private import math;

if(inXasyMode) settings.render=0;

if(prc0()) {
  if(settings.tex == "context") settings.prc=false;
  else {
    access embed;
    Embed=embed.embedplayer;
  }
}

// Useful lossy compression values.
restricted real Zero=0;
restricted real Low=0.0001;
restricted real Medium=0.001;
restricted real High=0.01;

restricted int PRCsphere=0;   // Renders slowly but produces smaller PRC files.
restricted int NURBSsphere=1; // Renders fast but produces larger PRC files.

struct render
{
  // PRC parameters:
  real compression;     // lossy compression parameter (0=no compression)
  real granularity;     // PRC rendering granularity

  bool closed;          // use one-sided rendering?
  bool tessellate;      // use tessellated mesh to store straight patches?

  bool3 merge;          // merge nodes before rendering, for faster but
                        // lower quality PRC rendering (the value default means
                        // merge opaque patches only).

  int sphere;           // PRC sphere type (PRCsphere or NURBSsphere).

  // General parameters:
  real margin;          // shrink amount for rendered openGL viewport, in bp.
  real tubegranularity; // granularity for rendering tubes 
  bool labelfill;       // fill subdivision cracks in unlighted labels

  bool partnames;       // assign part name indices to compound objects
  bool defaultnames;    // assign default names to unnamed objects

  static render defaultrender;
  
  void operator init(real compression=defaultrender.compression,
                     real granularity=defaultrender.granularity,
                     bool closed=defaultrender.closed,
                     bool tessellate=defaultrender.tessellate,
                     bool3 merge=defaultrender.merge,
                     int sphere=defaultrender.sphere,
                     real margin=defaultrender.margin,
                     real tubegranularity=defaultrender.tubegranularity,
                     bool labelfill=defaultrender.labelfill,
                     bool partnames=defaultrender.partnames,
                     bool defaultnames=defaultrender.defaultnames)
  {
    this.compression=compression;
    this.granularity=granularity;
    this.closed=closed;
    this.tessellate=tessellate;
    this.merge=merge;
    this.sphere=sphere;
    this.margin=margin;
    this.tubegranularity=tubegranularity;
    this.labelfill=labelfill;
    this.partnames=partnames;
    this.defaultnames=defaultnames;
  }
}

render operator init() {return render();}

render defaultrender=render.defaultrender=new render;
defaultrender.compression=High;
defaultrender.granularity=Medium;
defaultrender.closed=false;
defaultrender.tessellate=false;
defaultrender.merge=false;
defaultrender.margin=0.02;
defaultrender.tubegranularity=0.005;
defaultrender.sphere=NURBSsphere;
defaultrender.labelfill=true;
defaultrender.partnames=false;
defaultrender.defaultnames=true;

real defaultshininess=0.25;

real angleprecision=1e-5; // Precision for centering perspective projections.
int maxangleiterations=25;

string defaultembed3Doptions="3Dmenu";
string defaultembed3Dscript;
real defaulteyetoview=63mm/1000mm;

string partname(int i, render render=defaultrender) 
{
  return render.partnames ? string(i+1) : "";
}

triple O=(0,0,0);
triple X=(1,0,0), Y=(0,1,0), Z=(0,0,1);

// A translation in 3D space.
transform3 shift(triple v)
{
  transform3 t=identity(4);
  t[0][3]=v.x;
  t[1][3]=v.y;
  t[2][3]=v.z;
  return t;
}

// Avoid two parentheses.
transform3 shift(real x, real y, real z)
{
  return shift((x,y,z));
}

transform3 shift(transform3 t)
{
  transform3 T=identity(4);
  T[0][3]=t[0][3];
  T[1][3]=t[1][3];
  T[2][3]=t[2][3];
  return T;
}

// A 3D scaling in the x direction.
transform3 xscale3(real x)
{
  transform3 t=identity(4);
  t[0][0]=x;
  return t;
}

// A 3D scaling in the y direction.
transform3 yscale3(real y)
{
  transform3 t=identity(4);
  t[1][1]=y;
  return t;
}

// A 3D scaling in the z direction.
transform3 zscale3(real z)
{
  transform3 t=identity(4);
  t[2][2]=z;
  return t;
}

// A 3D scaling by s in the v direction.
transform3 scale(triple v, real s)
{
  v=unit(v);
  s -= 1;
  return new real[][] {
    {1+s*v.x^2, s*v.x*v.y, s*v.x*v.z, 0}, 
      {s*v.x*v.y, 1+s*v.y^2, s*v.y*v.z, 0}, 
        {s*v.x*v.z, s*v.y*v.z, 1+s*v.z^2, 0},
          {0, 0, 0, 1}};
}

// A transformation representing rotation by an angle in degrees about
// an axis v through the origin (in the right-handed direction).
transform3 rotate(real angle, triple v)
{
  if(v == O) abort("cannot rotate about the zero vector");
  v=unit(v);
  real x=v.x, y=v.y, z=v.z;
  real s=Sin(angle), c=Cos(angle), t=1-c;

  return new real[][] {
    {t*x^2+c,   t*x*y-s*z, t*x*z+s*y, 0},
      {t*x*y+s*z, t*y^2+c,   t*y*z-s*x, 0},
        {t*x*z-s*y, t*y*z+s*x, t*z^2+c,   0},
          {0,         0,         0,         1}};
}

// A transformation representing rotation by an angle in degrees about
// the line u--v (in the right-handed direction).
transform3 rotate(real angle, triple u, triple v)
{
  return shift(u)*rotate(angle,v-u)*shift(-u);
}

// Reflects about the plane through u, v, and w.
transform3 reflect(triple u, triple v, triple w)
{
  triple n=unit(cross(v-u,w-u));
  if(n == O)
    abort("points determining reflection plane cannot be colinear");

  return new real[][] {
    {1-2*n.x^2, -2*n.x*n.y, -2*n.x*n.z, u.x},
      {-2*n.x*n.y, 1-2*n.y^2, -2*n.y*n.z, u.y},
        {-2*n.x*n.z, -2*n.y*n.z, 1-2*n.z^2, u.z},
          {0, 0, 0, 1}
  }*shift(-u);
}

// Project u onto v.
triple project(triple u, triple v)
{
  v=unit(v);
  return dot(u,v)*v;
}

// Return a unit vector perpendicular to a given unit vector v.
triple perp(triple v)
{
  triple u=cross(v,Y);
  if(abs(u) > sqrtEpsilon) return unit(u);
  u=cross(v,Z);
  return (abs(u) > sqrtEpsilon) ? unit(u) : X;
}

// Return the transformation corresponding to moving the camera from the target
// (looking in the negative z direction) to the point 'eye' (looking at target),
// orienting the camera so that direction 'up' points upwards.
// Since, in actuality, we are transforming the points instead of the camera,
// we calculate the inverse matrix.
// Based on the gluLookAt implementation in the OpenGL manual.
transform3 look(triple eye, triple up=Z, triple target=O)
{
  triple f=unit(target-eye);
  if(f == O)
    f=-Z; // The eye is already at the origin: look down.

  triple s=cross(f,up);

  // If the eye is pointing either directly up or down, there is no
  // preferred "up" direction.  Pick one arbitrarily.
  s=s != O ? unit(s) : perp(f);

  triple u=cross(s,f);

  transform3 M={{ s.x,  s.y,  s.z, 0},
                { u.x,  u.y,  u.z, 0},
                {-f.x, -f.y, -f.z, 0},
                {   0,    0,    0, 1}};

  return M*shift(-eye);
}

// Return a matrix to do perspective distortion based on a triple v.
transform3 distort(triple v) 
{
  transform3 t=identity(4);
  real d=length(v);
  if(d == 0) return t;
  t[3][2]=-1/d;
  t[3][3]=0;
  return t;
}

projection operator * (transform3 t, projection P)
{
  projection P=P.copy();
  if(!P.absolute) {
    P.camera=t*P.camera;
    triple target=P.target;
    P.target=t*P.target;
    if(P.infinity)
      P.normal=t*(target+P.normal)-P.target;
    else
      P.normal=P.vector();
    P.calculate();
  }
  return P;
}

// With this, save() and restore() in plain also save and restore the
// currentprojection.
addSaveFunction(new restoreThunk() {
    projection P=currentprojection.copy();
    return new void() {
      currentprojection=P;
    };
  });

pair project(triple v, projection P=currentprojection)
{
  return project(v,P.t);
}

pair dir(triple v, triple dir, projection P)
{
  return unit(project(v+0.5dir,P)-project(v-0.5*dir,P));
}

// Uses the homogenous coordinate to perform perspective distortion.
// When combined with a projection to the XY plane, this effectively maps
// points in three space to a plane through target and
// perpendicular to the vector camera-target.
projection perspective(triple camera, triple up=Z, triple target=O,
                       real zoom=1, real angle=0, pair viewportshift=0,
                       bool showtarget=true, bool autoadjust=true,
                       bool center=autoadjust)
{
  if(camera == target)
    abort("camera cannot be at target");
  return projection(camera,up,target,zoom,angle,viewportshift,
                    showtarget,autoadjust,center,
                    new transformation(triple camera, triple up, triple target)
                    {return transformation(look(camera,up,target),
                                           distort(camera-target));});
}

projection perspective(real x, real y, real z, triple up=Z, triple target=O,
                       real zoom=1, real angle=0, pair viewportshift=0,
                       bool showtarget=true, bool autoadjust=true,
                       bool center=autoadjust)
{
  return perspective((x,y,z),up,target,zoom,angle,viewportshift,showtarget,
                     autoadjust,center);
}

projection orthographic(triple camera, triple up=Z, triple target=O,
                        real zoom=1, pair viewportshift=0,
                        bool showtarget=true, bool center=false)
{
  return projection(camera,up,target,zoom,viewportshift,showtarget,
                    center=center,new transformation(triple camera, triple up,
                                                     triple target) {
                      return transformation(look(camera,up,target));});
}

projection orthographic(real x, real y, real z, triple up=Z,
                        triple target=O, real zoom=1, pair viewportshift=0,
                        bool showtarget=true, bool center=false)
{
  return orthographic((x,y,z),up,target,zoom,viewportshift,showtarget,
                      center=center);
}

// Compute camera position with x axis below the horizontal at angle alpha,
// y axis below the horizontal at angle beta, and z axis up.
triple camera(real alpha, real beta)
{
  real denom=Tan(alpha+beta);
  real Tanalpha=Tan(alpha);
  real Tanbeta=Tan(beta);
  return (sqrt(Tanalpha/denom),sqrt(Tanbeta/denom),sqrt(Tanalpha*Tanbeta));
}

projection oblique(real angle=45)
{
  transform3 t=identity(4);
  real c2=Cos(angle)^2;
  real s2=1-c2;
  t[0][2]=-c2;
  t[1][2]=-s2;
  t[2][2]=1;
  t[2][3]=-1;
  return projection((c2,s2,1),up=Y,normal=(0,0,1),
                    new transformation(triple,triple,triple) {
                      return transformation(t);});
}

projection obliqueZ(real angle=45) {return oblique(angle);}

projection obliqueX(real angle=45)
{
  transform3 t=identity(4);
  real c2=Cos(angle)^2;
  real s2=1-c2;
  t[0][0]=-c2;
  t[1][0]=-s2;
  t[1][1]=0;
  t[0][1]=1;
  t[1][2]=1;
  t[2][2]=0;
  t[2][0]=1;
  t[2][3]=-1;
  return projection((1,c2,s2),normal=(1,0,0),
                    new transformation(triple,triple,triple) {
                      return transformation(t);});
}

projection obliqueY(real angle=45)
{
  transform3 t=identity(4);
  real c2=Cos(angle)^2;
  real s2=1-c2;
  t[0][1]=c2;
  t[1][1]=s2;
  t[1][2]=1;
  t[2][1]=-1;
  t[2][2]=0;
  t[2][3]=-1;
  return projection((c2,-1,s2),normal=(0,-1,0),
                    new transformation(triple,triple,triple) {
                      return transformation(t);});
}

projection oblique=oblique();
projection obliqueX=obliqueX(), obliqueY=obliqueY(), obliqueZ=obliqueZ();

projection LeftView=orthographic(-X,showtarget=true);
projection RightView=orthographic(X,showtarget=true);
projection FrontView=orthographic(-Y,showtarget=true);
projection BackView=orthographic(Y,showtarget=true);
projection BottomView=orthographic(-Z,up=-Y,showtarget=true);
projection TopView=orthographic(Z,up=Y,showtarget=true);

currentprojection=perspective(5,4,2);

projection projection() 
{
  projection P;
  real[] a=_projection();
  if(a.length == 0 || a[10] == 0.0) return currentprojection;
  int k=0;
  return a[0] == 1 ?
    orthographic((a[++k],a[++k],a[++k]),(a[++k],a[++k],a[++k]),
                 (a[++k],a[++k],a[++k]),a[++k],(a[k += 2],a[++k])) :
    perspective((a[++k],a[++k],a[++k]),(a[++k],a[++k],a[++k]),
                (a[++k],a[++k],a[++k]),a[++k],a[++k],(a[++k],a[++k]));
}

