/********************************************************************
* *
* Curve Rasterizing Algorithm *
* *
********************************************************************/
/**
* @author Zingl Alois
* @date 22.08.2016
* @version 1.2
*/
void plotLine(int x0, int y0, int x1, int y1)
{
int dx = abs(x1-x0), sx = x0<x1 ? 1 : -1;
int dy = -abs(y1-y0), sy = y0<y1 ? 1 : -1;
int err = dx+dy, e2; /* error value e_xy */
for (;;) { /* loop */
setPixel(x0,y0);
e2 = 2*err;
if (e2 >= dy) { /* e_xy+e_x > 0 */
if (x0 == x1) break;
err += dy; x0 += sx;
}
if (e2 <= dx) { /* e_xy+e_y < 0 */
if (y0 == y1) break;
err += dx; y0 += sy;
}
}
}
void plotLine3d (int x0, int y0, int z0, int x1, int y1, int z1)
{
int dx = abs(x1-x0), sx = x0 < x1 ? 1 : -1;
int dy = abs(y1-y0), sy = y0 < y1 ? 1 : -1;
int dz = abs(z1-z0), sz = z0 < z1 ? 1 : -1;
int dm = dx > dy && dx > dz ? dx : dy > dz ? dy : dz, i = dm; /* max diff */
x1 = y1 = z1 = dm/2; /* error offset */
for (;;) {
setPixel(x0,y0,z0);
if (i-- == 0) break;
x1 -= dx; if (x1 < 0) { x1 += dm; x0 += sx; }
y1 -= dy; if (y1 < 0) { y1 += dm; y0 += sy; }
z1 -= dz; if (z1 < 0) { z1 += dm; z0 += sz; }
}
}
void plotEllipse(int xm, int ym, int a, int b)
{
int x = -a, y = 0; /* II. quadrant from bottom left to top right */
long e2 = (long)b*b, err = (long)x*(2*e2+x)+e2; /* 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 >= (x*2+1)*(long)b*b) /* e_xy+e_x > 0 */
err += (++x*2+1)*(long)b*b;
if (e2 <= (y*2+1)*(long)a*a) /* e_xy+e_y < 0 */
err += (++y*2+1)*(long)a*a;
} while (x <= 0);
while (y++ < b) { /* too early stop of flat ellipses a=1, */
setPixel(xm, ym+y); /* -> finish tip of ellipse */
setPixel(xm, ym-y);
}
}
void plotOptimizedEllipse(int xm, int ym, int a, int b)
{
long x = -a, y = 0; /* II. quadrant from bottom left to top right */
long e2 = b, dx = (1+2*x)*e2*e2; /* error increment */
long 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*(long)b*b; } /* x step */
if (e2 <= dy) { y++; err += dy += 2*(long)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);
}
}
void plotCircle(int xm, int ym, int r)
{
int 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; /* e_xy+e_y < 0 */
if (r > x || err > y) /* e_xy+e_x > 0 or no 2nd y-step */
err += ++x*2+1; /* -> x-step now */
} while (x < 0);
}
void plotEllipseRect(int x0, int y0, int x1, int y1)
{ /* rectangular parameter enclosing the ellipse */
long a = abs(x1-x0), b = abs(y1-y0), b1 = b&1; /* diameter */
double dx = 4*(1.0-a)*b*b, dy = 4*(b1+1)*a*a; /* error increment */
double 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)/2; 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 step */
} 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--);
}
}
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 */
}
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 */
}
void plotQuadRationalBezierSeg(int x0, int y0, int x1, int y1,
int x2, int y2, float w)
{ /* plot a limited rational Bezier segment, squared weight */
int sx = x2-x1, sy = y2-y1; /* relative values for checks */
double dx = x0-x2, dy = y0-y2, xx = x0-x1, yy = y0-y1;
double 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*(long)sx+sy*(long)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 < 0.0) { /* 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 = sqrt(w); xy = 1.0/(w+1.0);
sx = floor((x0+2.0*w*x1+x2)*xy/2.0+0.5); /* subdivide curve in half */
sy = floor((y0+2.0*w*y1+y2)*xy/2.0+0.5);
dx = floor((w*x1+x0)*xy+0.5); dy = floor((y1*w+y0)*xy+0.5);
plotQuadRationalBezierSeg(x0,y0, dx,dy, sx,sy, cur);/* plot separately */
dx = floor((w*x1+x2)*xy+0.5); dy = 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 */
}
void plotQuadRationalBezier(int x0, int y0, int x1, int y1,
