/*****
* knot.cc
* Andy Hammerlindl 2005/02/10
*
* Describes a knot, a point and its neighbouring specifiers, used as an
* intermediate structure in solving paths.
*****/

#include "knot.h"
#include "util.h"

#include "angle.h"
#include "settings.h"

namespace camp {

/***** Debugging *****/
//bool tracing_solving=false;

template <typename T>
ostream& info(ostream& o, const char *name, cvector<T>& v)
{
 if (settings::verbose > 3) {
   o << name << ":\n\n";

   for(Int i=0; i < (Int) v.size(); ++i)
     o << v[i] << endl;

   o << endl;
 }
 return o;
}

ostream& info(ostream& o, string name, knotlist& l)
{
 if (settings::verbose > 3) {
   o << name << ":\n\n";

   for(Int i=0; i < (Int) l.size(); ++i)
     o << l[i] << endl;

   if (l.cyclic())
     o << "cyclic" << endl;

   o << endl;
 }
 return o;
}

#define INFO(x) info(cerr,#x,x)

/***** Auxillary computation functions *****/

// Computes the relative distance of a control point given the angles.
// The name is somewhat misleading as the velocity (with respect to the
// variable that parameterizes the path) is relative to the distance
// between the knots and even then, is actually three times this.
// The routine is based on the velocity function in Section 131 of the MetaPost
// code, but differs in that it automatically accounts for the tension and
// bounding with tension atleast.
double velocity(double theta, double phi, tension t)
{
 static const double VELOCITY_BOUND = 4.0;
 static const double a = sqrt(2.0);
 static const double b = 1.0/16.0;
 static const double c = 1.5*(sqrt(5.0)-1.0);
 static const double d = 1.5*(3.0-sqrt(5.0));

 double st = sin(theta), ct = cos(theta),
   sf = sin(phi),   cf = cos(phi);

 double denom = t.val * (3.0 + c*ct + d*cf);

 double r = denom != 0.0 ? (2.0 + a*(st - b*sf)*(sf - b*st)*(ct-cf)) / denom
   : VELOCITY_BOUND;

 //cerr << " velocity(" << theta << "," << phi <<")= " << r << endl;
 if (r >  VELOCITY_BOUND)
   r = VELOCITY_BOUND;

 // Apply boundedness condition for tension atleast cases.
 if (t.atleast)
   {
     double sine = sin(theta + phi);
     if ((st >= 0.0 && sf >= 0.0 && sine > 0.0) ||
         (st <= 0.0 && sf <= 0.0 && sine < 0.0))
       {
         double rmax = sf / sine;
         if (r > rmax)
           r = rmax;
       }
   }

 return r;
}

double niceAngle(pair z)
{
 return z.gety() == 0 ? z.getx() >= 0 ? 0 : PI
   : angle(z);
}

// Ensures an angle is in the range between -PI and PI.
double reduceAngle(double angle)
{
 return angle >  PI ? angle - 2.0*PI :
   angle < -PI ? angle + 2.0*PI :
   angle;
}


bool interesting(tension t)
{
 return t.val!=1.0 || t.atleast==true;
}

bool interesting(spec *s)
{
 return !s->open();
}

ostream& operator<<(ostream& out, const knot& k)
{
 if (interesting(k.tin))
   out << k.tin << " ";
 if (interesting(k.in))
   out << *k.in << " ";
 out << k.z;
 if (interesting(k.out))
   out << " " << *k.out;
 if (interesting(k.tout))
   out << " " << k.tout;
 return out;
}


eqn dirSpec::eqnOut(Int j, knotlist& l, cvector<double>&, cvector<double>&)
{
 // When choosing the control points, the path will come out the first knot
 // going straight to the next knot rotated by the angle theta.
 // Therefore, the angle theta we want is the difference between the
 // specified heading given and the heading to the next knot.
 double theta=reduceAngle(given-niceAngle(l[j+1].z-l[j].z));