// Map pair z to a triple by inverting the projection P onto the
// plane perpendicular to normal and passing through point.
triple invert(pair z, triple normal, triple point,
              projection P=currentprojection)
{
  transform3 t=P.t;
  real[][] A={{t[0][0]-z.x*t[3][0],t[0][1]-z.x*t[3][1],t[0][2]-z.x*t[3][2]},
              {t[1][0]-z.y*t[3][0],t[1][1]-z.y*t[3][1],t[1][2]-z.y*t[3][2]},
              {normal.x,normal.y,normal.z}};
  real[] b={z.x*t[3][3]-t[0][3],z.y*t[3][3]-t[1][3],dot(normal,point)};
  real[] x=solve(A,b,warn=false);
  return x.length > 0 ? (x[0],x[1],x[2]) : P.camera;
}

// Map pair to a triple on the projection plane.
triple invert(pair z, projection P=currentprojection)
{
  return invert(z,P.normal,P.target,P);
}

// Map pair dir to a triple direction at point v on the projection plane.
triple invert(pair dir, triple v, projection P=currentprojection)
{
  return invert(project(v,P)+dir,P.normal,v,P)-v;
}

pair xypart(triple v)
{
  return (v.x,v.y);
}

struct control {
  triple post,pre;
  bool active=false;
  bool straight=true;
  void operator init(triple post, triple pre, bool straight=false) {
    this.post=post;
    this.pre=pre;
    active=true;
    this.straight=straight;
  }
}

control nocontrol;
  
control operator * (transform3 t, control c) 
{
  control C;
  C.post=t*c.post;
  C.pre=t*c.pre;
  C.active=c.active;
  C.straight=c.straight;
  return C;
}

void write(file file, control c)
{
  write(file,".. controls ");
  write(file,c.post);
  write(file," and ");
  write(file,c.pre);
}
  
struct Tension {
  real out,in;
  bool atLeast;
  bool active;
  void operator init(real out=1, real in=1, bool atLeast=false,
                     bool active=true) {
    real check(real val) {
      if(val < 0.75) abort("tension cannot be less than 3/4");
      return val;
    }
    this.out=check(out);
    this.in=check(in);
    this.atLeast=atLeast;
    this.active=active;
  }
}

Tension operator init()
{
  return Tension();
}

Tension noTension;
noTension.active=false;
  
void write(file file, Tension t)
{
  write(file,"..tension ");
  if(t.atLeast) write(file,"atleast ");
  write(file,t.out);
  write(file," and ");
  write(file,t.in);
}
  
struct dir {
  triple dir;
  real gamma=1; // endpoint curl
  bool Curl;    // curl specified
  bool active() {
    return dir != O || Curl;
  }
  void init(triple v) {
    this.dir=v;
  }
  void init(real gamma) {
    if(gamma < 0) abort("curl cannot be less than 0");
    this.gamma=gamma;
    this.Curl=true;
  }
  void init(dir d) {
    dir=d.dir;
    gamma=d.gamma;
    Curl=d.Curl;
  }
  void default(triple v) {
    if(!active()) init(v);
  }
  void default(dir d) {
    if(!active()) init(d);
  }
  dir copy() {
    dir d=new dir;
    d.init(this);
    return d;
  }
}

void write(file file, dir d)
{
  if(d.dir != O) {
    write(file,"{"); write(file,unit(d.dir)); write(file,"}");
  } else if(d.Curl) {
    write(file,"{curl "); write(file,d.gamma); write(file,"}");
  }
}
  
dir operator * (transform3 t, dir d) 
{
  dir D=d.copy();
  D.init(unit(shiftless(t)*d.dir));
  return D;
}

void checkEmpty(int n) {
  if(n == 0)
    abort("nullpath3 has no points");
}

int adjustedIndex(int i, int n, bool cycles)
{
  checkEmpty(n);
  if(cycles)
    return i % n;
  else if(i < 0)
    return 0;
  else if(i >= n)
    return n-1;
  else
    return i;
}

struct flatguide3 {
  triple[] nodes;
  bool[] cyclic;     // true if node is really a cycle
  control[] control; // control points for segment starting at node
  Tension[] Tension; // Tension parameters for segment starting at node
  dir[] in,out;      // in and out directions for segment starting at node

  bool cyclic() {int n=cyclic.length; return n > 0 ? cyclic[n-1] : false;}
  bool precyclic() {int i=find(cyclic); return i >= 0 && i < cyclic.length-1;}
  
  int size() {
    return cyclic() ? nodes.length-1 : nodes.length;
  }
  
  void node(triple v, bool b=false) {
    nodes.push(v);
    control.push(nocontrol);
    Tension.push(noTension);
    in.push(new dir);
    out.push(new dir);
    cyclic.push(b);
  }

  void control(triple post, triple pre) {
    if(control.length > 0) {
      control c=control(post,pre,false);
      control[control.length-1]=c;
    }
  }

  void Tension(real out, real in, bool atLeast) {
    if(Tension.length > 0)
      Tension[Tension.length-1]=Tension(out,in,atLeast,true);
  }

  void in(triple v) {
    if(in.length > 0) {
      in[in.length-1].init(v);
    }
  }

  void out(triple v) {
    if(out.length > 0) {
      out[out.length-1].init(v);
    }
  }

  void in(real gamma) {
    if(in.length > 0) {
      in[in.length-1].init(gamma);
    }
  }

  void out(real gamma) {
    if(out.length > 0) {
      out[out.length-1].init(gamma);
    }
  }

  void cycleToken() {
    if(nodes.length > 0)
      node(nodes[0],true);
  }
  
  // Return true if outgoing direction at node i is known.
  bool solved(int i) {
    return out[i].active() || control[i].active;
  }
}

void write(file file, string s="", explicit flatguide3 x, suffix suffix=none)
{
  write(file,s);
  if(x.size() == 0) write(file,"<nullpath3>");
  else for(int i=0; i < x.nodes.length; ++i) {
      if(i > 0) write(file,endl);
      if(x.cyclic[i]) write(file,"cycle");
      else write(file,x.nodes[i]);
      if(i < x.nodes.length-1) {
        // Explicit control points trump other specifiers
        if(x.control[i].active)
          write(file,x.control[i]);
        else {
          write(file,x.out[i]);
          if(x.Tension[i].active) write(file,x.Tension[i]);
        }
        write(file,"..");
        if(!x.control[i].active) write(file,x.in[i]);
      }
    }
  write(file,suffix);
}

void write(string s="", flatguide3 x, suffix suffix=endl)
{
  write(stdout,s,x,suffix);
}

// A guide3 is most easily represented as something that modifies a flatguide3.
typedef void guide3(flatguide3);

restricted void nullpath3(flatguide3) {};

guide3 operator init() {return nullpath3;}

guide3 operator cast(triple v)
{
  return new void(flatguide3 f) {
    f.node(v);
  };
}

guide3 operator cast(cycleToken) {
  return new void(flatguide3 f) {
    f.cycleToken();
  };
}

guide3 operator controls(triple post, triple pre) 
{
  return new void(flatguide3 f) {
    f.control(post,pre);
  };
};
  
guide3 operator controls(triple v)
{
  return operator controls(v,v);
}

guide3 operator cast(tensionSpecifier t)
{
  return new void(flatguide3 f) {
    f.Tension(t.out, t.in, t.atLeast);
  };
}

guide3 operator cast(curlSpecifier spec)
{
  return new void(flatguide3 f) {
    if(spec.side == JOIN_OUT) f.out(spec.value);
    else if(spec.side == JOIN_IN) f.in(spec.value);
    else
      abort("invalid curl specifier");
  };
}

guide3 operator spec(triple v, int side)
{
  return new void(flatguide3 f) {
    if(side == JOIN_OUT) f.out(v);
    else if(side == JOIN_IN) f.in(v);
    else
      abort("invalid direction specifier");
  };
}
  
guide3 operator -- (... guide3[] g)
{
  return new void(flatguide3 f) {
    if(g.length > 0) {
      for(int i=0; i < g.length-1; ++i) {
        g[i](f);
        f.out(1);
        f.in(1);
      }
      g[g.length-1](f);
    }
  };
}

guide3 operator .. (... guide3[] g)
{
  return new void(flatguide3 f) {
    for(int i=0; i < g.length; ++i)
      g[i](f);
  };
}

typedef guide3 interpolate3(... guide3[]);

interpolate3 join3(tensionSpecifier t)
{
  return new guide3(... guide3[] a) {
    if(a.length == 0) return nullpath3;
    guide3 g=a[0];
    for(int i=1; i < a.length; ++i)
      g=g..t..a[i];
    return g;
  };
}

interpolate3 operator ::=join3(operator tension(1,true));
interpolate3 operator ---=join3(operator tension(infinity,true));

flatguide3 operator cast(guide3 g)
{
  flatguide3 f;
  g(f);
  return f;
}

flatguide3[] operator cast(guide3[] g)
{
  flatguide3[] p=new flatguide3[g.length];
  for(int i=0; i < g.length; ++i) {
    flatguide3 f;
    g[i](f);
    p[i]=f;
  }
  return p;
}

// A version of asin that tolerates numerical imprecision
real asin1(real x)
{
  return asin(min(max(x,-1),1));
}
  
// A version of acos that tolerates numerical imprecision
real acos1(real x)
{
  return acos(min(max(x,-1),1));
}
  
struct Controls {
  triple c0,c1;

  // 3D extension of John Hobby's control point formula
  // (cf. The MetaFont Book, page 131),
  // as described in John C. Bowman and A. Hammerlindl,
  // TUGBOAT: The Communications of the TeX Users Group 29:2 (2008).

  void operator init(triple v0, triple v1, triple d0, triple d1, real tout,
                     real tin, bool atLeast) {
    triple v=v1-v0;
    triple u=unit(v);
    real L=length(v);
    d0=unit(d0);
    d1=unit(d1);
    real theta=acos1(dot(d0,u));
    real phi=acos1(dot(d1,u));
    if(dot(cross(d0,v),cross(v,d1)) < 0) phi=-phi;
    c0=v0+d0*L*relativedistance(theta,phi,tout,atLeast);
    c1=v1-d1*L*relativedistance(phi,theta,tin,atLeast);
  }
}

private triple cross(triple d0, triple d1, triple reference)
{
  triple normal=cross(d0,d1);
  return normal == O ? reference : normal;
}
                                        
private triple dir(real theta, triple d0, triple d1, triple reference)
{
  triple normal=cross(d0,d1,reference);
  if(normal == O) return d1;
  return rotate(degrees(theta),dot(normal,reference) >= 0 ? normal : -normal)*
    d1;
}

private real angle(triple d0, triple d1, triple reference)
{
  real theta=acos1(dot(unit(d0),unit(d1)));
  return dot(cross(d0,d1,reference),reference) >= 0 ? theta : -theta;
}

// 3D extension of John Hobby's angle formula (The MetaFont Book, page 131).
// Notational differences: here psi[i] is the turning angle at z[i+1],
// beta[i] is the tension for segment i, and in[i] is the incoming
// direction for segment i (where segment i begins at node i).