int x2, int y2, float w)
{ /* plot any quadratic rational Bezier curve */
int x = x0-2*x1+x2, y = y0-2*y1+y2;
double 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 (fabs(xx*y) > fabs(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)/(double)x;
else { /* non-rational or rational case */
q = sqrt(4.0*w*w*(x0-x1)*(x2-x1)+(x2-x0)*(long)(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)*sqrt(q); /* weight P8 */
x = floor(xx+0.5); y = floor(yy+0.5); /* P4 */
yy = (xx-x0)*(y1-y0)/(x1-x0)+y0; /* intersect P3 | P0 P1 */
plotQuadRationalBezierSeg(x0,y0, x,floor(yy+0.5), x,y, ww);
yy = (xx-x2)*(y1-y2)/(x1-x2)+y2; /* intersect P4 | P1 P2 */
y1 = floor(yy+0.5); x0 = x1 = x; y0 = y; /* P0 = P4, P1 = P8 */
}
if ((y0-y1)*(long)(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 = sqrt(4.0*w*w*(y0-y1)*(y2-y1)+(y2-y0)*(long)(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)*sqrt(q); /* weight P7 */
x = floor(xx+0.5); y = floor(yy+0.5); /* P6 */
xx = (x1-x0)*(yy-y0)/(y1-y0)+x0; /* intersect P6 | P0 P1 */
plotQuadRationalBezierSeg(x0,y0, floor(xx+0.5),y, x,y, ww);
xx = (x1-x2)*(yy-y2)/(y1-y2)+x2; /* intersect P7 | P1 P2 */
x1 = floor(xx+0.5); x0 = x; y0 = y1 = y; /* P0 = P6, P1 = P7 */
}
plotQuadRationalBezierSeg(x0,y0, x1,y1, x2,y2, w*w); /* remaining */
}
void plotRotatedEllipse(int x, int y, int a, int b, float angle)
{ /* plot ellipse rotated by angle (radian) */
float xd = (long)a*a, yd = (long)b*b;
float s = sin(angle), zd = (xd-yd)*s; /* ellipse rotation */
xd = sqrt(xd-zd*s), yd = sqrt(yd+zd*s); /* surrounding rectangle */
a = xd+0.5; b = yd+0.5; zd = zd*a*b/(xd*yd); /* scale to integer */
plotRotatedEllipseRect(x-a,y-b, x+a,y+b, (long)(4*zd*cos(angle)));
}
void plotRotatedEllipseRect(int x0, int y0, int x1, int y1, long zd)
{ /* rectangle enclosing the ellipse, integer rotation angle */
int xd = x1-x0, yd = y1-y0;
float w = xd*(long)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 = floor(xd*w+0.5); yd = floor(yd*w+0.5); /* snap xe,ye 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);
}
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 */
}
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;
}
}
void plotLineAA(int x0, int y0, int x1, int y1)
{ /* draw a black (0) anti-aliased line on white (255) background */
int sx = x0 < x1 ? 1 : -1, sy = y0 < y1 ? 1 : -1, x2;
long dx = abs(x1-x0), dy = abs(y1-y0), err = dx*dx+dy*dy;
long e2 = err == 0 ? 1 : 0xffff7fl/sqrt(err); /* multiplication factor */
dx *= e2; dy *= e2; err = dx-dy; /* error value e_xy */
for ( ; ; ){ /* pixel loop */
setPixelAA(x0,y0,abs(err-dx+dy)>>16);
e2 = err; x2 = x0;
if (2*e2 >= -dx) { /* x step */
if (x0 == x1) break;
if (e2+dy < 0xff0000l) setPixelAA(x0,y0+sy,(e2+dy)>>16);
err -= dy; x0 += sx;
}
if (2*e2 <= dy) { /* y step */
if (y0 == y1) break;
if (dx-e2 < 0xff0000l) setPixelAA(x2+sx,y0,(dx-e2)>>16);
err += dx; y0 += sy;
}
}
void plotCircleAA(int xm, int ym, int r)
{ /* draw a black anti-aliased circle on white background */
int x = -r, y = 0; /* II. quadrant from bottom left to top right */
int i, x2, e2, err = 2-2*r; /* error of 1.step */
r = 1-err;
do {
i = 255*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 */
e2 = err; x2 = x; /* remember values */
if (err+y > 0) { /* x step */
i = 255*(err-2*x-1)/r; /* outward pixel */
if (i < 256) {
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 <= 0) { /* y step */
i = 255*(2*y+3-e2)/r; /* inward pixel */
if (i < 256) {
setPixelAA(xm-x2-1, ym+y, i);
setPixelAA(xm-y, ym-x2-1, i);
setPixelAA(xm+x2+1, ym-y, i);
setPixelAA(xm+y, ym+x2+1, i);
}
err += ++y*2+1;
}
} while (x < 0);
}
void plotEllipseRectAA(int x0, int y0, int x1, int y1)