 // Give a simple linear equation to ensure this theta is picked.
 return eqn(0.0,1.0,0.0,theta);
}

eqn dirSpec::eqnIn(Int j, knotlist& l, cvector<double>&, cvector<double>&)
{
 double theta=reduceAngle(given-niceAngle(l[j].z-l[j-1].z));
 return eqn(0.0,1.0,0.0,theta);
}

eqn curlSpec::eqnOut(Int j, knotlist& l, cvector<double>&,
                    cvector<double>& psi)
{
 double alpha=l[j].alpha();
 double beta=l[j+1].beta();

 double chi=alpha*alpha*gamma/(beta*beta);

 double C=alpha*chi+3-beta;
 double D=(3.0-alpha)*chi+beta;

 return eqn(0.0,C,D,-D*psi[j+1]);
}

eqn curlSpec::eqnIn(Int j, knotlist& l, cvector<double>&, cvector<double>&)
{
 double alpha=l[j-1].alpha();
 double beta=l[j].beta();

 double chi=beta*beta*gamma/(alpha*alpha);

 double A=(3-beta)*chi+alpha;
 double B=beta*chi+3-alpha;

 return eqn(A,B,0.0,0.0);
}

spec *controlSpec::outPartner(pair z)
{
 static curlSpec curl;
 return cz==z ? (spec *)&curl : (spec *)new dirSpec(z-cz);
}

spec *controlSpec::inPartner(pair z)
{
 static curlSpec curl;
 return cz==z ? (spec *)&curl : (spec *)new dirSpec(cz-z);
}

// Compute the displacement between points. The j-th result is the distance
// between knot j and knot j+1.
struct dzprop : public knotprop<pair> {
 dzprop(knotlist& l)
   : knotprop<pair>(l) {}

 pair solo(Int) { return pair(0,0); }
 pair start(Int j) { return l[j+1].z - l[j].z; }
 pair mid(Int j) { return l[j+1].z - l[j].z; }
 pair end(Int) { return pair(0,0); }
};

// Compute the distance between points, using the already computed dz.  This
// doesn't use the infomation in the knots, but the knotprop class is useful as
// it takes care of the iteration for us.
struct dprop : public knotprop<double> {
 cvector<pair>& dz;

 dprop(knotlist &l, cvector<pair>& dz)
   : knotprop<double>(l), dz(dz) {}

 double solo(Int j) { return length(dz[j]); }
 double start(Int j) { return length(dz[j]); }
 double mid(Int j) { return length(dz[j]); }
 double end(Int j) { return length(dz[j]); }
};

// Compute the turning angles (psi) between points, using the already computed
// dz.
struct psiprop : public knotprop<double> {
 cvector<pair>& dz;

 psiprop(knotlist &l, cvector<pair>& dz)
   : knotprop<double>(l), dz(dz) {}

 double solo(Int) { return 0; }

 // We set the starting and ending psi to zero.
 double start(Int) { return 0; }
 double end(Int) { return 0; }

 double mid(Int j) { return niceAngle(dz[j]/dz[j-1]); }
};

struct eqnprop : public knotprop<eqn> {
 cvector<double>& d;
 cvector<double>& psi;

 eqnprop(knotlist &l, cvector<double>& d, cvector<double>& psi)
   : knotprop<eqn>(l), d(d), psi(psi) {}

 eqn solo(Int) {
   assert(False);
   return eqn(0.0,1.0,0.0,0.0);
 }

 eqn start(Int j) {
   // Defer to the specifier, as it knows the specifics.
   return dynamic_cast<endSpec *>(l[j].out)->eqnOut(j,l,d,psi);
 }

 eqn end(Int j) {
   return dynamic_cast<endSpec *>(l[j].in)->eqnIn(j,l,d,psi);
 }

 eqn mid(Int j) {
   double lastAlpha = l[j-1].alpha();
   double thisAlpha = l[j].alpha();
   double thisBeta  = l[j].beta();
   double nextBeta  = l[j+1].beta();

   // Values based on the linear approximation of the curvature coming
   // into the knot with respect to theta[j-1] and theta[j].
   double inFactor = 1.0/(thisBeta*thisBeta*d[j-1]);
   double A = lastAlpha*inFactor;
   double B = (3.0 - lastAlpha)*inFactor;

   // Values based on the linear approximation of the curvature going out of
   // the knot with respect to theta[j] and theta[j+1].
   double outFactor = 1.0/(thisAlpha*thisAlpha*d[j]);
   double C = (3.0 - nextBeta)*outFactor;
   double D = nextBeta*outFactor;

   return eqn(A,B+C,D,-B*psi[j]-D*psi[j+1]);
 }
};