real[] theta(triple[] v, real[] alpha, real[] beta, 
             triple dir0, triple dirn, real g0, real gn, triple reference)
{
  real[] a,b,c,f,l,psi;
  int n=alpha.length;
  bool cyclic=v.cyclic;
  for(int i=0; i < n; ++i)
    l[i]=1/length(v[i+1]-v[i]);
  int i0,in;
  if(cyclic) {i0=0; in=n;}
  else {i0=1; in=n-1;}
  for(int i=0; i < in; ++i)
    psi[i]=angle(v[i+1]-v[i],v[i+2]-v[i+1],reference);
  if(cyclic) {
    l.cyclic=true;
    psi.cyclic=true;
  } else {
    psi[n-1]=0;
    if(dir0 == O) {
      real a0=alpha[0];
      real b0=beta[0];
      real chi=g0*(b0/a0)^2;
      a[0]=0;
      b[0]=3a0-a0/b0+chi;
      real C=chi*(3a0-1)+a0/b0;
      c[0]=C;
      f[0]=-C*psi[0];
    } else {
      a[0]=c[0]=0;
      b[0]=1;
      f[0]=angle(v[1]-v[0],dir0,reference);
    }
    if(dirn == O) {
      real an=alpha[n-1];
      real bn=beta[n-1];
      real chi=gn*(an/bn)^2;
      a[n]=chi*(3bn-1)+bn/an;
      b[n]=3bn-bn/an+chi;
      c[n]=f[n]=0;
    } else {
      a[n]=c[n]=0;
      b[n]=1;
      f[n]=angle(v[n]-v[n-1],dirn,reference);
    }
  }
  
  for(int i=i0; i < n; ++i) {
    real in=beta[i-1]^2*l[i-1];
    real A=in/alpha[i-1];
    a[i]=A;
    real B=3*in-A;
    real out=alpha[i]^2*l[i];
    real C=out/beta[i];
    b[i]=B+3*out-C;
    c[i]=C;
    f[i]=-B*psi[i-1]-C*psi[i];
  }
  
  return tridiagonal(a,b,c,f);
}

triple reference(triple[] v, int n, triple d0, triple d1)
{
  triple[] V=sequence(new triple(int i) {
      return cross(v[i+1]-v[i],v[i+2]-v[i+1]); 
    },n-1);
  if(n > 0) {
    V.push(cross(d0,v[1]-v[0]));
    V.push(cross(v[n]-v[n-1],d1));
  }

  triple max=V[0];
  real M=abs(max);
  for(int i=1; i < V.length; ++i) {
    triple vi=V[i];
    real a=abs(vi);
    if(a > M) {
      M=a;
      max=vi;
    }
  }

  triple reference;
  for(int i=0; i < V.length; ++i) {
    triple u=unit(V[i]);
    reference += dot(u,max) < 0 ? -u : u;
  }

  return reference;
}

// Fill in missing directions for n cyclic nodes.
void aim(flatguide3 g, int N) 
{
  bool cyclic=true;
  int start=0, end=0;
  
  // If the cycle contains one or more direction specifiers, break the loop.
  for(int k=0; k < N; ++k)
    if(g.solved(k)) {cyclic=false; end=k; break;}
  for(int k=N-1; k >= 0; --k)
    if(g.solved(k)) {cyclic=false; start=k; break;}
  while(start < N && g.control[start].active) ++start;
  
  int n=N-(start-end);
  if(n <= 1 || (cyclic && n <= 2)) return;

  triple[] v=new triple[cyclic ? n : n+1];
  real[] alpha=new real[n];
  real[] beta=new real[n];
  for(int k=0; k < n; ++k) {
    int K=(start+k) % N;
    v[k]=g.nodes[K];
    alpha[k]=g.Tension[K].out;
    beta[k]=g.Tension[K].in;
  }
  if(cyclic) {
    v.cyclic=true;
    alpha.cyclic=true;
    beta.cyclic=true;
  } else v[n]=g.nodes[(start+n) % N];
  int final=(end-1) % N;

  triple d0=g.out[start].dir;
  triple d1=g.in[final].dir;

  triple reference=reference(v,n,d0,d1);

  real[] theta=theta(v,alpha,beta,d0,d1,g.out[start].gamma,g.in[final].gamma,
                     reference);

  v.cyclic=true;
  theta.cyclic=true;
    
  for(int k=1; k < (cyclic ? n+1 : n); ++k) {
    triple w=dir(theta[k],v[k]-v[k-1],v[k+1]-v[k],reference);
    g.in[(start+k-1) % N].init(w);
    g.out[(start+k) % N].init(w);
  }

  if(g.out[start].dir == O)
    g.out[start].init(dir(theta[0],v[0]-g.nodes[(start-1) % N],v[1]-v[0],
                          reference));
  if(g.in[final].dir == O)
    g.in[final].init(dir(theta[n],v[n-1]-v[n-2],v[n]-v[n-1],reference));
}

// Fill in missing directions for the sequence of nodes i...n.
void aim(flatguide3 g, int i, int n) 
{
  int j=n-i;
  if(j > 1 || g.out[i].dir != O || g.in[i].dir != O) {
    triple[] v=new triple[j+1];
    real[] alpha=new real[j];
    real[] beta=new real[j];
    for(int k=0; k < j; ++k) {
      v[k]=g.nodes[i+k];
      alpha[k]=g.Tension[i+k].out;
      beta[k]=g.Tension[i+k].in;
    }
    v[j]=g.nodes[n];
    
    triple d0=g.out[i].dir;
    triple d1=g.in[n-1].dir;

    triple reference=reference(v,j,d0,d1);

    real[] theta=theta(v,alpha,beta,d0,d1,g.out[i].gamma,g.in[n-1].gamma,
                       reference);
    
    for(int k=1; k < j; ++k) {
      triple w=dir(theta[k],v[k]-v[k-1],v[k+1]-v[k],reference);
      g.in[i+k-1].init(w);
      g.out[i+k].init(w);
    }
    if(g.out[i].dir == O) {
      triple w=dir(theta[0],g.in[i].dir,v[1]-v[0],reference);
      if(i > 0) g.in[i-1].init(w);
      g.out[i].init(w);
    }
    if(g.in[n-1].dir == O) {
      triple w=dir(theta[j],g.out[n-1].dir,v[j]-v[j-1],reference);
      g.in[n-1].init(w);
      g.out[n].init(w);
    }
  }
}

private real Fuzz=10*realEpsilon;

triple XYplane(pair z) {return (z.x,z.y,0);}
triple YZplane(pair z) {return (0,z.x,z.y);}
triple ZXplane(pair z) {return (z.y,0,z.x);}

bool cyclic(guide3 g) {flatguide3 f; g(f); return f.cyclic();}
int size(guide3 g) {flatguide3 f; g(f); return f.size();}
int length(guide3 g) {flatguide3 f; g(f); return f.nodes.length-1;}

triple dir(path3 p)
{
  return dir(p,length(p));
}

triple dir(path3 p, path3 h)
{
  return 0.5*(dir(p)+dir(h));
}

// return the point on path3 p at arclength L
triple arcpoint(path3 p, real L)
{
  return point(p,arctime(p,L));
}

// return the direction on path3 p at arclength L
triple arcdir(path3 p, real L)
{
  return dir(p,arctime(p,L));
}

// return the time on path3 p at the relative fraction l of its arclength
real reltime(path3 p, real l)
{
  return arctime(p,l*arclength(p));
}

// return the point on path3 p at the relative fraction l of its arclength
triple relpoint(path3 p, real l)
{
  return point(p,reltime(p,l));
}

// return the direction of path3 p at the relative fraction l of its arclength
triple reldir(path3 p, real l)
{
  return dir(p,reltime(p,l));
}

// return the initial point of path3 p
triple beginpoint(path3 p)
{
  return point(p,0);
}

// return the point on path3 p at half of its arclength
triple midpoint(path3 p)
{
  return relpoint(p,0.5);
}

// return the final point of path3 p
triple endpoint(path3 p)
{
  return point(p,length(p));
}

path3 path3(triple v)
{
  triple[] point={v};
  return path3(point,point,point,new bool[] {false},false);
}

path3 path3(path p, triple plane(pair)=XYplane)
{
  int n=size(p);
  return path3(sequence(new triple(int i) {return plane(precontrol(p,i));},n),
               sequence(new triple(int i) {return plane(point(p,i));},n),
               sequence(new triple(int i) {return plane(postcontrol(p,i));},n),
               sequence(new bool(int i) {return straight(p,i);},n),
               cyclic(p));
}

path3[] path3(explicit path[] g, triple plane(pair)=XYplane)
{
  return sequence(new path3(int i) {return path3(g[i],plane);},g.length);
}

path3 invert(path p, triple normal, triple point,
             projection P=currentprojection)
{
  return path3(p,new triple(pair z) {return invert(z,normal,point,P);});
}

path3 invert(path p, triple point, projection P=currentprojection)
{
  return path3(p,new triple(pair z) {return invert(z,P.normal,point,P);});
}

path3 invert(path p, projection P=currentprojection)
{
  return path3(p,new triple(pair z) {return invert(z,P.normal,P.target,P);});
}

// Construct a path from a path3 by applying P to each control point.
path path(path3 p, pair P(triple)=xypart)
{
  int n=length(p);
  if(n < 0) return nullpath;
  guide g=P(point(p,0));
  if(n == 0) return g;
  for(int i=1; i < n; ++i)
    g=straight(p,i-1) ? g--P(point(p,i)) :
      g..controls P(postcontrol(p,i-1)) and P(precontrol(p,i))..P(point(p,i));
  
  if(straight(p,n-1))
    return cyclic(p) ? g--cycle : g--P(point(p,n));

  pair post=P(postcontrol(p,n-1));
  pair pre=P(precontrol(p,n));
  return cyclic(p) ? g..controls post and pre..cycle :
    g..controls post and pre..P(point(p,n));
}

void write(file file, string s="", explicit path3 x, suffix suffix=none)
{
  write(file,s);
  int n=length(x);
  if(n < 0) write("<nullpath3>");
  else {
    for(int i=0; i < n; ++i) {
      write(file,point(x,i));
      if(i < length(x)) {
        if(straight(x,i)) write(file,"--");
        else {
          write(file,".. controls ");
          write(file,postcontrol(x,i));
          write(file," and ");
          write(file,precontrol(x,i+1),newl);
          write(file," ..");
        }
      }
    }
    if(cyclic(x))
      write(file,"cycle",suffix);
    else
      write(file,point(x,n),suffix);
  }
}

void write(string s="", explicit path3 x, suffix suffix=endl)
{
  write(stdout,s,x,suffix);
}

void write(file file, string s="", explicit path3[] x, suffix suffix=none)
{
  write(file,s);
  if(x.length > 0) write(file,x[0]);
  for(int i=1; i < x.length; ++i) {
    write(file,endl);
    write(file," ^^");
    write(file,x[i]);
  }
  write(file,suffix);
}

void write(string s="", explicit path3[] x, suffix suffix=endl)
{
  write(stdout,s,x,suffix);
}

path3 solve(flatguide3 g)
{
  int n=g.nodes.length-1;

  // If duplicate points occur consecutively, add dummy controls (if absent).
  for(int i=0; i < n; ++i) {
    if(g.nodes[i] == g.nodes[i+1] && !g.control[i].active)
      g.control[i]=control(g.nodes[i],g.nodes[i],straight=true);
  }  
  
  // Fill in empty direction specifiers inherited from explicit control points.
  for(int i=0; i < n; ++i) {
    if(g.control[i].active) {
      g.out[i].init(g.control[i].post-g.nodes[i]);
      g.in[i].init(g.nodes[i+1]-g.control[i].pre);
    }
  }  
  
  // Propagate directions across nodes.
  for(int i=0; i < n; ++i) {
    int next=g.cyclic[i+1] ? 0 : i+1;
    if(g.out[next].active())
      g.in[i].default(g.out[next]);
    if(g.in[i].active()) {
      g.out[next].default(g.in[i]);
      g.out[i+1].default(g.in[i]);
    }
  }  
    
  // Compute missing 3D directions.
  // First, resolve cycles
  int i=find(g.cyclic);
  if(i > 0) {
    aim(g,i);
    // All other cycles can now be reduced to sequences.
    triple v=g.out[0].dir;
    for(int j=i; j <= n; ++j) {
      if(g.cyclic[j]) {
        g.in[j-1].default(v);
        g.out[j].default(v);
        if(g.nodes[j-1] == g.nodes[j] && !g.control[j-1].active)
          g.control[j-1]=control(g.nodes[j-1],g.nodes[j-1]);
      }
    }
  }
    
  // Next, resolve sequences.
  int i=0;
  int start=0;
  while(i < n) {
    // Look for a missing outgoing direction.
    while(i <= n && g.solved(i)) {start=i; ++i;}
    if(i > n) break;
    // Look for the end of the sequence.
    while(i < n && !g.solved(i)) ++i;
    
    while(start < i && g.control[start].active) ++start;
    
    if(start < i) 
      aim(g,start,i);
  }
  
  // Compute missing 3D control points.
  for(int i=0; i < n; ++i) {
    int next=g.cyclic[i+1] ? 0 : i+1;
    if(!g.control[i].active) {
      control c;
      if((g.out[i].Curl && g.in[i].Curl) ||
         (g.out[i].dir == O && g.in[i].dir == O)) {
        // fill in straight control points for path3 functions
        triple delta=(g.nodes[i+1]-g.nodes[i])/3;
        c=control(g.nodes[i]+delta,g.nodes[i+1]-delta,straight=true);
      } else {
        Controls C=Controls(g.nodes[i],g.nodes[next],g.out[i].dir,g.in[i].dir,
                            g.Tension[i].out,g.Tension[i].in,
                            g.Tension[i].atLeast);
        c=control(C.c0,C.c1);
      }
      g.control[i]=c;
    }
  }

