1, sy = y2-y1; /* relative values for checks */
var dx = x0-x2, dy = y0-y2, xx = x0-x1, yy = y0-y1;
var xy = xx*sy+yy*sx, cur = xx*sy-yy*sx, err, ed, f; /* curvature */
assert(xx*sx <= 0.0 && yy*sy <= 0.0); /* sign of gradient must not change */
if (cur != 0.0 && w > 0.0) { /* no straight line */
if (sx*sx+sy*sy > xx*xx+yy*yy) { /* begin with shorter part */
x2 = x0; x0 -= dx; y2 = y0; y0 -= dy; cur = -cur; /* swap P0 P2 */
}
xx = 2.0*(4.0*w*sx*xx+dx*dx); /* differences 2nd degree */
yy = 2.0*(4.0*w*sy*yy+dy*dy);
sx = x0 < x2 ? 1 : -1; /* x step direction */
sy = y0 < y2 ? 1 : -1; /* y step direction */
xy = -2.0*sx*sy*(2.0*w*xy+dx*dy);
if (cur*sx*sy < 0) { /* negated curvature? */
xx = -xx; yy = -yy; cur = -cur; xy = -xy;
}
dx = 4.0*w*(x1-x0)*sy*cur+xx/2.0+xy; /* differences 1st degree */
dy = 4.0*w*(y0-y1)*sx*cur+yy/2.0+xy;
if (w < 0.5 && dy > dx) { /* flat ellipse, algorithm fails */
cur = (w+1.0)/2.0; w = Math.sqrt(w); xy = 1.0/(w+1.0);
sx = Math.floor((x0+2.0*w*x1+x2)*xy/2.0+0.5); /* subdivide curve */
sy = Math.floor((y0+2.0*w*y1+y2)*xy/2.0+0.5);
dx = Math.floor((w*x1+x0)*xy+0.5); dy = Math.floor((y1*w+y0)*xy+0.5);
plotQuadRationalBezierSegAA(x0,y0, dx,dy, sx,sy, cur); /* plot apart */
dx = Math.floor((w*x1+x2)*xy+0.5); dy = Math.floor((y1*w+y2)*xy+0.5);
return plotQuadRationalBezierSegAA(sx,sy, dx,dy, x2,y2, cur);
}
err = dx+dy-xy; /* error 1st step */
do { /* pixel loop */
cur = Math.min(dx-xy,xy-dy); ed = Math.max(dx-xy,xy-dy);
ed += 2*ed*cur*cur/(4.*ed*ed+cur*cur); /* approximate error distance */
x1 = 255*Math.abs(err-dx-dy+xy)/ed;/* get blend value by pixel error */
if (x1 < 256) setPixelAA(x0,y0, x1); /* plot curve */
if (f = 2*err+dy < 0) { /* y step */
if (y0 == y2) return; /* last pixel -> curve finished */
if (dx-err < ed) setPixelAA(x0+sx,y0, 255*Math.abs(dx-err)/ed);
}
if (2*err+dx > 0) { /* x step */
if (x0 == x2) return; /* last pixel -> curve finished */
if (err-dy < ed) setPixelAA(x0,y0+sy, 255*Math.abs(err-dy)/ed);
x0 += sx; dx += xy; err += dy += yy;
}
if (f) { y0 += sy; dy += xy; err += dx += xx; } /* y step */
} while (dy < dx); /* gradient negates -> algorithm fails */
}
plotLineAA(x0,y0, x2,y2); /* plot remaining needle to end */
}
function plotQuadRationalBezierAA(x0, y0, x1, y1, x2, y2, w)
{ /* plot any anti-aliased quadratic rational Bezier curve */
var x = x0-2*x1+x2, y = y0-2*y1+y2;
var xx = x0-x1, yy = y0-y1, ww, t, q;
assert(w >= 0.0);
if (xx*(x2-x1) > 0) { /* horizontal cut at P4? */
if (yy*(y2-y1) > 0) /* vertical cut at P6 too? */
if (Math.abs(xx*y) > Math.abs(yy*x)) { /* which first? */
x0 = x2; x2 = xx+x1; y0 = y2; y2 = yy+y1; /* swap points */
} /* now horizontal cut at P4 comes first */
if (x0 == x2 || w == 1.0) t = (x0-x1)/x;
else { /* non-rational or rational case */
q = Math.sqrt(4.0*w*w*(x0-x1)*(x2-x1)+(x2-x0)*(x2-x0));
if (x1 < x0) q = -q;