// If the system of equations is homogeneous (ie. we are solving for x in
// Ax=0), there is no need to solve for theta; we can just use zeros for the
// thetas.  In fact, our general solving method may not work in this case.
// A common example of this is
//
//   a{curl 1}..{curl 1}b
//
// which arises when solving a one-length path a..b or in a larger path a
// section a--b.
bool homogeneous(cvector<eqn>& e)
{
 for(cvector<eqn>::iterator p=e.begin(); p!=e.end(); ++p)
   if (p->aug != 0)
     return false;
 return true;
}

// Checks whether the equation being solved will be solved as a straight
// path from the first point to the second.
bool straightSection(cvector<eqn>& e)
{
 return e.size()==2 && e.front().aug==0 && e.back().aug==0;
}

struct weqn : public eqn {
 double w;
 weqn(double pre, double piv, double post, double aug, double w=0)
   : eqn(pre,piv,post,aug), w(w) {}

 friend ostream& operator<< (ostream& out, const weqn we)
 {
   return out << (eqn &)we << " + " << we.w << " * theta[0]";
 }
};

weqn scale(weqn q) {
 assert(q.pre == 0 && q.piv != 0);
 return weqn(0,1,q.post/q.piv,q.aug/q.piv,q.w/q.piv);
}

/* Recalculate the equations in the form:
*   theta[j] + post * theta[j+1] = aug + w * theta[0]
*
* Used as the first step in solve cyclic equations.
*/
cvector<weqn> recalc(cvector<eqn>& e)
{
 Int n=(Int) e.size();
 cvector<weqn> we;
 weqn lasteqn(0,1,0,0,1);
 we.push_back(lasteqn); // As a placeholder.
 for (Int j=1; j < n; j++) {
   // Subtract a factor of the last equation so that the first entry is
   // zero, then procede to scale it.
   eqn& q=e[j];
   lasteqn=scale(weqn(0,q.piv-q.pre*lasteqn.post,q.post,
                      q.aug-q.pre*lasteqn.aug,-q.pre*lasteqn.w));
   we.push_back(lasteqn);
 }
 // To keep all of the infomation enocoded in the linear equations, we need
 // to augment the computation to replace our trivial start weqn with a
 // real one.  To do this, we take one more step in the iteration and
 // compute the weqn for j=n, since n=0 (mod n).
 {
   eqn& q=e[0];
   we.front()=scale(weqn(0,q.piv-q.pre*lasteqn.post,q.post,
                         q.aug-q.pre*lasteqn.aug,-q.pre*lasteqn.w));
 }
 return we;
}

double solveForTheta0(cvector<weqn>& we)
{
 // Solve for theta[0]=theta[n].
 // How we do this is essentially to write out the first equation as:
 //
 //   theta[n] = aug[0] + w[0]*theta[0] - post[0]*theta[1]
 //
 // and then use the next equation to substitute in for theta[1]:
 //
 //   theta[1] = aug[1] + w[1]*theta[0] - post[1]*theta[2]
 //
 // and so on until we have an equation just in terms of theta[0] and
 // theta[n] (which are the same theta).
 //
 // The loop invariant maintained is that after j iterations, we have
 //   theta[n]= a + b*theta[0] + c*theta[j]
 Int n=(Int) we.size();
 double a=0,b=0,c=1;
 for (Int j=0;j<n;++j) {
   weqn& q=we[j];
   a+=c*q.aug;
   b+=c*q.w;
   c=-c*q.post;
 }

 // After the iteration we have
 //
 //   theta[n] = a + b*theta[0] + c*theta[n]
 //
 // where theta[n]=theta[0], so
 return a/(1.0-(b+c));
}

cvector<double> backsubCyclic(cvector<weqn>& we, double theta0)
{
 Int n=(Int) we.size();
 cvector<double> thetas;
 double lastTheta=theta0;
 for (Int j=1;j<=n;++j)
   {
     weqn& q=we[n-j];
     assert(q.pre == 0 && q.piv == 1);
     double theta=-q.post*lastTheta+q.aug+q.w*theta0;
     thetas.push_back(theta);
     lastTheta=theta;
   }
 reverse(thetas.begin(),thetas.end());
 return thetas;
}