  // Convert to Knuth's format (control points stored with nodes)
  int n=g.nodes.length;
  bool cyclic;
  if(n > 0) {
    cyclic=g.cyclic[n-1];
    if(cyclic) --n;
  }
  triple[] pre=new triple[n];
  triple[] point=new triple[n];
  triple[] post=new triple[n];
  bool[] straight=new bool[n];
  if(n > 0) {
    for(int i=0; i < n-1; ++i) {
      point[i]=g.nodes[i];
      post[i]=g.control[i].post;
      pre[i+1]=g.control[i].pre;
      straight[i]=g.control[i].straight;
    }
    point[n-1]=g.nodes[n-1];
    if(cyclic) {
      pre[0]=g.control[n-1].pre;
      post[n-1]=g.control[n-1].post;
      straight[n-1]=g.control[n-1].straight;
    } else {
      pre[0]=point[0];
      post[n-1]=point[n-1];
      straight[n-1]=false;
    }
  }
  
  return path3(pre,point,post,straight,cyclic);
}

path nurb(path3 p, projection P, int ninterpolate=P.ninterpolate)
{
  triple f=P.camera;
  triple u=unit(P.normal);
  transform3 t=P.t;

  path nurb(triple v0, triple v1, triple v2, triple v3) {
    return nurb(project(v0,t),project(v1,t),project(v2,t),project(v3,t),
                dot(u,f-v0),dot(u,f-v1),dot(u,f-v2),dot(u,f-v3),ninterpolate);
  }

  path g;

  if(straight(p,0))
    g=project(point(p,0),t);

  int last=length(p);
  for(int i=0; i < last; ++i) {
    if(straight(p,i))
      g=g--project(point(p,i+1),t);
    else
      g=g&nurb(point(p,i),postcontrol(p,i),precontrol(p,i+1),point(p,i+1));
  }

  int n=length(g);
  if(cyclic(p)) g=g&cycle;

  return g;
}

path project(path3 p, projection P=currentprojection,
             int ninterpolate=P.ninterpolate)
{
  guide g;

  int last=length(p);
  if(last < 0) return g;
  
  transform3 t=P.t;

  if(ninterpolate == 1 || piecewisestraight(p)) {
    g=project(point(p,0),t);
    // Construct the path.
    int stop=cyclic(p) ? last-1 : last;
    for(int i=0; i < stop; ++i) {
      if(straight(p,i))
        g=g--project(point(p,i+1),t);
      else {
        g=g..controls project(postcontrol(p,i),t) and
          project(precontrol(p,i+1),t)..project(point(p,i+1),t);
      }
    }
  } else return nurb(p,P);
  
  if(cyclic(p))
    g=straight(p,last-1) ? g--cycle :
      g..controls project(postcontrol(p,last-1),t) and
      project(precontrol(p,last),t)..cycle;
  return g;
}

pair[] project(triple[] v, projection P=currentprojection)
{
  return sequence(new pair(int i) {return project(v[i],P.t);},v.length);
}

path[] project(explicit path3[] g, projection P=currentprojection)
{
  return sequence(new path(int i) {return project(g[i],P);},g.length);
}
  
guide3 operator cast(path3 p)
{
  int last=length(p);
  
  bool cyclic=cyclic(p);
  int stop=cyclic ? last-1 : last;
  return new void(flatguide3 f) {
    if(last >= 0) {
      f.node(point(p,0));
      for(int i=0; i < stop; ++i) {
        if(straight(p,i)) {
          f.out(1);
          f.in(1);
        } else
          f.control(postcontrol(p,i),precontrol(p,i+1));
        f.node(point(p,i+1));
      }
      if(cyclic) {
        if(straight(p,stop)) {
          f.out(1);
          f.in(1);
        } else
          f.control(postcontrol(p,stop),precontrol(p,last));
        f.cycleToken();
      }
    }
  };
}

// Return a unit normal vector to a planar path p (or O if the path is
// nonplanar).
triple normal(path3 p)
{
  triple normal;
  real fuzz=sqrtEpsilon*abs(max(p)-min(p));
  real absnormal;
  real theta;
  
  bool Cross(triple a, triple b) {
    if(abs(a) >= fuzz && abs(b) >= fuzz) {
      triple n=cross(unit(a),unit(b));
      real absn=abs(n);
      if(absn < sqrtEpsilon) return false;
      n=unit(n);
      if(absnormal > 0 &&
         abs(normal-n) > sqrtEpsilon && abs(normal+n) > sqrtEpsilon)
        return true;
      else {
        int sign=dot(n,normal) >= 0 ? 1 : -1;
        theta += sign*asin1(absn);
        if(absn > absnormal) {
          absnormal=absn;
          normal=n;
          theta=sign*theta;
        }
      }
    }
    return false;
  }
  
  int L=length(p);
  if(L <= 0) return O;

  triple zi=point(p,0);
  triple v0=zi-precontrol(p,0);
  for(int i=0; i < L; ++i) {
    triple c0=postcontrol(p,i);
    triple c1=precontrol(p,i+1);
    triple zp=point(p,i+1);
    triple v1=c0-zi;
    triple v2=c1-c0;
    triple v3=zp-c1;
    if(Cross(v0,v1) || Cross(v1,v2) || Cross(v2,v3)) return O;
    v0=v3;
    zi=zp;
  }
  return theta >= 0 ? normal : -normal;
}

// Return a unit normal vector to a polygon with vertices in p.
triple normal(triple[] p)
{
  triple normal;
  real fuzz=sqrtEpsilon*abs(maxbound(p)-minbound(p));
  real absnormal;
  real theta;
  
  bool Cross(triple a, triple b) {
    if(abs(a) >= fuzz && abs(b) >= fuzz) {
      triple n=cross(unit(a),unit(b));
      real absn=abs(n);
      n=unit(n);
      if(absnormal > 0 && absn > sqrtEpsilon &&
         abs(normal-n) > sqrtEpsilon && abs(normal+n) > sqrtEpsilon)
        return true;
      else {
        int sign=dot(n,normal) >= 0 ? 1 : -1;
        theta += sign*asin1(absn);
        if(absn > absnormal) {
          absnormal=absn;
          normal=n;
          theta=sign*theta;
        }
      }
    }
    return false;
  }
  
  if(p.length <= 0) return O;

  triple zi=p[0];
  triple v0=zi-p[p.length-1];
  for(int i=0; i < p.length-1; ++i) {
    triple zp=p[i+1];
    triple v1=zp-zi;
    if(Cross(v0,v1)) return O;
    v0=v1;
    zi=zp;
  }
  return theta >= 0 ? normal : -normal;
}

// Transforms that map XY plane to YX, YZ, ZY, ZX, and XZ planes.
restricted transform3 XY=identity4;
restricted transform3 YX=rotate(-90,O,Z);
restricted transform3 YZ=rotate(90,O,Z)*rotate(90,O,X);
restricted transform3 ZY=rotate(-90,O,X)*YZ;
restricted transform3 ZX=rotate(-90,O,Z)*rotate(-90,O,Y);
restricted transform3 XZ=rotate(-90,O,Y)*ZX;

private transform3 flip(transform3 t, triple X, triple Y, triple Z,
                        projection P)
{
  static transform3 flip(triple v) {
    static real s(real x) {return x > 0 ? -1 : 1;}
    return scale(s(v.x),s(v.y),s(v.z));
  }

  triple u=unit(P.normal);
  triple up=unit(perp(P.up,u));
  bool upright=dot(Z,u) >= 0;
  if(dot(Y,up) < 0) {
    t=flip(Y)*t;
    upright=!upright;
  }
  return upright ? t : flip(X)*t;
}

restricted transform3 XY(projection P=currentprojection)
{
  return flip(XY,X,Y,Z,P);
}

restricted transform3 YX(projection P=currentprojection)
{
  return flip(YX,Y,X,Z,P);
}

restricted transform3 YZ(projection P=currentprojection)
{
  return flip(YZ,Y,Z,X,P);
}

restricted transform3 ZY(projection P=currentprojection)
{
  return flip(ZY,Z,Y,X,P);
}

restricted transform3 ZX(projection P=currentprojection)
{
  return flip(ZX,Z,X,Y,P);
}

restricted transform3 XZ(projection P=currentprojection)
{
  return flip(XZ,X,Z,Y,P);
}

// Transform3 that projects in direction dir onto plane with normal n
// through point O.
transform3 planeproject(triple n, triple O=O, triple dir=n)
{
  real a=n.x, b=n.y, c=n.z;
  real u=dir.x, v=dir.y, w=dir.z;
  real delta=1.0/(a*u+b*v+c*w);
  real d=-(a*O.x+b*O.y+c*O.z)*delta;
  return new real[][] {
    {(b*v+c*w)*delta,-b*u*delta,-c*u*delta,-d*u},
      {-a*v*delta,(a*u+c*w)*delta,-c*v*delta,-d*v},
        {-a*w*delta,-b*w*delta,(a*u+b*v)*delta,-d*w},
          {0,0,0,1}
  };
}

// Transform3 that projects in direction dir onto plane defined by p.
transform3 planeproject(path3 p, triple dir=O)
{
  triple n=normal(p);
  return planeproject(n,point(p,0),dir == O ? n : dir);
}

// Transform for projecting onto plane through point O with normal cross(u,v).
transform transform(triple u, triple v, triple O=O,
                    projection P=currentprojection)
{
  transform3 t=P.t;
  static real[] O={0,0,0,1};
  real[] tO=t*O;
  real tO3=tO[3];
  real factor=1/tO3^2;
  real[] x=(tO3*t[0]-tO[0]*t[3])*factor;
  real[] y=(tO3*t[1]-tO[1]*t[3])*factor;
  triple x=(x[0],x[1],x[2]);
  triple y=(y[0],y[1],y[2]);
  u=unit(u);
  v=unit(v);
  return (0,0,dot(u,x),dot(v,x),dot(u,y),dot(v,y));
}

// Project Label onto plane through point O with normal cross(u,v).
Label project(Label L, triple u, triple v, triple O=O,
              projection P=currentprojection) {
  Label L=L.copy();
  L.position=project(O,P.t);
  L.transform(transform(u,v,O,P)); 
  return L;
}

path3 operator cast(guide3 g) {return solve(g);}
path3 operator cast(triple v) {return path3(v);}

guide3[] operator cast(triple[] v)
{
  return sequence(new guide3(int i) {return v[i];},v.length);
}

path3[] operator cast(triple[] v)
{
  return sequence(new path3(int i) {return v[i];},v.length);
}

path3[] operator cast(guide3[] g)
{
  return sequence(new path3(int i) {return solve(g[i]);},g.length);
}

guide3[] operator cast(path3[] g)
{
  return sequence(new guide3(int i) {return g[i];},g.length);
}

void write(file file, string s="", explicit guide3[] x, suffix suffix=none)
{
  write(file,s,(path3[]) x,suffix);
}

void write(string s="", explicit guide3[] x, suffix suffix=endl) 
{
  write(stdout,s,(path3[]) x,suffix);
}

triple point(explicit guide3 g, int t) {
  flatguide3 f;
  g(f);
  int n=f.size();
  return f.nodes[adjustedIndex(t,n,f.cyclic())];
}

triple[] dirSpecifier(guide3 g, int t)
{
  flatguide3 f;
  g(f);
  int n=f.size();
  checkEmpty(n);
  if(f.cyclic()) t=t % n;
  else if(t < 0 || t >= n-1) return new triple[];
  return new triple[] {f.out[t].dir,f.in[t].dir};
}

triple[] controlSpecifier(guide3 g, int t) {
  flatguide3 f;
  g(f);
  int n=f.size();
  checkEmpty(n);
  if(f.cyclic()) t=t % n;
  else if(t < 0 || t >= n-1) return new triple[];
  control c=f.control[t];
  if(c.active) return new triple[] {c.post,c.pre};
  else return new triple[];
}