t = (2.0*w*(x0-x1)-x0+x2+q)/(2.0*(1.0-w)*(x2-x0)); /* t at P4 */
}
q = 1.0/(2.0*t*(1.0-t)*(w-1.0)+1.0); /* sub-divide at t */
xx = (t*t*(x0-2.0*w*x1+x2)+2.0*t*(w*x1-x0)+x0)*q; /* = P4 */
yy = (t*t*(y0-2.0*w*y1+y2)+2.0*t*(w*y1-y0)+y0)*q;
ww = t*(w-1.0)+1.0; ww *= ww*q; /* squared weight P3 */
w = ((1.0-t)*(w-1.0)+1.0)*Math.sqrt(q); /* weight P8 */
x = Math.floor(xx+0.5); y = Math.floor(yy+0.5); /* P4 */
yy = (xx-x0)*(y1-y0)/(x1-x0)+y0; /* intersect P3 | P0 P1 */
plotQuadRationalBezierSegAA(x0,y0, x,Math.floor(yy+0.5), x,y, ww);
yy = (xx-x2)*(y1-y2)/(x1-x2)+y2; /* intersect P4 | P1 P2 */
y1 = Math.floor(yy+0.5); x0 = x1 = x; y0 = y; /* P0 = P4, P1 = P8 */
}
if ((y0-y1)*(y2-y1) > 0) { /* vertical cut at P6? */
if (y0 == y2 || w == 1.0) t = (y0-y1)/(y0-2.0*y1+y2);
else { /* non-rational or rational case */
q = Math.sqrt(4.0*w*w*(y0-y1)*(y2-y1)+(y2-y0)*(y2-y0));
if (y1 < y0) q = -q;
t = (2.0*w*(y0-y1)-y0+y2+q)/(2.0*(1.0-w)*(y2-y0)); /* t at P6 */
}
q = 1.0/(2.0*t*(1.0-t)*(w-1.0)+1.0); /* sub-divide at t */
xx = (t*t*(x0-2.0*w*x1+x2)+2.0*t*(w*x1-x0)+x0)*q; /* = P6 */
yy = (t*t*(y0-2.0*w*y1+y2)+2.0*t*(w*y1-y0)+y0)*q;
ww = t*(w-1.0)+1.0; ww *= ww*q; /* squared weight P5 */
w = ((1.0-t)*(w-1.0)+1.0)*Math.sqrt(q); /* weight P7 */
x = Math.floor(xx+0.5); y = Math.floor(yy+0.5); /* P6 */
xx = (x1-x0)*(yy-y0)/(y1-y0)+x0; /* intersect P6 | P0 P1 */
plotQuadRationalBezierSegAA(x0,y0, Math.floor(xx+0.5),y, x,y, ww);
xx = (x1-x2)*(yy-y2)/(y1-y2)+x2; /* intersect P7 | P1 P2 */
x1 = Math.floor(xx+0.5); x0 = x; y0 = y1 = y; /* P0 = P6, P1 = P7 */
}
plotQuadRationalBezierSegAA(x0,y0, x1,y1, x2,y2, w*w); /* remaining part */
}
function plotRotatedEllipseAA(x, y, a, b, angle)
{ /* plot ellipse rotated by angle (radian) */
var xd = a*a, yd = b*b;
var s = Math.sin(angle), zd = (xd-yd)*s; /* ellipse rotation */
xd = Math.sqrt(xd-zd*s), yd = Math.sqrt(yd+zd*s); /* surrounding rect*/
a = Math.floor(xd+0.5); b = Math.floor(yd+0.5); zd = zd*a*b/(xd*yd);
plotRotatedEllipseRectAA(x-a,y-b, x+a,y+b, (4*zd*Math.cos(angle)));
}
function plotRotatedEllipseRectAA(x0, y0, x1, y1, zd)
{ /* rectangle enclosing the ellipse, integer rotation angle */
var xd = x1-x0, yd = y1-y0, w = xd*yd;
if (w != 0.0) w = (w-zd)/(w+w); /* squared weight of P1 */
assert(w <= 1.0 && w >= 0.0); /* limit angle to |zd|<=xd*yd */
xd = Math.floor(xd*w+0.5); yd = Math.floor(yd*w+0.5); /* snap to int */
plotQuadRationalBezierSegAA(x0,y0+yd, x0,y0, x0+xd,y0, 1.0-w);
plotQuadRationalBezierSegAA(x0,y0+yd, x0,y1, x1-xd,y1, w);
plotQuadRationalBezierSegAA(x1,y1-yd, x1,y1, x1-xd,y1, 1.0-w);
plotQuadRationalBezierSegAA(x1,y1-yd, x1,y0, x0+xd,y0, w);
}
function plotCubicBezierSegAA(x0, y0, x1, y1, x2, y2, x3, y3)
{ /* plot limited anti-aliased cubic Bezier segment */
var f, fx, fy, leg = 1;
var sx = x0 < x3 ? 1 : -1, sy = y0 < y3 ? 1 : -1; /* step direction */
var xc = -Math.abs(x0+x1-x2-x3), xa = xc-4*sx*(x1-x2), xb = sx*(x0-x1-x2+x3);