// For the non-cyclic equations, do row operation to put the matrix into
// reduced echelon form, ie. calculates equivalent equations but with pre=0 and
// piv=1 for each eqn.
struct ref : public knotprop<eqn> {
 cvector<eqn>& e;
 eqn lasteqn;

 ref(knotlist& l, cvector<eqn>& e)
   : knotprop<eqn>(l), e(e), lasteqn(0,1,0,0) {}

 // Scale the equation so that the pivot (diagonal) entry is one, and save
 // the new equation as lasteqn.
 eqn scale(eqn q) {
   assert(q.pre == 0 && q.piv != 0);
   return lasteqn = eqn(0,1,q.post/q.piv,q.aug/q.piv);
 }

 eqn start(Int j) {
   return scale(e[j]);
 }
 eqn mid(Int j) {
   // Subtract a factor of the last equation so that the first entry is
   // zero, then procede to scale it.
   eqn& q=e[j];
   return scale(eqn(0,q.piv-q.pre*lasteqn.post,q.post,
                    q.aug-q.pre*lasteqn.aug));
 }
 // The end case is the same as the middle case.
};

// Once the matrix is in reduced echelon form, we can solve for the values by
// back-substitution.  This algorithm works from the bottom-up, so backCompute
// must be used to get the answer.
struct backsub : public knotprop<double> {
 cvector<eqn>& e;
 double lastTheta;

 backsub(knotlist& l, cvector<eqn>& e)
   : knotprop<double>(l), e(e) {}

 double end(Int j) {
   eqn& q=e[j];
   assert(q.pre == 0 && q.piv == 1 && q.post == 0);
   double theta=q.aug;
   lastTheta=theta;
   return theta;
 }

 double mid(Int j) {
   eqn& q=e[j];
   assert(q.pre == 0 && q.piv == 1);
   double theta=-q.post*lastTheta+q.aug;
   lastTheta=theta;
   return theta;
 }

 // start is the same as mid.
};

// Once the equations have been determined, solve for the thetas.
cvector<double> solveThetas(knotlist& l, cvector<eqn>& e)
{
 if (homogeneous(e))
   // We are solving Ax=0, so a solution is zero for every theta.
   return cvector<double>(e.size(),0);
 else if (l.cyclic()) {
   // The knotprop template is unusually unhelpful in this case, so I
   // won't use it here. The algorithm breaks into three passes on the
   // object.  The old Asymptote code used a two-pass method, but I
   // implemented this to stay closer to the MetaPost source code.
   // This might be something to look at for optimization.
   cvector<weqn> we=recalc(e);
   INFO(we);
   double theta0=solveForTheta0(we);
   return backsubCyclic(we, theta0);
 }
 else { /* Non-cyclic case. */
   /* First do row operations to get it into reduced echelon form. */
   cvector<eqn> el=ref(l,e).compute();

   /* Then, do back substitution. */
   return backsub(l,el).backCompute();
 }
}

// Once thetas have been solved, determine the first control point of every
// join.
struct postcontrolprop : public knotprop<pair> {
 cvector<pair>& dz;
 cvector<double>& psi;
 cvector<double>& theta;

 postcontrolprop(knotlist& l, cvector<pair>& dz,
                 cvector<double>& psi, cvector<double>& theta)
   : knotprop<pair>(l), dz(dz), psi(psi), theta(theta) {}

 double phi(Int j) {
   /* The third angle: psi + theta + phi = 0 */
   return -psi[j] - theta[j];
 }

 double vel(Int j) {
   /* Use the standard velocity function. */
   return velocity(theta[j],phi(j+1),l[j].tout);
 }

 // start is the same as mid.

 pair mid(Int j) {
   // Put a control point at the relative distance determined by the velocity,
   // and at an angle determined by theta.
   return l[j].z + vel(j)*expi(theta[j])*dz[j];
 }

 // The end postcontrol is the same as the last knot.
 pair end(Int j) {
   return l[j].z;
 }
};

// Determine the first control point of every join.
struct precontrolprop : public knotprop<pair> {
 cvector<pair>& dz;
 cvector<double>& psi;
 cvector<double>& theta;

 precontrolprop(knotlist& l, cvector<pair>& dz,
                cvector<double>& psi, cvector<double>& theta)
   : knotprop<pair>(l), dz(dz), psi(psi), theta(theta) {}

 double phi(Int j) {
   return -psi[j] - theta[j];
 }

 double vel(Int j) {
   return velocity(phi(j),theta[j-1],l[j].tin);
 }

 // The start precontrol is the same as the first knot.
 pair start(Int j) {
   return l[j].z;
 }
 pair mid(Int j) {
   return l[j].z - vel(j)*expi(-phi(j))*dz[j-1];
 }