tensionSpecifier tensionSpecifier(guide3 g, int t)
{
  flatguide3 f;
  g(f);
  int n=f.size();
  checkEmpty(n);
  if(f.cyclic()) t=t % n;
  else if(t < 0 || t >= n-1) return operator tension(1,1,false);
  Tension T=f.Tension[t];
  return operator tension(T.out,T.in,T.atLeast);
}

real[] curlSpecifier(guide3 g, int t)
{
  flatguide3 f;
  g(f);
  int n=f.size();
  checkEmpty(n);
  if(f.cyclic()) t=t % n;
  else if(t < 0 || t >= n-1) return new real[];
  return new real[] {f.out[t].gamma,f.in[t].gamma};
}

guide3 reverse(guide3 g)
{
  flatguide3 f;
  bool cyclic=cyclic(g);
  g(f);

  if(f.precyclic())
    return reverse(solve(g));

  int n=f.size();
  checkEmpty(n);
  guide3 G;
  if(n >= 0) {
    int start=cyclic ? n : n-1;
    for(int i=start; i > 0; --i) {
      G=G..f.nodes[i];
      control c=f.control[i-1];
      if(c.active)
        G=G..operator controls(c.pre,c.post);
      else {
        dir in=f.in[i-1];
        triple d=in.dir;
        if(d != O) G=G..operator spec(-d,JOIN_OUT);
        else if(in.Curl) G=G..operator curl(in.gamma,JOIN_OUT);
        dir out=f.out[i-1];
        triple d=out.dir;
        if(d != O) G=G..operator spec(-d,JOIN_IN);
        else if(out.Curl) G=G..operator curl(out.gamma,JOIN_IN);
      }
    }
    if(cyclic) G=G..cycle;
    else G=G..f.nodes[0];
  }
  return G;
}

triple intersectionpoint(path3 p, path3 q, real fuzz=-1)
{
  real[] t=intersect(p,q,fuzz);
  if(t.length == 0) abort("paths do not intersect");
  return point(p,t[0]);
}

// return an array containing all intersection points of p and q
triple[] intersectionpoints(path3 p, path3 q, real fuzz=-1)
{
  real[][] t=intersections(p,q,fuzz);
  return sequence(new triple(int i) {return point(p,t[i][0]);},t.length);
}

triple[] intersectionpoints(explicit path3[] p, explicit path3[] q,
                            real fuzz=-1)
{
  triple[] v;
  for(int i=0; i < p.length; ++i)
    for(int j=0; j < q.length; ++j)
      v.append(intersectionpoints(p[i],q[j],fuzz));
  return v;
}

path3 operator &(path3 p, cycleToken tok)
{
  int n=length(p);
  if(n < 0) return nullpath3;
  triple a=point(p,0);
  triple b=point(p,n);
  return subpath(p,0,n-1)..controls postcontrol(p,n-1) and precontrol(p,n)..
    cycle;
}

// return the point on path3 p at arclength L
triple arcpoint(path3 p, real L)
{
  return point(p,arctime(p,L));
}

// return the point on path3 p at arclength L
triple arcpoint(path3 p, real L)
{
  return point(p,arctime(p,L));
}

// return the direction on path3 p at arclength L
triple arcdir(path3 p, real L)
{
  return dir(p,arctime(p,L));
}

// return the time on path3 p at the relative fraction l of its arclength
real reltime(path3 p, real l)
{
  return arctime(p,l*arclength(p));
}

// return the point on path3 p at the relative fraction l of its arclength
triple relpoint(path3 p, real l)
{
  return point(p,reltime(p,l));
}

// return the direction of path3 p at the relative fraction l of its arclength
triple reldir(path3 p, real l)
{
  return dir(p,reltime(p,l));
}

// return the point on path3 p at half of its arclength
triple midpoint(path3 p)
{
  return relpoint(p,0.5);
}

real relative(Label L, path3 g)
{
  return L.position.relative ? reltime(g,L.relative()) : L.relative();
}

// return the linear transformation that maps X,Y,Z to u,v,w.
transform3 transform3(triple u, triple v, triple w=cross(u,v)) 
{
  return new real[][] {
    {u.x,v.x,w.x,0},
      {u.y,v.y,w.y,0},
        {u.z,v.z,w.z,0},
          {0,0,0,1}
  };
}

// return the rotation that maps Z to a unit vector u about cross(u,Z),
transform3 align(triple u)
{
  real a=u.x;
  real b=u.y;
  real c=u.z;
  real d=a^2+b^2;

  if(d != 0) {
    d=sqrt(d);
    real e=1/d;
    return new real[][] {
      {-b*e,-a*c*e,a,0},
        {a*e,-b*c*e,b,0},
          {0,d,c,0},
            {0,0,0,1}};
  }
  return c >= 0 ? identity(4) : diagonal(1,-1,-1,1);
}

// return a rotation that maps X,Y to the projection plane.
transform3 transform3(projection P=currentprojection)
{
  triple w=unit(P.normal);
  triple v=unit(perp(P.up,w));
  if(v == O) v=cross(perp(w),w);
  triple u=cross(v,w);
  return u != O ? transform3(u,v,w) : identity(4);
}

triple[] triples(real[] x, real[] y, real[] z)
{
  if(x.length != y.length || x.length != z.length)
    abort("arrays have different lengths");
  return sequence(new triple(int i) {return (x[i],y[i],z[i]);},x.length);
}

path3[] operator cast(path3 p)
{
  return new path3[] {p};
}

path3[] operator cast(guide3 g)
{
  return new path3[] {(path3) g};
}

path3[] operator ^^ (path3 p, path3  q) 
{
  return new path3[] {p,q};
}

path3[] operator ^^ (path3 p, explicit path3[] q) 
{
  return concat(new path3[] {p},q);
}

path3[] operator ^^ (explicit path3[] p, path3 q) 
{
  return concat(p,new path3[] {q});
}

path3[] operator ^^ (explicit path3[] p, explicit path3[] q) 
{
  return concat(p,q);
}

path3[] operator * (transform3 t, explicit path3[] p) 
{
  return sequence(new path3(int i) {return t*p[i];},p.length);
}

triple[] operator * (transform3 t, triple[] v) 
{
  return sequence(new triple(int i) {return t*v[i];},v.length);
}

triple[][] operator * (transform3 t, triple[][] v) 
{
  triple[][] V=new triple[v.length][];
  for(int i=0; i < v.length; ++i) {
    triple[] vi=v[i];
    V[i]=sequence(new triple(int j) {return t*vi[j];},vi.length);
  }
  return V;
}

triple min(explicit path3[] p)
{
  checkEmpty(p.length);
  triple minp=min(p[0]);
  for(int i=1; i < p.length; ++i)
    minp=minbound(minp,min(p[i]));
  return minp;
}

triple max(explicit path3[] p)
{
  checkEmpty(p.length);
  triple maxp=max(p[0]);
  for(int i=1; i < p.length; ++i)
    maxp=maxbound(maxp,max(p[i]));
  return maxp;
}

path3 randompath3(int n, bool cumulate=true, interpolate3 join=operator ..)
{
  guide3 g;
  triple w;
  for(int i=0; i <= n; ++i) {
    triple z=(unitrand()-0.5,unitrand()-0.5,unitrand()-0.5);
    if(cumulate) w += z; 
    else w=z;
    g=join(g,w);
  }
  return g;
}

path3[] box(triple v1, triple v2)
{
  return
    (v1.x,v1.y,v1.z)--
    (v1.x,v1.y,v2.z)--
    (v1.x,v2.y,v2.z)--
    (v1.x,v2.y,v1.z)--
    (v1.x,v1.y,v1.z)--
    (v2.x,v1.y,v1.z)--
    (v2.x,v1.y,v2.z)--
    (v2.x,v2.y,v2.z)--
    (v2.x,v2.y,v1.z)--
    (v2.x,v1.y,v1.z)^^
    (v2.x,v2.y,v1.z)--
    (v1.x,v2.y,v1.z)^^
    (v1.x,v2.y,v2.z)--
    (v2.x,v2.y,v2.z)^^
    (v2.x,v1.y,v2.z)--
    (v1.x,v1.y,v2.z);
}

restricted path3[] unitbox=box(O,(1,1,1));
restricted path3 unitcircle3=X..Y..-X..-Y..cycle;
restricted path3 unitsquare3=O--X--X+Y--Y--cycle;

path3 circle(triple c, real r, triple normal=Z)
{
  path3 p=normal == Z ? unitcircle3 : align(unit(normal))*unitcircle3;
  return shift(c)*scale3(r)*p;
}

// return an arc centered at c from triple v1 to v2 (assuming |v2-c|=|v1-c|),
// drawing in the given direction.
// The normal must be explicitly specified if c and the endpoints are colinear.
path3 arc(triple c, triple v1, triple v2, triple normal=O, bool direction=CCW)
{
  v1 -= c;
  real r=abs(v1);
  v1=unit(v1);
  v2=unit(v2-c);

  if(normal == O) {
    normal=cross(v1,v2);
    if(normal == O) abort("explicit normal required for these endpoints");
  }

  transform3 T;
  bool align=normal != Z;
  if(align) {
    T=align(unit(normal));
    transform3 Tinv=transpose(T);
    v1=Tinv*v1;
    v2=Tinv*v2;
  }
  
  string invalidnormal="invalid normal vector";
  real fuzz=sqrtEpsilon*max(abs(v1),abs(v2));
  if(abs(v1.z) > fuzz || abs(v2.z) > fuzz)
    abort(invalidnormal);
  
  real[] t1=intersect(unitcircle3,O--2*(v1.x,v1.y,0));
  real[] t2=intersect(unitcircle3,O--2*(v2.x,v2.y,0));
  
  if(t1.length == 0 || t2.length == 0)
    abort(invalidnormal);

  real t1=t1[0];
  real t2=t2[0];
  int n=length(unitcircle3);
  if(direction) {
    if(t1 >= t2) t1 -= n;
  } else if(t2 >= t1) t2 -= n;

  path3 p=subpath(unitcircle3,t1,t2);
  if(align) p=T*p;
  return shift(c)*scale3(r)*p;
}

// return an arc centered at c with radius r from c+r*dir(theta1,phi1) to
// c+r*dir(theta2,phi2) in degrees, drawing in the given direction
// relative to the normal vector cross(dir(theta1,phi1),dir(theta2,phi2)).
// The normal must be explicitly specified if c and the endpoints are colinear.
path3 arc(triple c, real r, real theta1, real phi1, real theta2, real phi2,
          triple normal=O, bool direction)
{
  return arc(c,c+r*dir(theta1,phi1),c+r*dir(theta2,phi2),normal,direction);
}

// return an arc centered at c with radius r from c+r*dir(theta1,phi1) to
// c+r*dir(theta2,phi2) in degrees, drawing drawing counterclockwise
// relative to the normal vector cross(dir(theta1,phi1),dir(theta2,phi2))
// iff theta2 > theta1 or (theta2 == theta1 and phi2 >= phi1).
// The normal must be explicitly specified if c and the endpoints are colinear.
path3 arc(triple c, real r, real theta1, real phi1, real theta2, real phi2,
          triple normal=O)
{
  return arc(c,r,theta1,phi1,theta2,phi2,normal,
             theta2 > theta1 || (theta2 == theta1 && phi2 >= phi1) ? CCW : CW);
}

private real epsilon=1000*realEpsilon;

// Return a representation of the plane through point O with normal cross(u,v).
path3 plane(triple u, triple v, triple O=O)
{
  return O--O+u--O+u+v--O+v--cycle;
}