var yc = -Math.abs(y0+y1-y2-y3), ya = yc-4*sy*(y1-y2), yb = sy*(y0-y1-y2+y3);
var ab, ac, bc, ba, xx, xy, yy, dx, dy, ex, px, py, ed, ip, 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 */
return plotQuadBezierSegAA(x0,y0, (3*x1-x0)>>1,(3*y1-y0)>>1, x3,y3);
x1 = (x1-x0)*(x1-x0)+(y1-y0)*(y1-y0)+1; /* line lengths */
x2 = (x2-x3)*(x2-x3)+(y2-y3)*(y2-y3)+1;
exit:
do { /* loop over both ends */
ab = xa*yb-xb*ya; ac = xa*yc-xc*ya; bc = xb*yc-xc*yb;
ip = 4*ab*bc-ac*ac; /* self intersection loop at all? */
ex = ab*(ab+ac-3*bc)+ac*ac; /* P0 part of self-intersection loop? */
f = ex > 0 ? 1 : Math.floor(Math.sqrt(1+1024/x1)); /* calc resolution */
ab *= f; ac *= f; bc *= f; ex *= f*f; /* increase resolution */
xy = 9*(ab+ac+bc)/8; ba = 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*ba))/4;
yy = 3*(3*ab*(3*xb*xb-xa*xa-2*xa*xc)-xa*(3*ac*(xa+xb)+xa*ba))/4;
xy = xa*ya*(6*ab+6*ac-3*bc+ba); ac = ya*ya; ba = xa*xa;
xy = 3*(xy+9*f*(ba*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; ba = -ba;
} /* init differences of 3rd degree */
ab = 6*ya*ac; ac = -6*xa*ac; bc = 6*ya*ba; ba = -6*xa*ba;
dx += xy; ex = dx+dy; dy += xy; /* error of 1st step */
loop:
for (fx = fy = f; ; ) {
if (x0 == x3 || y0 == y3) break exit;
y1 = Math.min(Math.abs(xy-dx),Math.abs(dy-xy));
ed = Math.max(Math.abs(xy-dx),Math.abs(dy-xy)); /* approximate err */
ed = f*(ed+2*ed*y1*y1/(4*ed*ed+y1*y1));
y1 = 255*Math.abs(ex-(f-fx+1)*dx-(f-fy+1)*dy+f*xy)/ed;
if (y1 < 256) setPixelAA(x0, y0, y1); /* plot curve */
px = Math.abs(ex-(f-fx+1)*dx+(fy-1)*dy); /* pixel varensity x move */
py = Math.abs(ex+(fx-1)*dx-(f-fy+1)*dy); /* pixel varensity y move */
y2 = y0;
do { /* move sub-steps of one pixel */
if (ip >= -EP) /* intersection possible? -> check.. */
if (dx+xx > xy || dy+yy < xy) break loop; /* two x or y steps */
y1 = 2*ex+dx; /* save value for test of y step */
if (2*ex+dy > 0) { /* x sub-step */
fx--; ex += dx += xx; dy += xy += ac; yy += bc; xx += ab;
} else if (y1 > 0) break loop; /* tiny nearly cusp */
if (y1 <= 0) { /* y sub-step */
fy--; ex += dy += yy; dx += xy += bc; xx += ac; yy += ba;
}
} while (fx > 0 && fy > 0); /* pixel complete? */
if (2*fy <= f) { /* x+ anti-aliasing pixel */
if (py < ed) setPixelAA(x0+sx, y0, 255*py/ed); /* plot curve */
y0 += sy; fy += f; /* y step */
}
if (2*fx <= f) { /* y+ anti-aliasing pixel */
if (px < ed) setPixelAA(x0, y2+sy, 255*px/ed); /* plot curve */
x0 += sx; fx += f; /* x step */
}
}
if (2*ex < dy && 2*fy <= f+2) { /* round x+ approximation pixel */
if (py < ed) setPixelAA(x0+sx, y0, 255*py/ed); /* plot curve */
y0 += sy;
}
if (2*ex > dx && 2*fx <= f+2) { /* round y+ approximation pixel */
if (px < ed) setPixelAA(x0, y2+sy, 255*px/ed); /* plot curve */
x0 += sx;
}
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 */
plotLineAA(x0,y0, x3,y3); /* remaining part in case of cusp or crunode */
}