 // end is the same as mid.
};

// Puts solved controls into a protopath starting at the given index.
// By convention, the first knot is not coded, as it is assumed to be coded by
// the previous section (or it is the first breakpoint and encoded as a special
// case).
struct encodeControls : public knoteffect {
 protopath& p;
 Int k;
 cvector<pair>& pre;
 cvector<pair>& post;

 encodeControls(protopath& p, Int k,
                cvector<pair>& pre, knotlist& l, cvector<pair>& post)
   : knoteffect(l), p(p), k(k), pre(pre), post(post) {}

 void encodePre(Int j) {
   p.pre(k+j)=pre[j];
 }
 void encodePoint(Int j) {
   p.point(k+j)=l[j].z;
 }
 void encodePost(Int j) {
   p.post(k+j)=post[j];
 }

 void solo(Int) {
#if 0
   encodePoint(j);
#endif
 }
 void start(Int j) {
#if 0
   encodePoint(j);
#endif
   encodePost(j);
 }
 void mid(Int j) {
   encodePre(j);
   encodePoint(j);
   encodePost(j);
 }
 void end(Int j) {
   encodePre(j);
   encodePoint(j);
 }
};

void encodeStraight(protopath& p, Int k, knotlist& l)
{
 pair a=l.front().z;
 double at=l.front().tout.val;
 pair b=l.back().z;
 double bt=l.back().tin.val;
 pair step=(b-a)/3.0;

 if (at==1.0 && bt==1.0) {
   p.straight(k)=true;
   p.post(k)=a+step;
   p.pre(k+1)=b-step;
   p.point(k+1)=b;
 }
 else {
   p.post(k)=a+step/at;
   p.pre(k+1)=b-step/bt;
   p.point(k+1)=b;
 }
}

void solveSection(protopath& p, Int k, knotlist& l)
{
 if (l.length()>0) {
   info(cerr, "solving section", l);

   // Calculate useful properties.
   cvector<pair>   dz  =  dzprop(l)   .compute();
   cvector<double> d   =   dprop(l,dz).compute();
   cvector<double> psi = psiprop(l,dz).compute();

   INFO(dz); INFO(d); INFO(psi);

   // Build and solve the linear equations for theta.
   cvector<eqn>        e = eqnprop(l,d,psi).compute();
   INFO(e);

   if (straightSection(e))
     // Handle straight section as special case.
     encodeStraight(p,k,l);
   else {
     cvector<double> theta = solveThetas(l,e);
     INFO(theta);

     // Calculate the control points.
     cvector<pair> post = postcontrolprop(l,dz,psi,theta).compute();
     cvector<pair> pre  =  precontrolprop(l,dz,psi,theta).compute();

     // Encode the results into the protopath.
     encodeControls(p,k,pre,l,post).exec();
   }
 }
}

// Find the first breakpoint in the knotlist, ie. where we can start solving a
// non-cyclic section.  If the knotlist is fully cyclic, then this returns
// NOBREAK.
// This must be called with a knot that has all of its implicit specifiers in
// place.
const Int NOBREAK=-1;
Int firstBreakpoint(knotlist& l)
{
 for (Int j=0;j<l.size();++j)
   if (!l[j].out->open())
     return j;
 return NOBREAK;
}

// Once a breakpoint, a, is found, find where the next breakpoint after it is.
// This must be called with a knot that has all of its implicit specifiers in
// place, so that breakpoint can be identified by either an in or out specifier
// that is not open.
Int nextBreakpoint(knotlist& l, Int a)
{
 // This is guaranteed to terminate if a is the index of a breakpoint.  If the
 // path is non-cyclic it will stop at or before the last knot which must be a
 // breakpoint.  If the path is cyclic, it will stop at or before looping back
 // around to a which is a breakpoint.
 Int j=a+1;
 while (l[j].in->open())
   ++j;
 return j;
}