// PRC/OpenGL support

include three_light;

void draw(frame f, path3 g, material p=currentpen, light light=nolight,
          string name="", render render=defaultrender,
          projection P=currentprojection);

void begingroup3(frame f, string name="", render render=defaultrender,
                 triple center=O, int interaction=0)
{
  _begingroup3(f,name,render.compression,render.granularity,render.closed,
               render.tessellate,render.merge == false,
               render.merge == true,center,interaction);
}

void begingroup3(picture pic=currentpicture, string name="",
                 render render=defaultrender,
                 triple center=O, int interaction=0)
{
  pic.add(new void(frame f, transform3, picture pic, projection) {
      if(is3D())
        begingroup3(f,name,render,center,interaction);
      if(pic != null)
        begingroup(pic);
    },true);
}

void endgroup3(picture pic=currentpicture)
{
  pic.add(new void(frame f, transform3, picture pic, projection) {
      if(is3D())
        endgroup3(f);
      if(pic != null)
        endgroup(pic);
    },true);
}

void addPath(picture pic, path3 g, pen p)
{
  if(size(g) > 0)
    pic.addBox(min(g),max(g),min3(p),max3(p));
}

include three_surface;
include three_margins;

pair min(path3 p, projection P) 
{
  path3 q=P.T.modelview*p;
  if(P.infinity)
    return xypart(min(q));
  return maxratio(q)/P.T.projection[3][2];
}

pair max(path3 p, projection P) 
{
  path3 q=P.T.modelview*p;
  if(P.infinity)
    return xypart(max(q));
  return minratio(q)/P.T.projection[3][2];
}

pair min(frame f, projection P) 
{
  frame g=P.T.modelview*f;
  if(P.infinity)
    return xypart(min3(g));
  return maxratio(g)/P.T.projection[3][2];
}

pair max(frame f, projection P) 
{
  frame g=P.T.modelview*f;
  if(P.infinity)
    return xypart(max3(g));
  return minratio(g)/P.T.projection[3][2];
}

void draw(picture pic=currentpicture, Label L="", path3 g,
          align align=NoAlign, material p=currentpen, margin3 margin=NoMargin3,
          light light=nolight, string name="", render render=defaultrender)
{
  pen q=(pen) p;
  pic.add(new void(frame f, transform3 t, picture pic, projection P) {
      path3 G=margin(t*g,q).g;
      if(is3D()) {
        draw(f,G,p,light,name,render,null);
        if(pic != null && size(G) > 0)
          pic.addBox(min(G,P),max(G,P),min(q),max(q));
      }
      if(pic != null)
        draw(pic,project(G,P),q);
    },true);
  Label L=L.copy();
  L.align(align);
  if(L.s != "") {
    L.p(q);
    label(pic,L,g);
  }
  addPath(pic,g,q);
}

include three_tube;

draw=new void(frame f, path3 g, material p=currentpen,
              light light=nolight, string name="",
              render render=defaultrender,
              projection P=currentprojection) {
  pen q=(pen) p;
  if(is3D()) {
    p=material(p);
    real width=linewidth(q);
    void drawthick(path3 g) {
      if(settings.thick) {
        if(width > 0) {
          bool prc=prc();
          void cylinder(transform3) {};
          void sphere(transform3, bool half) {};
          void disk(transform3) {};
          void pipe(path3, path3);
          if(prc) {
            cylinder=new void(transform3 t) {drawPRCcylinder(f,t,p,light);};
            sphere=new void(transform3 t, bool half)
              {drawPRCsphere(f,t,half,p,light,render);};
            disk=new void(transform3 t) {draw(f,t*unitdisk,p,light,render);};
            pipe=new void(path3 center, path3 g)
              {drawPRCtube(f,center,g,p,light);};
          }
          real linecap=linecap(q);
          real r=0.5*width;
          bool open=!cyclic(g);
          int L=length(g);
          triple g0=point(g,0);
          triple gL=point(g,L);
          if(open && L > 0) {
            if(linecap == 2) {
              g0 -= r*dir(g,0);
              gL += r*dir(g,L);
              g=g0..g..gL;
              L += 2;
            }
          }
          tube T=tube(g,width,render,cylinder,sphere,pipe);
          path3 c=T.center;
          if(L >= 0) {
            if(open) {
              int Lc=length(c);
              triple c0=point(c,0);
              triple cL=point(c,Lc);
              triple dir0=dir(g,0);
              triple dirL=dir(g,L);
              triple dirc0=dir(c,0);
              triple dircL=dir(c,Lc);
              transform3 t0=shift(g0)*align(-dir0);
              transform3 tL=shift(gL)*align(dirL);
              transform3 tc0=shift(c0)*align(-dirc0);
              transform3 tcL=shift(cL)*align(dircL);
              if(linecap == 0 || linecap == 2) {
                transform3 scale2r=scale(r,r,1);
                T.s.append(t0*scale2r*unitdisk);
                disk(tc0*scale2r);
                if(L > 0) {
                  T.s.append(tL*scale2r*unitdisk);
                  disk(tcL*scale2r);
                }
              } else if(linecap == 1) {
                transform3 scale3r=scale3(r);
                T.s.append(t0*scale3r*
                           (dir0 != O ? unithemisphere : unitsphere));
                sphere(tc0*scale3r,half=straight(c,0));
                if(L > 0) {
                  T.s.append(tL*scale3r*
                             (dirL != O ? unithemisphere : unitsphere));
                  sphere(tcL*scale3r,half=straight(c,Lc-1));
                }
              }
            }
            if(opacity(q) == 1)
              _draw(f,c,q);
          }
          for(patch s : T.s.s)
            draw3D(f,s,p,light,prc=false);
        } else _draw(f,g,q);
      } else _draw(f,g,q);
    }
    bool group=q != nullpen && (name != "" || render.defaultnames);
    if(group)
      begingroup3(f,name == "" ? "curve" : name,render);
    if(linetype(q).length == 0) drawthick(g);
    else {
      real[] dash=linetype(adjust(q,arclength(g),cyclic(g)));
      if(sum(dash) > 0) {
        dash.cyclic=true;
        real offset=offset(q);
        real L=arclength(g);
        int i=0;
        real l=offset;
        while(l <= L) {
          real t1=arctime(g,l);
          l += dash[i];
          real t2=arctime(g,min(l,L));
          drawthick(subpath(g,t1,t2));
          ++i;
          l += dash[i];
          ++i;
        }
      }
    }
    if(group)
      endgroup3(f);
  } else draw(f,project(g,P),q);
};

void draw(frame f, explicit path3[] g, material p=currentpen,
          light light=nolight, string name="",
          render render=defaultrender, projection P=currentprojection)
{
  bool group=g.length > 1 && (name != "" || render.defaultnames);
  if(group)
    begingroup3(f,name == "" ? "curves" : name,render);
  for(int i=0; i < g.length; ++i)
    draw(f,g[i],p,light,partname(i,render),render,P);
  if(group)
    endgroup3(f);
}

void draw(picture pic=currentpicture, explicit path3[] g,
          material p=currentpen, margin3 margin=NoMargin3, light light=nolight,
          string name="", render render=defaultrender)
{
  bool group=g.length > 1 && (name != "" || render.defaultnames);
  if(group)
    begingroup3(pic,name == "" ? "curves" : name,render);
  for(int i=0; i < g.length; ++i)
    draw(pic,g[i],p,margin,light,partname(i,render),render);
  if(group)
      endgroup3(pic);
}

include three_arrows;

void draw(picture pic=currentpicture, Label L="", path3 g, 
          align align=NoAlign, material p=currentpen, arrowbar3 arrow,
          arrowbar3 bar=None, margin3 margin=NoMargin3, light light=nolight,
          light arrowheadlight=currentlight, string name="",
          render render=defaultrender)
{
  bool group=arrow != None || bar != None;
  if(group)
    begingroup3(pic,name,render);
  bool drawpath=arrow(pic,g,p,margin,light,arrowheadlight);
  if(bar(pic,g,p,margin,light,arrowheadlight) && drawpath)
    draw(pic,L,g,align,p,margin,light,render);
  if(group)
    endgroup3(pic);
  if(L.s != "")
    label(pic,L,g,align,(pen) p);
}

void draw(frame f, path3 g, material p=currentpen, arrowbar3 arrow,
          light light=nolight, light arrowheadlight=currentlight,
          string name="", render render=defaultrender,
          projection P=currentprojection)
{
  picture pic;
  bool group=arrow != None;
  if(group)
    begingroup3(f,name,render);
  if(arrow(pic,g,p,NoMargin3,light,arrowheadlight))
    draw(f,g,p,light,render,P);
  add(f,pic.fit());
  if(group)
    endgroup3(f);
}

void add(picture pic=currentpicture, void d(picture,transform3),
         bool exact=false)
{
  pic.add(d,exact);
}

// Fit the picture src using the identity transformation (so user
// coordinates and truesize coordinates agree) and add it about the point
// position to picture dest.
void add(picture dest, picture src, triple position, bool group=true,
         bool above=true)
{
  dest.add(new void(picture f, transform3 t) {
      f.add(shift(t*position)*src,group,above);
    });
}

void add(picture src, triple position, bool group=true, bool above=true)
{
  add(currentpicture,src,position,group,above);
}

// Align an arrow pointing to b from the direction dir. The arrow is
// 'length' PostScript units long.
void arrow(picture pic=currentpicture, Label L="", triple b, triple dir,
           real length=arrowlength, align align=NoAlign,
           pen p=currentpen, arrowbar3 arrow=Arrow3, margin3 margin=EndMargin3,
           light light=nolight, light arrowheadlight=currentlight,
           string name="", render render=defaultrender)
{
  Label L=L.copy();
  if(L.defaultposition) L.position(0);
  L.align(L.align,dir);
  L.p(p);
  picture opic;
  marginT3 margin=margin(b--b,p); // Extract margin.begin and margin.end
  triple a=(margin.begin+length+margin.end)*unit(dir);
  draw(opic,L,a--O,align,p,arrow,margin,light,arrowheadlight,name,render);
  add(pic,opic,b);
}

void arrow(picture pic=currentpicture, Label L="", triple b, pair dir,
           real length=arrowlength, align align=NoAlign,
           pen p=currentpen, arrowbar3 arrow=Arrow3, margin3 margin=EndMargin3,
           light light=nolight, light arrowheadlight=currentlight,
           string name="", render render=defaultrender,
           projection P=currentprojection)
{
  arrow(pic,L,b,invert(dir,b,P),length,align,p,arrow,margin,light,
        arrowheadlight,name,render);
}

triple min3(picture pic, projection P=currentprojection)
{
  return pic.min3(P);
}
  
triple max3(picture pic, projection P=currentprojection)
{
  return pic.max3(P);
}
  
triple size3(picture pic, bool user=false, projection P=currentprojection)
{
  transform3 t=pic.calculateTransform3(P);
  triple M=pic.max(t);
  triple m=pic.min(t);
  if(!user) return M-m;
  t=inverse(t);
  return t*M-t*m;
}

triple point(frame f, triple dir)
{
  triple m=min3(f);
  triple M=max3(f);
  return m+realmult(rectify(dir),M-m);
}

triple point(picture pic=currentpicture, triple dir, bool user=true,
             projection P=currentprojection)
{
  triple min = pic.userMin(), max = pic.userMax();
  triple v=min+realmult(rectify(dir),max-min);
  return user ? v : pic.calculateTransform3(P)*v;
}

triple truepoint(picture pic=currentpicture, triple dir, bool user=true,
                 projection P=currentprojection)
{
  transform3 t=pic.calculateTransform3(P);
  triple m=pic.min(t);
  triple M=pic.max(t);
  triple v=m+realmult(rectify(dir),M-m);
  return user ? inverse(t)*v : v;
}

void add(picture dest=currentpicture, object src, pair position=0, pair align=0,
         bool group=true, filltype filltype=NoFill, bool above=true)
{
  if(prc())
    label(dest,src,position,align);
  else if(settings.render == 0)
    plain.add(dest,src,position,align,group,filltype,above);
}

private struct viewpoint {
  triple target,camera,up;
  real angle;
  void operator init(string s) {
    s=replace(s,'\n'," ");
    string[] S=split(s);
    int pos(string s, string key) {
      int pos=find(s,key);
      if(pos < 0) return -1;
      pos += length(key);
      while(substr(s,pos,1) == " ") ++pos;
      if(substr(s,pos,1) == "=")
        return pos+1;
      return -1;
    }
    triple C2C=X;
    real ROO=1;
    real ROLL=0;
    angle=30;
    int pos;
    for(int k=0; k < S.length; ++k) {
      if((pos=pos(S[k],"COO")) >= 0)
        target=((real) substr(S[k],pos),(real) S[++k],(real) S[++k]);
      else if((pos=pos(S[k],"C2C")) >= 0)
        C2C=((real) substr(S[k],pos),(real) S[++k],(real) S[++k]);
      else if((pos=pos(S[k],"ROO")) >= 0)
        ROO=(real) substr(S[k],pos);
      else if((pos=pos(S[k],"ROLL")) >= 0)
        ROLL=(real) substr(S[k],pos);
      else if((pos=pos(S[k],"AAC")) >= 0)
        angle=(real) substr(S[k],pos);
    }
    camera=target+ROO*C2C;
    triple u=unit(target-camera);
    triple w=unit(Z-u.z*u);
    up=rotate(ROLL,O,u)*w;
  }
}

projection perspective(string s)
{
  viewpoint v=viewpoint(s);
  projection P=perspective(v.camera,v.up,v.target);
  P.angle=v.angle;
  P.absolute=true;
  return P;
}