function plotCubicBezierAA(x0, y0, x1, y1, x2, y2, x3, y3)
{ /* plot any cubic Bezier curve */
var n = 0, i = 0;
var xc = x0+x1-x2-x3, xa = xc-4*(x1-x2);
var xb = x0-x1-x2+x3, xd = xb+4*(x1+x2);
var yc = y0+y1-y2-y3, ya = yc-4*(y1-y2);
var yb = y0-y1-y2+y3, yd = yb+4*(y1+y2);
var fx0 = x0, fx1, fx2, fx3, fy0 = y0, fy1, fy2, fy3;
var t1 = xb*xb-xa*xc, t2, t = new Array(5);
/* sub-divide curve at gradient sign changes */
if (xa == 0) { /* horizontal */
if (Math.abs(xc) < 2*Math.abs(xb)) t[n++] = xc/(2.0*xb); /* one change */
} else if (t1 > 0.0) { /* two changes */
t2 = Math.sqrt(t1);
t1 = (xb-t2)/xa; if (Math.abs(t1) < 1.0) t[n++] = t1;
t1 = (xb+t2)/xa; if (Math.abs(t1) < 1.0) t[n++] = t1;
}
t1 = yb*yb-ya*yc;
if (ya == 0) { /* vertical */
if (Math.abs(yc) < 2*Math.abs(yb)) t[n++] = yc/(2.0*yb); /* one change */
} else if (t1 > 0.0) { /* two changes */
t2 = Math.sqrt(t1);
t1 = (yb-t2)/ya; if (Math.abs(t1) < 1.0) t[n++] = t1;
t1 = (yb+t2)/ya; if (Math.abs(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 = Math.floor(fx3+0.5); y3 = Math.floor(fy3+0.5); /* scale bounds */
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 */
plotCubicBezierSegAA(x0,y0, x0+fx1,y0+fy1, x0+fx2,y0+fy2, x3,y3);
x0 = x3; y0 = y3; fx0 = fx3; fy0 = fy3; t1 = t2;
}
}
function plotLineWidth(x0, y0, x1, y1, th)
{ /* plot an anti-aliased line of width th pixel */
var dx = Math.abs(x1-x0), sx = x0 < x1 ? 1 : -1;
var dy = Math.abs(y1-y0), sy = y0 < y1 ? 1 : -1;
var err, e2 = Math.sqrt(dx*dx+dy*dy); /* length */
if (th <= 1 || e2 == 0) return plotLineAA(x0,y0, x1,y1); /* assert */
dx *= 255/e2; dy *= 255/e2; th = 255*(th-1); /* scale values */
if (dx < dy) { /* steep line */
x1 = Math.round((e2+th/2)/dy); /* start offset */
err = x1*dy-th/2; /* shift error value to offset width */
for (x0 -= x1*sx; ; y0 += sy) {
setPixelAA(x1 = x0, y0, err); /* aliasing pre-pixel */
for (e2 = dy-err-th; e2+dy < 255; e2 += dy)
setPixel(x1 += sx, y0); /* pixel on the line */
setPixelAA(x1+sx, y0, e2); /* aliasing post-pixel */
if (y0 == y1) break;
err += dx; /* y-step */
if (err > 255) { err -= dy; x0 += sx; } /* x-step */
}
} else { /* flat line */
y1 = Math.round((e2+th/2)/dx); /* start offset */
err = y1*dx-th/2; /* shift error value to offset width */
for (y0 -= y1*sy; ; x0 += sx) {
setPixelAA(x0, y1 = y0, err); /* aliasing pre-pixel */
for (e2 = dx-err-th; e2+dx < 255; e2 += dx)
setPixel(x0, y1 += sy); /* pixel on the line */
setPixelAA(x0, y1+sy, e2); /* aliasing post-pixel */
if (x0 == x1) break;
err += dy; /* x-step */
if (err > 255) { err -= dx; y0 += sy; } /* y-step */
}
}
}
function plotEllipseRectWidth(x0, y0, x1, y1, th)
{ /* draw anti-aliased ellipse inside rectangle with thick line */
var a = Math.abs(x1-x0), b = Math.abs(y1-y0), b1 = b&1; /* outer diameter */
var a2 = a-2*th, b2 = b-2*th; /* inner diameter */
var dx = 4*(a-1)*b*b, dy = 4*(b1-1)*a*a; /* error increment */