{ /* draw a black anti-aliased rectangular ellipse on white background */
long a = abs(x1-x0), b = abs(y1-y0), b1 = b&1; /* diameter */
float dx = 4*(a-1.0)*b*b, dy = 4*(b1+1)*a*a; /* error increment */
float ed, i, err = b1*a*a-dx+dy; /* error of 1.step */
bool f;
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)/2; y1 = y0-b1; /* starting pixel */
a = 8*a*a; b1 = 8*b*b;
for (;;) { /* approximate ed=sqrt(dx*dx+dy*dy) */
i = min(dx,dy); ed = 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*fabs(err+dx-dy); /* get intensity value by pixel error */
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 < 255) {
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 < 255) {
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*fabs(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;
}
}
void plotQuadBezierSegAA(int x0, int y0, int x1, int y1, int x2, int y2)
{ /* draw an limited anti-aliased 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, ed, 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*(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 = fmin(dx+xy,-xy-dy);
ed = fmax(dx+xy,-xy-dy); /* approximate error distance */
ed += 2*ed*cur*cur/(4*ed*ed+cur*cur);
setPixelAA(x0,y0, 255*fabs(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*fabs(err-dy)/ed);
x0 += sx; dx -= xy; err += dy += yy;
}
if (y1) { /* y step */
if (cur < ed) setPixelAA(x1+sx,y0, 255*fabs(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 */
}
void plotQuadRationalBezierSegAA(int x0, int y0, int x1, int y1,
int x2, int y2, float w)
{ /* draw an anti-aliased rational quadratic Bezier segment, squared weight */
int sx = x2-x1, sy = y2-y1; /* relative values for checks */
double dx = x0-x2, dy = y0-y2, xx = x0-x1, yy = y0-y1;
double xy = xx*sy+yy*sx, cur = xx*sy-yy*sx, err, ed; /* curvature */
bool f;
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*(long)sx+sy*(long)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+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 = sqrt(w); xy = 1.0/(w+1.0);
sx = floor((x0+2.0*w*x1+x2)*xy/2.0+0.5); /* subdivide curve in half */
sy = floor((y0+2.0*w*y1+y2)*xy/2.0+0.5);
dx = floor((w*x1+x0)*xy+0.5); dy = floor((y1*w+y0)*xy+0.5);
plotQuadRationalBezierSegAA(x0,y0, dx,dy, sx,sy, cur); /* plot apart */
dx = floor((w*x1+x2)*xy+0.5); dy = 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 = fmin(dx-xy,xy-dy); ed = fmax(dx-xy,xy-dy);
ed += 2*ed*cur*cur/(4.*ed*ed+cur*cur); /* approximate error distance */
x1 = 255*fabs(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*fabs(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*fabs(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 */
}
void plotCubicBezierSegAA(int x0, int y0, float x1, float y1,
float x2, float y2, int x3, int y3)
{ /* plot limited anti-aliased 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, 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 */
sx = floor((3*x1-x0+1)/2); sy = floor((3*y1-y0+1)/2); /* new midpoint */
return plotQuadBezierSegAA(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;
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 : sqrt(1+1024/x1); /* calculate 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 */
for (fx = fy = f; x0 != x3 && y0 != y3; ) {
y1 = fmin(fabs(xy-dx),fabs(dy-xy));
ed = fmax(fabs(xy-dx),fabs(dy-xy)); /* approximate error distance */
ed = f*(ed+2*ed*y1*y1/(4*ed*ed+y1*y1));
y1 = 255*fabs(ex-(f-fx+1)*dx-(f-fy+1)*dy+f*xy)/ed;
if (y1 < 256) setPixelAA(x0, y0, y1); /* plot curve */
px = fabs(ex-(f-fx+1)*dx+(fy-1)*dy); /* pixel intensity x move */
py = fabs(ex+(fx-1)*dx-(f-fy+1)*dy); /* pixel intensity y move */
y2 = y0;
do { /* move sub-steps of one pixel */