projection absorthographic(triple camera=Z, triple target=O, real roll=0)
{
  triple u=unit(target-camera);
  triple w=unit(Z-u.z*u);
  triple up=rotate(roll,O,u)*w;
  projection P=
    projection(camera,up,target,1,0,false,false,
               new transformation(triple camera, triple up, triple target)
               {return transformation(look(camera,up,target));});
  P.absolute=true;
  return P;
}

projection absperspective(triple camera=Z, triple target=O, real roll=0,
                          real angle=30)
{
  triple u=unit(target-camera);
  triple w=unit(Z-u.z*u);
  triple up=rotate(roll,O,u)*w;
  projection P=perspective(camera,up,target);
  P.angle=angle;
  P.absolute=true;
  return P;
}

private string Format(real x)
{
  assert(abs(x) < 1e17,"Number too large: "+string(x));
  return format("%.9f",x,"C");
}

private string Format(triple v, string sep=" ")
{
  return Format(v.x)+sep+Format(v.y)+sep+Format(v.z);
}

private string Format(real[] c)
{
  return Format((c[0],c[1],c[2]));
}

private string format(triple v, string sep=" ")
{
  return string(v.x)+sep+string(v.y)+sep+string(v.z);
}

private string Format(transform3 t, string sep=" ")
{
  return
    Format(t[0][0])+sep+Format(t[1][0])+sep+Format(t[2][0])+sep+
    Format(t[0][1])+sep+Format(t[1][1])+sep+Format(t[2][1])+sep+
    Format(t[0][2])+sep+Format(t[1][2])+sep+Format(t[2][2])+sep+
    Format(t[0][3])+sep+Format(t[1][3])+sep+Format(t[2][3]);
}

string lightscript(light light) {
  string script="for(var i=scene.lights.count-1; i >= 0; i--)
  scene.lights.removeByIndex(i);"+'\n\n';
  for(int i=0; i < light.position.length; ++i) {
    string Li="L"+string(i);
    real[] diffuse=light.diffuse[i];
    script += Li+"=scene.createLight();"+'\n'+
      Li+".direction.set("+format(-light.position[i],",")+");"+'\n'+
      Li+".color.set("+format((diffuse[0],diffuse[1],diffuse[2]),",")+");"+'\n';
  }
  // Work around initialization bug in Adobe Reader 8.0:
  return script +"
scene.lightScheme=scene.LIGHT_MODE_HEADLAMP;
scene.lightScheme=scene.LIGHT_MODE_FILE;
";
}

void writeJavaScript(string name, string preamble, string script) 
{
  file out=output(name);
  write(out,preamble);
  if(script != "") {
    write(out,endl);
    file in=input(script);
    while(true) {
      string line=in;
      if(eof(in)) break;
      write(out,line,endl);
    }
  }
  close(out);
  if(settings.verbose > 1) write("Wrote "+name);
  if(!settings.inlinetex)
    file3.push(name);
}

pair viewportmargin(pair lambda)
{
  return maxbound(0.5*(viewportsize-lambda),viewportmargin);
}

string embed3D(string prefix, string label=prefix, string text=label,
               frame f, string format="",
               real width=0, real height=0,
               string options="", string script="",
               light light=currentlight, projection P=currentprojection,
               real viewplanesize=0)
{
  if(!prc(format) || Embed == null) return "";

  if(width == 0) width=settings.paperwidth;
  if(height == 0) height=settings.paperheight;

  if(script == "") script=defaultembed3Dscript;

  if(P.infinity) {
    if(viewplanesize==0) {
      triple lambda=max3(f)-min3(f);
      pair margin=viewportmargin((lambda.x,lambda.y));
      viewplanesize=(max(lambda.x+2*margin.x,lambda.y+2*margin.y))/P.zoom;
    }
  } else
    if(!P.absolute) P.angle=2*aTan(Tan(0.5*P.angle));

  shipout3(prefix,f);

  string name=prefix+".js";
  // Adobe Reader doesn't appear to support user-specified viewport lights.
  bool lightscript=light.on() && !light.viewport;
  if(lightscript)
    writeJavaScript(name,lightscript(light),script);

  if(!settings.inlinetex && !prconly())
    file3.push(prefix+".prc");

  static transform3 flipxz=xscale3(-1)*zscale3(-1);
  transform3 inv=inverse(flipxz*P.T.modelview);

  string options3="3Dlights="+
    (light.on() ? (light.viewport ? "Headlamp" : "File") : "None");
  if(defaultembed3Doptions != "") options3 += ","+defaultembed3Doptions;

  if((settings.render < 0 || !settings.embed) && settings.auto3D)
    options3 += ",activate=pagevisible";
  options3 += ",3Dtoolbar="+(settings.toolbar ? "true" : "false")+
    ",label="+label+
    (P.infinity ? ",3Dortho="+Format(1/viewplanesize) :
     ",3Daac="+Format(P.angle))+
    ",3Dc2w="+Format(inv)+
    ",3Droo="+Format(abs(P.vector()))+
    ",3Dpsob="+(P.infinity ? "Max" : "H")+
    ",3Dbg="+Format(light.background());
  if(options != "") options3 += ","+options;
  if(settings.inlinetex)
    prefix=jobname(prefix);
  if(lightscript)
    options3 += ",add3Djscript="+prefix+".js";
  options3 += ",add3Djscript=asylabels.js";

  return text == "" ? Embed(prefix+".prc","",options3,width,height) :
    "\hbox to 0pt{"+text+"\hss}"+Embed(prefix+".prc","\phantom{"+text+"}",
                                       options3);
}

struct scene
{
  frame f;
  transform3 t;
  projection P;
  bool adjusted;
  real width,height;
  pair viewportmargin;
  transform3 T=identity4;
  picture pic2;
  
  void operator init(frame f, real width, real height,
                     projection P=currentprojection) {
    this.f=f;
    this.t=identity4;
    this.P=P;
    this.width=width;
    this.height=height;
  }
  
  void operator init(picture pic, real xsize=pic.xsize, real ysize=pic.ysize,
                     bool keepAspect=pic.keepAspect, bool is3D=true,
                     projection P=currentprojection) {
    real xsize3=pic.xsize3, ysize3=pic.ysize3, zsize3=pic.zsize3;
    bool warn=true;
        
    if(xsize3 == 0 && ysize3 == 0 && zsize3 == 0) {
      xsize3=ysize3=zsize3=max(xsize,ysize);
      warn=false;
    }

    if(P.absolute)
      this.P=P.copy();
    else if(P.showtarget)
      draw(pic,P.target,nullpen);

    t=pic.scaling(xsize3,ysize3,zsize3,keepAspect,warn);
    adjusted=false;
    triple m=pic.min(t);
    triple M=pic.max(t);

    if(!P.absolute) {
      this.P=t*P;
      if(this.P.center && settings.render != 0) {
        triple target=0.5*(m+M);
        this.P.target=target;
        this.P.calculate();
      }
      if(this.P.autoadjust || this.P.infinity) 
        adjusted=adjusted | this.P.adjust(m,M);
    }

    bool scale=xsize != 0 || ysize != 0;
    bool scaleAdjust=scale && this.P.autoadjust;
    bool noAdjust=(this.P.absolute || !scaleAdjust);

    if(pic.bounds3.exact && noAdjust)
      this.P.bboxonly=false;
    
    f=pic.fit3(t,pic.bounds3.exact ? pic2 : null,this.P);

    if(!pic.bounds3.exact) {
      if(noAdjust)
        this.P.bboxonly=false;

      transform3 s=pic.scale3(f,xsize3,ysize3,zsize3,keepAspect);
      t=s*t;
      this.P=s*this.P;
      f=pic.fit3(t,pic2,this.P);
    }

    if(is3D || scale) {
      pic2.bounds.exact=true;
      transform s=pic2.scaling(xsize,ysize,keepAspect);

      pair m2=pic2.min(s);
      pair M2=pic2.max(s);
      pair lambda=M2-m2;
      viewportmargin=viewportmargin(lambda);
      width=ceil(lambda.x+2*viewportmargin.x);
      height=ceil(lambda.y+2*viewportmargin.y);

      if(!this.P.absolute) {
        if(scaleAdjust) {
          pair v=(s.xx,s.yy);
          transform3 T=this.P.t;
          pair x=project(X,T);
          pair y=project(Y,T);
          pair z=project(Z,T);
          real f(pair a, pair b) {
            return b == 0 ? (0.5*(a.x+a.y)) :
              (b.x^2*a.x+b.y^2*a.y)/(b.x^2+b.y^2);
          }
          transform3 s=keepAspect ? scale3(min(f(v,x),f(v,y),f(v,z))) :
            xscale3(f(v,x))*yscale3(f(v,y))*zscale3(f(v,z));
          s=shift(this.P.target)*s*shift(-this.P.target);
          t=s*t;
          this.P=s*this.P;
          this.P.bboxonly=false;
          if(!is3D) pic2.erase();
          f=pic.fit3(t,is3D ? null : pic2,this.P);
        }

        if(this.P.autoadjust || this.P.infinity)
          adjusted=adjusted | this.P.adjust(min3(f),max3(f));
      }
    }
  }

  // Choose the angle to be just large enough to view the entire image.
  real angle(projection P) {
    T=identity4;
    real h=-0.5*P.target.z;
    pair r,R;
    real diff=realMax;
    pair s;
    int i;
    do {
      r=minratio(f);
      R=maxratio(f);
      pair lasts=s;
      if(P.autoadjust) {
        s=r+R;
        if(s != 0) {
          transform3 t=shift(h*s.x,h*s.y,0);
          f=t*f;
          T=t*T;
          adjusted=true;
        }
      }
      diff=abs(s-lasts);
      ++i;
    } while (diff > angleprecision && i < maxangleiterations);
    real aspect=width > 0 ? height/width : 1;
    real rx=-r.x*aspect;
    real Rx=R.x*aspect;
    real ry=-r.y;
    real Ry=R.y;
    if(!P.autoadjust) {
      if(rx > Rx) Rx=rx;
      else rx=Rx;
      if(ry > Ry) Ry=ry;
      else ry=Ry;
    }
    return (1+angleprecision)*max(aTan(rx)+aTan(Rx),aTan(ry)+aTan(Ry));
  }
}

object embed(string prefix=outprefix(), string label=prefix,
             string text=label, scene S, string format="", bool view=true,
             string options="", string script="", light light=currentlight)
{
  object F;
  transform3 modelview;
  projection P=S.P;
  transform3 tinv=inverse(S.t);

  projection Q;
  triple orthoshift;
  modelview=P.T.modelview;
  transform3 inv;
  if(P.absolute) {
    Q=modelview*P;
    inv=inverse(modelview);
  } else {
    triple target=P.target;
    S.f=modelview*S.f;
    P=modelview*P;
    Q=P.copy();

    if(P.infinity) {
      triple m=min3(S.f);
      triple M=max3(S.f);
      triple lambda=M-m;
      S.viewportmargin=viewportmargin((lambda.x,lambda.y));
      S.width=ceil(lambda.x+2*S.viewportmargin.x);
      S.height=ceil(lambda.y+2*S.viewportmargin.y);
      orthoshift=(-0.5(m.x+M.x),-0.5*(m.y+M.y),0);
      S.f=shift(orthoshift)*S.f; // Eye will be at (0,0,0)
      inv=inverse(modelview);
    } else {
      if(P.angle == 0) {
        P.angle=S.angle(P);
        modelview=S.T*modelview;
        if(S.viewportmargin.y != 0)
          P.angle=2*aTan(Tan(0.5*P.angle)-S.viewportmargin.y/P.target.z);
      }
      inv=inverse(modelview);
      Q.angle=P.angle;
      if(settings.verbose > 0) {
        if(S.adjusted) 
          write("adjusting camera to ",tinv*inv*P.camera);
        target=inv*P.target;
      }
      P=S.T*P;
    }
    if(settings.verbose > 0) {
      if((P.center && settings.render != 0) || (!P.infinity && P.autoadjust))
        write("adjusting target to ",tinv*target);
    }
  }
  light Light=modelview*light;