var i = a+b2, err = b1*a*a, dx2, dy2, e2, ed;
/* thick line correction */
if (th < 1.5) return plotEllipseRectAA(x0,y0, x1,y1);
if ((th-1)*(2*b-th) > a*a) b2 = Math.sqrt(a*(b-a)*i*a2)/(a-th);
if ((th-1)*(2*a-th) > b*b) { a2 = Math.sqrt(b*(a-b)*i*b2)/(b-th); th = (a-a2)/2; }
if (a == 0 || b == 0) return plotLine(x0,y0, x1,y1);
if (x0 > x1) { x0 = x1; x1 += a; } /* if called with swapped points */
if (y0 > y1) y0 = y1; /* .. exchange them */
if (b2 <= 0) th = a; /* filled ellipse */
e2 = th-Math.floor(th); th = x0+th-e2;
dx2 = 4*(a2+2*e2-1)*b2*b2; dy2 = 4*(b1-1)*a2*a2; e2 = dx2*e2;
y0 += (b+1)>>1; y1 = y0-b1; /* starting pixel */
a = 8*a*a; b1 = 8*b*b; a2 = 8*a2*a2; b2 = 8*b2*b2;
do {
for (;;) {
if (err < 0 || x0 > x1) { i = x0; break; }
i = Math.min(dx,dy); ed = Math.max(dx,dy);
if (y0 == y1+1 && 2*err > dx && a > b1) ed = a/4; /* x-tip */
else ed += 2*ed*i*i/(4*ed*ed+i*i+1)+1;/* approx ed=sqrt(dx*dx+dy*dy) */
i = 255*err/ed; /* outside anti-aliasing */
setPixelAA(x0,y0, i); setPixelAA(x0,y1, i);
setPixelAA(x1,y0, i); setPixelAA(x1,y1, i);
if (err+dy+a < dx) { i = x0+1; break; }
x0++; x1--; err -= dx; dx -= b1; /* x error increment */
}
for (; i < th && 2*i <= x0+x1; i++) { /* fill line pixel */
setPixel(i,y0); setPixel(x0+x1-i,y0);
setPixel(i,y1); setPixel(x0+x1-i,y1);
}
while (e2 > 0 && x0+x1 >= 2*th) { /* inside anti-aliasing */
i = Math.min(dx2,dy2); ed = Math.max(dx2,dy2);
if (y0 == y1+1 && 2*e2 > dx2 && a2 > b2) ed = a2/4; /* x-tip */
else ed += 2*ed*i*i/(4*ed*ed+i*i); /* approximation */
i = 255-255*e2/ed; /* get intensity value by pixel error */
setPixelAA(th,y0, i); setPixelAA(x0+x1-th,y0, i);
setPixelAA(th,y1, i); setPixelAA(x0+x1-th,y1, i);
if (e2+dy2+a2 < dx2) break;
th++; e2 -= dx2; dx2 -= b2; /* x error increment */
}
e2 += dy2 += a2;
y0++; y1--; err += dy += a; /* y step */
} while (x0 < x1);
if (y0-y1 <= b)
{
if (err > dy+a) { y0--; y1++; err -= dy -= a; }
for (; y0-y1 <= b; err += dy += a) { /* too early stop of flat ellipses */
i = 255*4*err/b1; /* -> finish tip of ellipse */
setPixelAA(x0,y0, i); setPixelAA(x1,y0++, i);
setPixelAA(x0,y1, i); setPixelAA(x1,y1--, i);
}
}
}
function plotQuadRationalBezierWidthSeg(x0, y0, x1, y1, x2, y2, w, th)
{ /* plot a limited rational Bezier segment of thickness th, squared weight */
var sx = x2-x1, sy = y2-y1; /* relative values for checks */
var dx = x0-x2, dy = y0-y2, xx = x0-x1, yy = y0-y1;
var xy = xx*sy+yy*sx, cur = xx*sy-yy*sx, err, e2, ed; /* curvature */
assert(xx*sx <= 0.0 && yy*sy <= 0.0); /* sign of gradient must not change */
if (cur != 0.0 && w > 0.0) { /* no straight line */
if (sx*sx+sy*sy > xx*xx+yy*yy) { /* begin with longer part */
x2 = x0; x0 -= dx; y2 = y0; y0 -= dy; cur = -cur; /* swap P0 P2 */
}
xx = 2.0*(4.0*w*sx*xx+dx*dx); /* differences 2nd degree */
yy = 2.0*(4.0*w*sy*yy+dy*dy);
sx = x0 < x2 ? 1 : -1; /* x step direction */