if (ip >= -EP) /* intersection possible? -> check.. */
if (dx+xx > xy || dy+yy < xy) goto exit; /* 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) goto exit; /* 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 */
}
}
break; /* finish curve by line */
exit:
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 */
}
void plotLineWidth(int x0, int y0, int x1, int y1, float wd)
{ /* plot an anti-aliased line of width wd */
int dx = abs(x1-x0), sx = x0 < x1 ? 1 : -1;
int dy = abs(y1-y0), sy = y0 < y1 ? 1 : -1;
int err = dx-dy, e2, x2, y2; /* error value e_xy */
float ed = dx+dy == 0 ? 1 : sqrt((float)dx*dx+(float)dy*dy);
for (wd = (wd+1)/2; ; ) { /* pixel loop */
setPixelColor(x0, y0, max(0,255*(abs(err-dx+dy)/ed-wd+1)));
e2 = err; x2 = x0;
if (2*e2 >= -dx) { /* x step */
for (e2 += dy, y2 = y0; e2 < ed*wd && (y1 != y2 || dx > dy); e2 += dx)
setPixelColor(x0, y2 += sy, max(0,255*(abs(e2)/ed-wd+1)));
if (x0 == x1) break;
e2 = err; err -= dy; x0 += sx;
}
if (2*e2 <= dy) { /* y step */
for (e2 = dx-e2; e2 < ed*wd && (x1 != x2 || dx < dy); e2 += dy)
setPixelColor(x2 += sx, y0, max(0,255*(abs(e2)/ed-wd+1)));
if (y0 == y1) break;
err += dx; y0 += sy;
}
}
}
void plotQuadSpline(int n, int x[], int y[])
{ /* plot quadratic spline, destroys input arrays x,y */
#define M_MAX 6
float mi = 1, m[M_MAX]; /* diagonal constants of matrix */
int i, x0, y0, x1, y1, x2 = x[n], y2 = y[n];
assert(n > 1); /* need at least 3 points P[0]..P[n] */
x[1] = x0 = 8*x[1]-2*x[0]; /* first row of matrix */
y[1] = y0 = 8*y[1]-2*y[0];
for (i = 2; i < n; i++) { /* forward sweep */
if (i-2 < M_MAX) m[i-2] = mi = 1.0/(6.0-mi);
x[i] = x0 = floor(8*x[i]-x0*mi+0.5); /* store yi */
y[i] = y0 = floor(8*y[i]-y0*mi+0.5);
}
x1 = floor((x0-2*x2)/(5.0-mi)+0.5); /* correction last row */
y1 = floor((y0-2*y2)/(5.0-mi)+0.5);
for (i = n-2; i > 0; i--) { /* back substitution */
if (i <= M_MAX) mi = m[i-1];
x0 = floor((x[i]-x1)*mi+0.5); /* next corner */
y0 = floor((y[i]-y1)*mi+0.5);
plotQuadBezier((x0+x1)/2,(y0+y1)/2, x1,y1, x2,y2);
x2 = (x0+x1)/2; x1 = x0;
y2 = (y0+y1)/2; y1 = y0;
}
plotQuadBezier(x[0],y[0], x1,y1, x2,y2);
}
void plotCubicSpline(int n, int x[], int y[])
{ /* plot cubic spline, destroys input arrays x,y */
#define M_MAX 6
float mi = 0.25, m[M_MAX]; /* diagonal constants of matrix */
int x3 = x[n-1], y3 = y[n-1], x4 = x[n], y4 = y[n];
int i, x0, y0, x1, y1, x2, y2;
assert(n > 2); /* need at least 4 points P[0]..P[n] */
x[1] = x0 = 12*x[1]-3*x[0]; /* first row of matrix */
y[1] = y0 = 12*y[1]-3*y[0];
for (i = 2; i < n; i++) { /* foreward sweep */
if (i-2 < M_MAX) m[i-2] = mi = 0.25/(2.0-mi);
x[i] = x0 = floor(12*x[i]-2*x0*mi+0.5);
y[i] = y0 = floor(12*y[i]-2*y0*mi+0.5);
}
x2 = floor((x0-3*x4)/(7-4*mi)+0.5); /* correct last row */
y2 = floor((y0-3*y4)/(7-4*mi)+0.5);
plotCubicBezier(x3,y3, (x2+x4)/2,(y2+y4)/2, x4,y4, x4,y4);
if (n-3 < M_MAX) mi = m[n-3];
x1 = floor((x[n-2]-2*x2)*mi+0.5);
y1 = floor((y[n-2]-2*y2)*mi+0.5);
for (i = n-3; i > 0; i--) { /* back substitution */
if (i <= M_MAX) mi = m[i-1];
x0 = floor((x[i]-2*x1)*mi+0.5);
y0 = floor((y[i]-2*y1)*mi+0.5);
x4 = floor((x0+4*x1+x2+3)/6.0); /* reconstruct P[i] */
y4 = floor((y0+4*y1+y2+3)/6.0);
plotCubicBezier(x4,y4,
floor((2*x1+x2)/3+0.5),floor((2*y1+y2)/3+0.5),
floor((x1+2*x2)/3+0.5),floor((y1+2*y2)/3+0.5),
x3,y3);
x3 = x4; y3 = y4; x2 = x1; y2 = y1; x1 = x0; y1 = y0;
}
x0 = x[0]; x4 = floor((3*x0+7*x1+2*x2+6)/12.0); /* reconstruct P[1] */
y0 = y[0]; y4 = floor((3*y0+7*y1+2*y2+6)/12.0);
plotCubicBezier(x4,y4, floor((2*x1+x2)/3+0.5),floor((2*y1+y2)/3+0.5),
floor((x1+2*x2)/3+0.5),floor((y1+2*y2)/3+0.5), x3,y3);
plotCubicBezier(x0,y0, x0,y0, (x0+x1)/2,(y0+y1)/2, x4,y4);
}
