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% nurbstobezier.mp
% Troy Henderson
% 2007
prologues := 1;
% Evaluate a cubic polynomial of the "standard" Bezier form at t
vardef evalbezier(expr p,t) =
save _a,_b,_c,_d;
numeric _a,_b,_c,_d;
_a:=(1-t)**3;
_b:=3*((1-t)**2)*t;
_c:=3*(1-t)*(t**2);
_d:=t**3;
(point 0 of p)*_a + (postcontrol 0 of p)*_b + (precontrol 1 of p)*_c + (point 1 of p)*_d
enddef;
% Evaluate the derivative of a cubic polynomial of the "standard"
% Bezier form at t
vardef evalbezierderivative(expr p,t) =
save _a,_b,_c;
pair _a,_b,_c;
_a:=3*((point 1 of p) - 3*(precontrol 1 of p) + 3*(postcontrol 0 of p) -(point 0 of p));
_b:=6*((precontrol 1 of p) - 2*(postcontrol 0 of p) + (point 0 of p));
_c:=3*((postcontrol 0 of p) - (point 0 of p));
_a*(t**2) + _b*t + _c
enddef;
% Evaluate a rational function of the "standard" cubic NURBS form at t
vardef evalnurbs(expr p,w,t) =
save _q,_r;
path _q,_r;
_q:=((cyanpart w)*(point 0 of p)).. controls ((magentapart w)*(postcontrol 0 of p)) and ((yellowpart w)*(precontrol 1 of p)) .. ((blackpart w)*(point 1 of p));
_r:=(cyanpart w,0) .. controls (magentapart w,0) and (yellowpart w,0) .. (blackpart w,0);
evalbezier(_q,t)/(xpart evalbezier(_r,t))
enddef;
% Evaluate the derivative of a rational function of the "standard"
% cubic NURBS form at t
vardef evalnurbsderivative(expr p,w,t) =
save _a,_b,_c,_d,_q,_r;
pair _a,_b;
numeric _c,_d;
path _q,_r;
_q:=((cyanpart w)*(point 0 of p)) .. controls ((magentapart w)*(postcontrol 0 of p)) and ((yellowpart w)*(precontrol 1 of p)) .. ((blackpart w)*(point 1 of p));
_r:=(cyanpart w,0) .. controls (magentapart w,0) and (yellowpart w,0) .. (blackpart w,0);
_a:=evalbezier(_q,t);
_b:=evalbezierderivative(_q,t);
_c:=xpart evalbezier(_r,t);
_d:=xpart evalbezierderivative(_r,t);
(_b*_c-_a*_d)/(_c**2)
enddef;
% Fit a cubic polynomial of the "standard" Bezier form to a
% rational function of the
% "standard" cubic NURBS form with function and derivative agreement
% at tmin and tmax
vardef nurbstobezier(expr p,w,tmin,tmax) =
save _a,_b,_c,_d,_e;
pair _a,_b,_c,_d;
numeric _e;
_e:=(tmax-tmin)/3;
_a:=evalnurbs(p,w,tmin);
_b:=_a + _e*evalnurbsderivative(p,w,tmin);
_d:=evalnurbs(p,w,tmax);
_c:=_d - _e*evalnurbsderivative(p,w,tmax);
_a .. controls _b and _c .. _d
enddef;
% Reparameterize a cubic polynomial of the "standard" Bezier form by mapping
% the interval [tmin,tmax] to [0,1]
vardef beziertobezier(expr p,tmin,tmax) =
nurbstobezier(p,(1,1,1,1),tmin,tmax)
enddef;
% Evalute the L^2[0,1] norm of a cubic polynomial of the "standard"
% Bezier form
vardef beziernorm(expr p) =
save _a,_b,_c,_d,_i,_xabs,_yabs,_A,_B,_C,_D,_I;
numeric _a,_b,_c,_d,_i,_xabs,_yabs,_A,_B,_C,_D,_I;
_xabs:=max(abs(xpart point 0 of p),abs(xpart postcontrol 0 of p),abs(xpart precontrol 1 of p),abs(xpart point 1 of p));
_yabs:=max(abs(ypart point 0 of p),abs(ypart postcontrol 0 of p),abs(ypart precontrol 1 of p),abs(ypart point 1 of p));
if (_xabs > 0):
_a:=xpart((point 1 of p) - 3*(precontrol 1 of p) + 3*(postcontrol 0 of p) - (point 0 of p))/_xabs;
_b:=3*xpart((precontrol 1 of p) - 2*(postcontrol 0 of p) + (point 0 of p))/_xabs;
_c:=3*xpart((postcontrol 0 of p) - (point 0 of p))/_xabs;
_d:=xpart(point 0 of p)/_xabs;
_i:=(_a**2)/7 + ((_b)**2 + 2*_a*_c)/5 + (_a*_b + 2*_b*_d + (_c**2))/3 + (_a*_d + _b*_c)/2 + (_c*_d + (_d**2));
else:
_i:=0;
fi;
if (_yabs > 0):
_A:=ypart((point 1 of p) - 3*(precontrol 1 of p) + 3*(postcontrol 0 of p) - (point 0 of p))/_yabs;
_B:=3*ypart((precontrol 1 of p) - 2*(postcontrol 0 of p) + (point 0 of p))/_yabs;
_C:=3*ypart((postcontrol 0 of p) - (point 0 of p))/_yabs;
_D:=ypart(point 0 of p)/_yabs;
_I:=(_A**2)/7 + ((_B)**2 + 2*_A*_C)/5 + (_A*_B + 2*_B*_D + (_C**2))/3 + (_A*_D + _B*_C)/2 + (_C*_D + (_D**2));
else:
_I:=0;
fi;
(_xabs*sqrt(_i)) ++ (_yabs*sqrt(_I))
enddef;
% Fit a cubic Bezier spline to a rational function of the "standard"
% cubic NURBS form by iteratively refining the Bezier curve.