sy = y0 < y2 ? 1 : -1; /* y step direction */
xy = -2.0*sx*sy*(2.0*w*xy+dx*dy);
if (cur*sx*sy < 0) { /* negated curvature? */
xx = -xx; yy = -yy; cur = -cur; xy = -xy;
}
dx = 4.0*w*(x1-x0)*sy*cur+xx/2.0; /* differences 1st degree */
dy = 4.0*w*(y0-y1)*sx*cur+yy/2.0;
if (w < 0.5 && (dx+xx <= 0 || dy+yy >= 0)) {/* flat ellipse, algo fails */
cur = (w+1.0)/2.0; w = fsqrt(w); xy = 1.0/(w+1.0);
sx = Math.floor((x0+2.0*w*x1+x2)*xy/2.0+0.5); /* subdivide curve */
sy = Math.floor((y0+2.0*w*y1+y2)*xy/2.0+0.5); /* plot separately */
dx = Math.floor((w*x1+x0)*xy+0.5); dy = Math.floor((y1*w+y0)*xy+0.5);
plotQuadRationalBezierWidthSeg(x0,y0, dx,dy, sx,sy, cur, th);
dx = Math.floor((w*x1+x2)*xy+0.5); dy = Math.floor((y1*w+y2)*xy+0.5);
return plotQuadRationalBezierWidthSeg(sx,sy, dx,dy, x2,y2, cur, th);
}
fail:
for (err = 0; dy+2*yy < 0 && dx+2*xx > 0; ) /* loop of steep/flat curve */
if (dx+dy+xy < 0) { /* steep curve */
do {
ed = -dy-2*dy*dx*dx/(4.*dy*dy+dx*dx); /* approximate sqrt */
w = (th-1)*ed; /* scale line width */
x1 = Math.floor((err-ed-w/2)/dy); /* start offset */
e2 = err-x1*dy-w/2; /* error value at offset */
x1 = x0-x1*sx; /* start point */
setPixelAA(x1, y0, 255*e2/ed); /* aliasing pre-pixel */
for (e2 = -w-dy-e2; e2-dy < ed; e2 -= dy)
setPixel(x1 += sx, y0); /* pixel on thick line */
setPixelAA(x1+sx, y0, 255*e2/ed); /* aliasing post-pixel */
if (y0 == y2) return; /* last pixel -> curve finished */
y0 += sy; dy += xy; err += dx; dx += xx; /* y step */
if (2*err+dy > 0) { /* e_x+e_xy > 0 */
x0 += sx; dx += xy; err += dy; dy += yy; /* x step */
}
if (x0 != x2 && (dx+2*xx <= 0 || dy+2*yy >= 0))
if (Math.abs(y2-y0) > Math.abs(x2-x0)) break fail;
else break; /* other curve near */
} while (dx+dy+xy < 0); /* gradient still steep? */
/* change from steep to flat curve */
for (cur = err-dy-w/2, y1 = y0; cur < ed; y1 += sy, cur += dx) {
for (e2 = cur, x1 = x0; e2-dy < ed; e2 -= dy)
setPixel(x1 -= sx, y1); /* pixel on thick line */
setPixelAA(x1-sx, y1, 255*e2/ed); /* aliasing post-pixel */
}
} else { /* flat curve */
do {
ed = dx+2*dx*dy*dy/(4.*dx*dx+dy*dy); /* approximate sqrt */
w = (th-1)*ed; /* scale line width */
y1 = Math.floor((err+ed+w/2)/dx); /* start offset */
e2 = y1*dx-w/2-err; /* error value at offset */
y1 = y0-y1*sy; /* start point */
setPixelAA(x0, y1, 255*e2/ed); /* aliasing pre-pixel */
for (e2 = dx-e2-w; e2+dx < ed; e2 += dx)
setPixel(x0, y1 += sy); /* pixel on thick line */
setPixelAA(x0, y1+sy, 255*e2/ed); /* aliasing post-pixel */
if (x0 == x2) return; /* last pixel -> curve finished */
x0 += sx; dx += xy; err += dy; dy += yy; /* x step */
if (2*err+dx < 0) { /* e_y+e_xy < 0 */
y0 += sy; dy += xy; err += dx; dx += xx; /* y step */
}
if (y0 != y2 && (dx+2*xx <= 0 || dy+2*yy >= 0))
if (Math.abs(y2-y0) <= Math.abs(x2-x0)) break fail;
else break; /* other curve near */
} while (dx+dy+xy >= 0); /* gradient still flat? */