  if(prefix == "") prefix=outprefix();
  bool prc=prc(format);
  bool preview=settings.render > 0 && !prconly();
  if(prc) {
    // The media9.sty package cannot handle spaces or dots in filenames.
    string dir=stripfile(prefix);
    prefix=dir+replace(stripdirectory(prefix),
                       new string[][]{{" ","_"},{".","_"}});
    if((settings.embed || nativeformat() == "pdf") && !prconly())
      prefix += "+"+(string) file3.length;
  } else
    preview=false;
  if(preview || (!prc && settings.render != 0)) {
    frame f=S.f;
    triple m,M;
    real zcenter;
    real r;
    if(P.absolute) {
      f=modelview*f;
      m=min3(f);
      M=max3(f);
      r=0.5*abs(M-m);
      zcenter=0.5*(M.z+m.z);
    } else {
      m=min3(f);
      M=max3(f);
      zcenter=P.target.z;
      r=P.distance(m,M);
    }
    M=(M.x,M.y,zcenter+r);
    m=(m.x,m.y,zcenter-r);

    if(P.infinity) {
      triple margin=(S.viewportmargin.x,S.viewportmargin.y,0);
      M += margin; 
      m -= margin;
    } else if(M.z >= 0) abort("camera too close");

    shipout3(prefix,f,preview ? nativeformat() : format,
             S.width-defaultrender.margin,S.height-defaultrender.margin,
             P.infinity ? 0 : 2aTan(Tan(0.5*P.angle)*P.zoom),
             P.zoom,m,M,P.viewportshift,
             tinv*inv*shift(0,0,zcenter),Light.background(),Light.position,
             Light.diffuse,Light.ambient,Light.specular,
             Light.viewport,view && !preview);
    if(!preview) return F;
  }

  string image;
  if((preview || (prc && settings.render == 0)) && settings.embed) {
    image=prefix;
    if(settings.inlinetex) image += "_0";
    if(!preview && !shipped && !S.pic2.empty2()) {
      transform T=S.pic2.scaling(S.width,S.height);
      shipout(image,S.pic2.fit(T),newframe,nativeformat(),false,false,null);
    }
    
    image += "."+nativeformat();
    if(!settings.inlinetex) file3.push(image);
    image=graphic(image,"hiresbb");
  }
  if(prc) {
    if(P.viewportshift != 0) {
    if(!P.infinity)
      warning("offaxis",
              "PRC does not support off-axis projections; use pan instead of
shift");

    triple lambda=max3(S.f)-min3(S.f);
    Q.target -= (P.viewportshift.x*lambda.x/P.zoom,
                   P.viewportshift.y*lambda.y/P.zoom,0);
    }

    real viewplanesize=0;
    if(P.absolute) {
      if(P.infinity) {
        S.f=modelview*S.f;
        triple lambda=max3(S.f)-min3(S.f);
        pair margin=viewportmargin((lambda.x,lambda.y));
        viewplanesize=(max(lambda.x+2*margin.x,lambda.y+2*margin.y))/Q.zoom;
        S.f=inv*S.f;
      }
      Q=inv*Q;
    } else {
      if(P.infinity) {
        triple lambda=max3(S.f)-min3(S.f);
        pair margin=viewportmargin((lambda.x,lambda.y));
        viewplanesize=(max(lambda.x+2*margin.x,lambda.y+2*margin.y))/(Q.zoom);
        transform3 t=inv*shift(-orthoshift);
        Q=t*Q;
        S.f=t*S.f;
      } else {
        Q=inv*Q;
        S.f=inv*S.f;
      }
    }
    F.L=embed3D(prefix,label,text=image,S.f,format,
                S.width-2,S.height-2,options,script,light,Q,viewplanesize);
  }
  return F;
}

object embed(string prefix=outprefix(), string label=prefix,
             string text=label, picture pic, string format="",
             real xsize=pic.xsize, real ysize=pic.ysize,
             bool keepAspect=pic.keepAspect, bool view=true, string options="",
             string script="", light light=currentlight,
             projection P=currentprojection)
{
  bool is3D=is3D(format);
  scene S=scene(pic,xsize,ysize,keepAspect,is3D,P);
  if(is3D)
    return embed(prefix,label,text,S,format,view,options,script,light);
  else {
    object F;
    transform T=S.pic2.scaling(xsize,ysize,keepAspect);
    F.f=pic.fit(scale(S.t[0][0])*T);
    add(F.f,S.pic2.fit());
    return F;
  }
}

object embed(string prefix=outprefix(), string label=prefix,
             string text=label,
             frame f, string format="", real width=0, real height=0,
             bool view=true, string options="", string script="",
             light light=currentlight, projection P=currentprojection)
{
  if(is3D(format))
    return embed(label,text,prefix,scene(f,width,height,P),format,view,options,
                 script,light);
  else {
    object F;
    F.f=f;
    return F;
  }
}

embed3=new object(string prefix, frame f, string format, string options,
                  string script, light light, projection P) {
  return embed(prefix=prefix,f,format,options,script,light,P);
};

frame embedder(object embedder(string prefix, string format),
               string prefix, string format, bool view, light light)
{
  frame f;
  bool prc=prc(format);
  if(!prc && settings.render != 0 && !view) {
    static int previewcount=0;
    bool keep=prefix != "";
    prefix=outprefix(prefix)+"+"+(string) previewcount;
    ++previewcount;
    format=nativeformat();
    if(!keep) file3.push(prefix+"."+format);
  }
  object F=embedder(prefix,format);
  if(prc)
    label(f,F.L);
  else {
    if(settings.render == 0) {
      add(f,F.f);
      if(light.background != nullpen)
        box(f,light.background,Fill,above=false);
    } else if(!view)
      label(f,graphic(prefix,"hiresbb"));
  }
  return f;
}

currentpicture.fitter=new frame(string prefix, picture pic, string format,
                                real xsize, real ysize, bool keepAspect,
                                bool view, string options, string script,
                                light light, projection P) {
  frame f;
  bool empty3=pic.empty3();
  if(!empty3) f=embedder(new object(string prefix, string format) {
      return embed(prefix=prefix,pic,format,xsize,ysize,keepAspect,view,
                   options,script,light,P);
    },prefix,format,view,light);
  if(is3D(format) || empty3) add(f,pic.fit2(xsize,ysize,keepAspect));
  return f;
};

frame embedder(string prefix, frame f, string format, real width, real height,
               bool view, string options, string script, light light,
               projection P)
{
  return embedder(new object(string prefix, string format) {
      return embed(prefix=prefix,f,format,width,height,view,options,script,
                   light,P);
    },prefix,format,view,light);
}

projection[][] ThreeViewsUS={{TopView},
                             {FrontView,RightView}};

projection[][] SixViewsUS={{null,TopView},
                           {LeftView,FrontView,RightView,BackView},
                           {null,BottomView}};

projection[][] ThreeViewsFR={{RightView,FrontView},
                             {null,TopView}};

projection[][] SixViewsFR={{null,BottomView},
                           {RightView,FrontView,LeftView,BackView},
                           {null,TopView}};

projection[][] ThreeViews={{FrontView,TopView,RightView}};

projection[][] SixViews={{FrontView,TopView,RightView},
                         {BackView,BottomView,LeftView}};

void addViews(picture dest, picture src, projection[][] views=SixViewsUS,
              bool group=true, filltype filltype=NoFill)
{
  frame[][] F=array(views.length,new frame[]);
  pair[][] M=array(views.length,new pair[]);
  pair[][] m=array(views.length,new pair[]);
  
  for(int i=0; i < views.length; ++i) {
    projection[] viewsi=views[i];
    frame[] Fi=F[i];
    pair[] Mi=M[i];
    pair[] mi=m[i];
    for(projection P : viewsi) {
      if(P != null) {
        frame f=src.fit(P);
        mi.push(min(f));
        Mi.push(max(f));
        Fi.push(f);
      } else {
        pair Infinity=(infinity,infinity);
        mi.push(Infinity);
        Mi.push(-Infinity);
        Fi.push(newframe);
      }
    }
  }
  
  real[] my=new real[views.length];
  real[] My=new real[views.length];
  
  int Nj=0;
  for(int i=0; i < views.length; ++i) {
    my[i]=minbound(m[i]).y;
    My[i]=maxbound(M[i]).y;
    Nj=max(Nj,views[i].length);
  }
  
  real[] mx=array(Nj,infinity);
  real[] Mx=array(Nj,-infinity);
  for(int i=0; i < views.length; ++i) {
    pair[] mi=m[i];
    pair[] Mi=M[i];
    for(int j=0; j < views[i].length; ++j) {
      mx[j]=min(mx[j],mi[j].x);
      Mx[j]=max(Mx[j],Mi[j].x);
    }
  }

  if(group) begingroup(dest);

  real y;
  for(int i=0; i < views.length; ++i) {
    real x;
    pair[] mi=m[i];
    for(int j=0; j < views[i].length; ++j) {
      if(size(F[i][j]) != 0)
        add(dest,shift(x-mx[j],y+my[i])*F[i][j],filltype);
      x += (Mx[j]-mx[j]);
    }
    y -= (My[i]-my[i]);
  }

  if(group) endgroup(dest);
}

void addViews(picture src, projection[][] views=SixViewsUS, bool group=true,
              filltype filltype=NoFill)
{
  addViews(currentpicture,src,views,group,filltype);
}

void addStereoViews(picture dest, picture src, bool group=true,
                    filltype filltype=NoFill, real eyetoview=defaulteyetoview,
                    bool leftright=true, projection P=currentprojection)
{
  triple v=P.vector();
  triple h=0.5*abs(v)*eyetoview*unit(cross(P.up,v));
  projection leftEye=P.copy();
  leftEye.camera -= h;
  leftEye.calculate();
  projection rightEye=P.copy();
  rightEye.camera += h;
  rightEye.calculate();
  addViews(dest,src,leftright ?
           new projection[][] {{leftEye,rightEye}} :
           new projection[][] {{rightEye,leftEye}},group,filltype);
}

void addStereoViews(picture src, bool group=true,
                    filltype filltype=NoFill,
                    real eyetoview=defaulteyetoview, bool leftright=true,
                    projection P=currentprojection)
{
  addStereoViews(currentpicture,src,group,filltype,eyetoview,leftright,P);
}

// Fit an array of 3D pictures simultaneously using the sizing of picture all.
frame[] fit3(string prefix="", picture[] pictures, picture all,
             string format="", bool view=true, string options="",
             string script="", light light=currentlight,
             projection P=currentprojection)
{
  frame[] out;
  scene S=scene(all,P);
  triple m=all.min(S.t);
  triple M=all.max(S.t);
  out=new frame[pictures.length];
  int i=0;
  bool reverse=settings.reverse;
  settings.animating=true;

  for(picture pic : pictures) {
    picture pic2;
    frame f=pic.fit3(S.t,pic2,S.P);
    if(settings.interrupt) break;
    add(f,pic2.fit2());
    draw(f,m,nullpen);
    draw(f,M,nullpen);
    out[i]=f;
    ++i;
  }

  while(!settings.interrupt) {
    for(int i=settings.reverse ? pictures.length-1 : 0;
        i >= 0 && i < pictures.length && !settings.interrupt;
        settings.reverse ? --i : ++i) {
      frame f=embedder(prefix,out[i],format,S.width,S.height,view,options,
                       script,light,S.P);
      if(!settings.loop) out[i]=f;
    }   
    if(!settings.loop) break;
  }
  
  settings.animating=false;
  settings.interrupt=false;
  settings.reverse=reverse;

  return out;
}

// Fit an array of pictures simultaneously using the size of the first picture.
fit=new frame[](string prefix="", picture[] pictures, string format="",
                bool view=true, string options="", string script="",
                projection P=currentprojection) {
  if(pictures.length == 0)
    return new frame[];

  picture all;
  size(all,pictures[0]);
  for(picture pic : pictures)
    add(all,pic);

  return all.empty3() ? fit2(pictures,all) :
  fit3(prefix,pictures,all,format,view,options,script,P);
};

// Add frame src to picture dest about position.
void add(picture dest=currentpicture, frame src, triple position)
{
  if(is3D(src)) {
    dest.add(new void(frame f, transform3 t, picture, projection) {
        add(f,shift(t*position)*src);
      },true);
  } else {
    dest.add(new void(frame, transform3 t, picture pic, projection P) {
        if(pic != null) {
          pic.add(new void(frame f, transform T) {
              add(f,T*shift(project(t*position,P))*src);
            },true);
        }
      },true);
  }
  dest.addBox(position,position,min3(src),max3(src));
}

exitfcn currentexitfunction=atexit();

void exitfunction()
{
  if(currentexitfunction != null) currentexitfunction();
  if(!settings.keep)
    for(int i=0; i < file3.length; ++i)
      delete(file3[i]);
  file3=new string[];
}

atexit(exitfunction);