// Write out the controls for section of the form
//   a.. control b and c ..d
void writeControls(protopath& p, Int a, knotlist& l)
{
 // By convention, the first point will already be encoded.
 p.straight(a)=dynamic_cast<controlSpec *>(l[a].out)->straight;
 p.post(a)=dynamic_cast<controlSpec *>(l[a].out)->cz;
 p.pre(a+1)=dynamic_cast<controlSpec *>(l[a+1].in)->cz;
 p.point(a+1)=l[a+1].z;
}

// Solves a path that has all of its specifiers laid out explicitly.
path solveSpecified(knotlist& l)
{
 protopath p(l.size(),l.cyclic());

 Int first=firstBreakpoint(l);
 if (first==NOBREAK)
   /* We are solving a fully cyclic path, so do it in one swoop. */
   solveSection(p,0,l);
 else {
   // Encode the first point.
   p.point(first)=l[first].z;

   // If the path is cyclic, we should stop where we started (modulo the
   // length of the path); otherwise, just stop at the end.
   Int last=l.cyclic() ? first+l.length()
     : l.length();
   Int a=first;
   while (a!=last) {
     if (l[a].out->controlled()) {
       assert(l[a+1].in->controlled());

       // Controls are already picked, just write them out.
       writeControls(p,a,l);
       ++a;
     }
     else {
       // Find the section a to b and solve it, putting the result (starting
       // from index a into our protopath.
       Int b=nextBreakpoint(l,a);
       subknotlist section(l,a,b);
       solveSection(p,a,section);
       a=b;
     }
   }

   // For a non-cyclic path, the end control points need to be set.
   p.controlEnds();
 }

 return p.fix();
}

/* If a knot is open on one side and restricted on the other, this replaces the
* open side with a restriction determined by the restriction on the other
* side. After this, any knot will either have two open specs or two
* restrictions.
*/
struct partnerUp : public knoteffect {
 partnerUp(knotlist& l)
   : knoteffect(l) {}

 void mid(Int j) {
   knot& k=l[j];
   if (k.in->open() && !k.out->open())
     k.in=k.out->inPartner(k.z);
   else if (!k.in->open() && k.out->open())
     k.out=k.in->outPartner(k.z);
 }
};

/* Ensures a non-cyclic path has direction specifiers at the ends, adding curls
* if there are none.
*/
void curlEnds(knotlist& l)
{
 static curlSpec endSpec;

 if (!l.cyclic()) {
   if (l.front().in->open())
     l.front().in=&endSpec;
   if (l.back().out->open())
     l.back().out=&endSpec;
 }
}

/* If a point occurs twice in a row in a knotlist, write in controls
* between the two knots at that point (unless it already has controls).
*/
struct controlDuplicates : public knoteffect {
 controlDuplicates(knotlist& l)
   : knoteffect(l) {}

 void solo(Int) { /* One point ==> no duplicates */ }
 // start is the same as mid.
 void mid(Int j) {
   knot &k1=l[j];
   knot &k2=l[j+1];
   if (!k1.out->controlled() && k1.z==k2.z) {
     k1.out=k2.in=new controlSpec(k1.z,true);
   }
 }
 void end(Int) { /* No next point to compare with. */ }
};

path solve(knotlist& l)
{
 if (l.empty())
   return path();
 else {
   info(cerr, "input knotlist", l);
   curlEnds(l);
   controlDuplicates(l).exec();
   partnerUp(l).exec();
   info(cerr, "specified knotlist", l);
   return solveSpecified(l);
 }
}

// Code for Testing
#if 0
path solveSimple(cvector<pair>& z)
{
 // The two specifiers used: an open spec and a curl spec for the ends.
 spec open;

//  curlSpec curl;
//  curlSpec curly(2.0);
//  dirSpec E(0);
//  dirSpec N(PI/2.0);

 controlSpec here(pair(150,150));

 // Encode the knots as open in the knotlist.
 cvector<knot> nodes;
 for (cvector<pair>::iterator p=z.begin(); p!=z.end(); ++p) {
   knot k;
   k.z=*p;
   k.in=k.out=&open;

   nodes.push_back(k);
 }

 // Substitute in a curl spec for the ends.
 //nodes.front().out=nodes.back().in=&curl;

 // Test direction specifiers.
 //nodes.front().tout=2;
 //nodes.front().out=nodes.back().in=&curly;

 //nodes[0].out=nodes[0].in=&E;
 nodes[1].out=nodes[2].in=&here;

 simpleknotlist l(nodes,false);
 return solve(l);
}
#endif

} // namespace camp

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