/* change from flat to steep curve */
for (cur = -err+dx-w/2, x1 = x0; cur < ed; x1 += sx, cur -= dy) {
for (e2 = cur, y1 = y0; e2+dx < ed; e2 += dx)
setPixel(x1, y1 -= sy); /* pixel on thick line */
setPixelAA(x1, y1-sy, 255*e2/ed); /* aliasing post-pixel */
}
}
}
plotLineWidth(x0,y0, x2,y2, th); /* confusing error values */
}
function plotQuadRationalBezierWidth(x0, y0, x1, y1, x2, y2, w, th)
{ /* plot any anti-aliased quadratic rational Bezier curve */
var x = x0-2*x1+x2, y = y0-2*y1+y2;
var xx = x0-x1, yy = y0-y1, ww, t, q;
assert(w >= 0.0);
if (xx*(x2-x1) > 0) { /* horizontal cut at P4? */
if (yy*(y2-y1) > 0) /* vertical cut at P6 too? */
if (Math.abs(xx*y) > Math.abs(yy*x)) { /* which first? */
x0 = x2; x2 = xx+x1; y0 = y2; y2 = yy+y1; /* swap points */
} /* now horizontal cut at P4 comes first */
if (x0 == x2 || w == 1.0) t = (x0-x1)/x;
else { /* non-rational or rational case */
q = Math.sqrt(4.0*w*w*(x0-x1)*(x2-x1)+(x2-x0)*(x2-x0));
if (x1 < x0) q = -q;
t = (2.0*w*(x0-x1)-x0+x2+q)/(2.0*(1.0-w)*(x2-x0)); /* t at P4 */
}
q = 1.0/(2.0*t*(1.0-t)*(w-1.0)+1.0); /* sub-divide at t */
xx = (t*t*(x0-2.0*w*x1+x2)+2.0*t*(w*x1-x0)+x0)*q; /* = P4 */
yy = (t*t*(y0-2.0*w*y1+y2)+2.0*t*(w*y1-y0)+y0)*q;
ww = t*(w-1.0)+1.0; ww *= ww*q; /* squared weight P3 */
w = ((1.0-t)*(w-1.0)+1.0)*Math.sqrt(q); /* weight P8 */
x = Math.floor(xx+0.5); y = Math.floor(yy+0.5); /* P4 */
yy = (xx-x0)*(y1-y0)/(x1-x0)+y0; /* intersect P3 | P0 P1 */
plotQuadRationalBezierWidthSeg(x0,y0, x,Math.floor(yy+0.5), x,y, ww, th);
yy = (xx-x2)*(y1-y2)/(x1-x2)+y2; /* intersect P4 | P1 P2 */
y1 = Math.floor(yy+0.5); x0 = x1 = x; y0 = y; /* P0 = P4, P1 = P8 */
}
if ((y0-y1)*(y2-y1) > 0) { /* vertical cut at P6? */
if (y0 == y2 || w == 1.0) t = (y0-y1)/(y0-2.0*y1+y2);
else { /* non-rational or rational case */
q = Math.sqrt(4.0*w*w*(y0-y1)*(y2-y1)+(y2-y0)*(y2-y0));
if (y1 < y0) q = -q;
t = (2.0*w*(y0-y1)-y0+y2+q)/(2.0*(1.0-w)*(y2-y0)); /* t at P6 */
}
q = 1.0/(2.0*t*(1.0-t)*(w-1.0)+1.0); /* sub-divide at t */
xx = (t*t*(x0-2.0*w*x1+x2)+2.0*t*(w*x1-x0)+x0)*q; /* = P6 */
yy = (t*t*(y0-2.0*w*y1+y2)+2.0*t*(w*y1-y0)+y0)*q;
ww = t*(w-1.0)+1.0; ww *= ww*q; /* squared weight P5 */
w = ((1.0-t)*(w-1.0)+1.0)*Math.sqrt(q); /* weight P7 */
x = Math.floor(xx+0.5); y = Math.floor(yy+0.5); /* P6 */
xx = (x1-x0)*(yy-y0)/(y1-y0)+x0; /* intersect P6 | P0 P1 */
plotQuadRationalBezierWidthSeg(x0,y0, Math.floor(xx+0.5),y, x,y, ww, th);
xx = (x1-x2)*(yy-y2)/(y1-y2)+x2; /* intersect P7 | P1 P2 */
x1 = Math.floor(xx+0.5); x0 = x; y0 = y1 = y; /* P0 = P6, P1 = P7 */
}
plotQuadRationalBezierWidthSeg(x0,y0, x1,y1, x2,y2, w*w, th);
}
function plotRotatedEllipseWidth(x, y, a, b, angle, th)
{ /* plot ellipse rotated by angle (radian) */
var xd = a*a, yd = b*b;
var s = Math.sin(angle), zd = (xd-yd)*s; /* ellipse rotation */
xd = Math.sqrt(xd-zd*s), yd = Math.sqrt(yd+zd*s); /* surrounding rect*/