% p is a 4 point path containing the 4 cubic NURBS (2D) control points
% w is a cmykcolor containing the 4 cubic NURBS weights
% EPS is the tolerance to stop refining each branch of the Bezier spline
vardef fitnurbswithbezier(expr p,w,EPS) =
save _a,_b,_c,_e,_error,_k,_q;
numeric _a,_b,_c,_error,_k;
path _q,_q[],_e;
_a:=0;
_b:=1;
_k:=1/sqrt(2);
_q:=(point 0 of p);
_q[4]:=nurbstobezier(p,w,_a,_b);
forever:
exitunless(_a<1);
_q[1]:=_q[4];
_c:=_b-_k*((_b-_a)**2);
_q[2]:=beziertobezier(_q[1],_a,_c);
_q[3]:=nurbstobezier(p,w,_a,_c);
_q[4]:=_q[3];
_e:=((point 0 of _q[2])-(point 0 of _q[3])) .. controls ((postcontrol 0 of _q[2])-(postcontrol 0 of _q[3])) and ((precontrol 1 of _q[2])-(precontrol 1 of _q[3])) .. ((point 1 of _q[2])-(point 1 of _q[3]));
_error:=beziernorm(_e)/beziernorm(_q[3]);
show _error;
if (_error > EPS):
_b:=_c;
else:
_q[2]:=beziertobezier(_q[1],_c,_b);
_q[3]:=nurbstobezier(p,w,_c,_b);
_e:=((point 0 of _q[2])-(point 0 of _q[3])) .. controls ((postcontrol 0 of _q[2])-(postcontrol 0 of _q[3])) and ((precontrol 1 of _q[2])-(precontrol 1 of _q[3])) .. ((point 1 of _q[2])-(point 1 of _q[3]));
_error:=beziernorm(_e)/beziernorm(_q[3]);
if (_error > EPS):
_q:=_q .. controls (postcontrol 0 of _q[4]) and (precontrol 1 of _q[4]) .. (point 1 of _q[4]);
_a:=_c;
_q[4]:=_q[3];
else:
_q:=_q .. controls (postcontrol 0 of _q[1]) and (precontrol 1 of _q[1]) .. (point 1 of _q[1]);
_a:=_b;
_q[4]:=nurbstobezier(p,w,_a,1);
fi;
_b:=1;
fi;
endfor;
_q
enddef;
% This macro is used to provide a path to draw the NURBS
% It returns a path of length N passing through N+1 equally spaced
% (in time) points along the NURBS connected by line segments
vardef samplednurbs(expr p,w,N) =
save _a,_b,_c,_d,_n,_t,_q;
numeric _a,_b,_c,_d,_n,_t;
path _q;
_q:=(point 0 of p);
for _n=1 upto N:
_t:=_n/N;
_a:=(cyanpart w)*((1-_t)**3);
_b:=3*(magentapart w)*((1-_t)**2)*_t;
_c:=3*(yellowpart w)*(1-_t)*(_t**2);
_d:=(blackpart w)*(_t**3);
_q:=_q .. ((_a*(point 0 of p)+_b*(postcontrol 0 of p)+_c*(precontrol 1 of p)+_d*(point 1 of p))/(_a+_b+_c+_d));
endfor;
( _q )
enddef;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Here's where the fun begins %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
beginfig(0);
% p contains the 4 control points of the rational function of the
% "standard" cubic NURBS form
path p;
p:=(297.63725,297.63725) .. controls (132.98871,286.67885) and (180.62535,152.16249) .. (429.54399,226.31157);
% w contains the 4 weights for the rational function of the
% "standard" cubic NURBS form
cmykcolor w;
w:=(2.15756,1.6709,0.8598,1.34647);
% EPS represents the minimum "acceptable error" to stop refining any
% given branch of the Bezier
Err:=0.040;
% q represents the Bezier spline fit to the rational function of the
% "standard" cubic NURBS form
path q;
q:=fitnurbswithbezier(p,w,Err);
% q:=fitnurbswithbezier(reverse p,(blackpart w,yellowpart w,magentapart w,cyanpart w),Err);
% draw the NURBS by sampling it at many points and connecting the
% samples via line segments
draw samplednurbs(p,w,20) withcolor red withpen pencircle scaled 2bp;
% draw the Bezier spline and its knots
draw q;
for n=0 upto length q:
draw fullcircle scaled 2 shifted point n of q withcolor blue;
endfor;
endfig;
end
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