a = Math.floor(xd+0.5); b = Math.floor(yd+0.5); zd = zd*a*b/(xd*yd);
plotRotatedEllipseRectWidth(x-a,y-b, x+a,y+b, (4*zd*Math.cos(angle)), th);
}
function plotRotatedEllipseRectWidth(x0, y0, x1, y1, zd, th)
{ /* rectangle enclosing the ellipse, integer rotation angle */
var xd = x1-x0, yd = y1-y0, w = xd*yd;
if (w != 0.0) w = (w-zd)/(w+w); /* squared weight of P1 */
assert(w <= 1.0 && w >= 0.0); /* limit angle to |zd|<=xd*yd */
xd = Math.floor(xd*w+0.5); yd = Math.floor(yd*w+0.5); /* snap to int */
plotQuadRationalBezierWidthSeg(x0,y0+yd, x0,y0, x0+xd,y0, 1.0-w, th);
plotQuadRationalBezierWidthSeg(x0,y0+yd, x0,y1, x1-xd,y1, w, th);
plotQuadRationalBezierWidthSeg(x1,y1-yd, x1,y1, x1-xd,y1, 1.0-w, th);
plotQuadRationalBezierWidthSeg(x1,y1-yd, x1,y0, x0+xd,y0, w, th);
}
function plotCubicBezierWidth(x0, y0, x1, y1, x2, y2, x3, y3, th)
{ /* plot any cubic Bezier curve */
var n = 0, i = 0;
var xc = x0+x1-x2-x3, xa = xc-4*(x1-x2);
var xb = x0-x1-x2+x3, xd = xb+4*(x1+x2);
var yc = y0+y1-y2-y3, ya = yc-4*(y1-y2);
var yb = y0-y1-y2+y3, yd = yb+4*(y1+y2);
var fx0 = x0, fx1, fx2, fx3, fy0 = y0, fy1, fy2, fy3;
var t1 = xb*xb-xa*xc, t2, t = new Array(7);
/* sub-divide curve at gradient sign changes */
if (xa == 0) { /* horizontal */
if (Math.abs(xc) < 2*Math.abs(xb)) t[n++] = xc/(2.0*xb); /* one change */
} else if (t1 > 0.0) { /* two changes */
t2 = Math.sqrt(t1);
t1 = (xb-t2)/xa; if (Math.abs(t1) < 1.0) t[n++] = t1;
t1 = (xb+t2)/xa; if (Math.abs(t1) < 1.0) t[n++] = t1;
}
t1 = yb*yb-ya*yc;
if (ya == 0) { /* vertical */
if (Math.abs(yc) < 2*Math.abs(yb)) t[n++] = yc/(2.0*yb); /* one change */
} else if (t1 > 0.0) { /* two changes */
t2 = Math.sqrt(t1);
t1 = (yb-t2)/ya; if (Math.abs(t1) < 1.0) t[n++] = t1;
t1 = (yb+t2)/ya; if (Math.abs(t1) < 1.0) t[n++] = t1;
}
t1 = 2*(xa*yb-xb*ya); t2 = xa*yc-xc*ya; /* divide at inflection point */
i = t2*t2-2*t1*(xb*yc-xc*yb);
if (i > 0) {
i = Math.sqrt(i);
t[n] = (t2+i)/t1; if (Math.abs(t[n]) < 1.0) n++;
t[n] = (t2-i)/t1; if (Math.abs(t[n]) < 1.0) n++;
}
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 points */
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 = Math.floor(fx3+0.5); y3 = Math.floor(fy3+0.5); /* scale bounds */
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 */
plotCubicBezierSegWidth(x0,y0, x0+fx1,y0+fy1, x0+fx2,y0+fy2, x3,y3, th);
x0 = x3; y0 = y3; fx0 = fx3; fy0 = fy3; t1 = t2;
}
}
function plotCubicBezierSegWidth(x0,y0, x1,y1, x2,y2, x3,y3, th)
{ /* split cubic Bezier segment in two quadratic segments */
var x = Math.floor((x0+3*x1+3*x2+x3+4)/8),
y = Math.floor((y0+3*y1+3*y2+y3+4)/8);
plotQuadRationalBezierWidthSeg(x0,y0,
Math.floor((x0+3*x1+2)/4),Math.floor((y0+3*y1+2)/4), x,y, 1,th);
plotQuadRationalBezierWidthSeg(x,y,
Math.floor((3*x2+x3+2)/4),Math.floor((3*y2+y3+2)/4), x3,y3, 1,th);
}
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