real camerafactor=2;       // Factor used for camera adjustment.
pair viewportsize=0;       // Horizontal and vertical viewport limits.

restricted bool Aspect=true;
restricted bool IgnoreAspect=false;

typedef real[][] transform3;
restricted transform3 identity4=identity(4);

// A uniform 3D scaling.
transform3 scale3(real s)
{
  transform3 t=identity(4);
  t[0][0]=t[1][1]=t[2][2]=s;
  return t;
}

// Simultaneous 3D scalings in the x, y, and z directions.
transform3 scale(real x, real y, real z)
{
  transform3 t=identity(4);
  t[0][0]=x;
  t[1][1]=y;
  t[2][2]=z;
  return t;
}

transform3 shiftless(transform3 t)
{
  transform3 T=copy(t);
  T[0][3]=T[1][3]=T[2][3]=0;
  return T;
}

// A coordinate in "flex space." A linear combination of user and true-size
// coordinates.
struct coord {
  real user,truesize;
  bool finite=true;

  // Build a coord.
  static coord build(real user, real truesize) {
    coord c=new coord;
    c.user=user;
    c.truesize=truesize;
    return c;
  }

  // Deep copy of coordinate.  Users may add coords to the picture, but then
  // modify the struct. To prevent this from yielding unexpected results, deep
  // copying is used.
  coord copy() {
    return build(user, truesize);
  }
  
  void clip(real min, real max) {
    user=min(max(user,min),max);
    truesize=0;
  }
}

struct coords2 {
  coord[] x,y;
  void erase() {
    x.delete();
    y.delete();
  }
  // Only a shallow copy of the individual elements of x and y
  // is needed since, once entered, they are never modified.
  coords2 copy() {
    coords2 c=new coords2;
    c.x=copy(x);
    c.y=copy(y);
    return c;
  }
  void append(coords2 c) {
    x.append(c.x);
    y.append(c.y);
  }
  void push(pair user, pair truesize) {
    x.push(coord.build(user.x,truesize.x));
    y.push(coord.build(user.y,truesize.y));
  }
  void push(coord cx, coord cy) {
    x.push(cx);
    y.push(cy);
  }
  void push(transform t, coords2 c1, coords2 c2) {
    for(int i=0; i < c1.x.length; ++i) {
      coord cx=c1.x[i], cy=c2.y[i];
      pair tinf=shiftless(t)*((finite(cx.user) ? 0 : 1),
                              (finite(cy.user) ? 0 : 1));
      pair z=t*(cx.user,cy.user);
      pair w=(cx.truesize,cy.truesize);
      w=length(w)*unit(shiftless(t)*w);
      coord Cx,Cy;
      Cx.user=(tinf.x == 0 ? z.x : infinity);
      Cy.user=(tinf.y == 0 ? z.y : infinity);
      Cx.truesize=w.x;
      Cy.truesize=w.y;
      push(Cx,Cy);
    }
  }
  void xclip(real min, real max) {
    for(int i=0; i < x.length; ++i) 
      x[i].clip(min,max);
  }
  void yclip(real min, real max) {
    for(int i=0; i < y.length; ++i) 
      y[i].clip(min,max);
  }
}
  
struct coords3 {
  coord[] x,y,z;
  void erase() {
    x.delete();
    y.delete();
    z.delete();
  }
  // Only a shallow copy of the individual elements of x and y
  // is needed since, once entered, they are never modified.
  coords3 copy() {
    coords3 c=new coords3;
    c.x=copy(x);
    c.y=copy(y);
    c.z=copy(z);
    return c;
  }
  void append(coords3 c) {
    x.append(c.x);
    y.append(c.y);
    z.append(c.z);
  }
  void push(triple user, triple truesize) {
    x.push(coord.build(user.x,truesize.x));
    y.push(coord.build(user.y,truesize.y));
    z.push(coord.build(user.z,truesize.z));
  }
  void push(coord cx, coord cy, coord cz) {
    x.push(cx);
    y.push(cy);
    z.push(cz);
  }
  void push(transform3 t, coords3 c1, coords3 c2, coords3 c3) {
    for(int i=0; i < c1.x.length; ++i) {
      coord cx=c1.x[i], cy=c2.y[i], cz=c3.z[i];
      triple tinf=shiftless(t)*((finite(cx.user) ? 0 : 1),
                                (finite(cy.user) ? 0 : 1),
                                (finite(cz.user) ? 0 : 1));
      triple z=t*(cx.user,cy.user,cz.user);
      triple w=(cx.truesize,cy.truesize,cz.truesize);
      w=length(w)*unit(shiftless(t)*w);
      coord Cx,Cy,Cz;
      Cx.user=(tinf.x == 0 ? z.x : infinity);
      Cy.user=(tinf.y == 0 ? z.y : infinity);
      Cz.user=(tinf.z == 0 ? z.z : infinity);
      Cx.truesize=w.x;
      Cy.truesize=w.y;
      Cz.truesize=w.z;
      push(Cx,Cy,Cz);
    }
  }
}
  
bool operator <= (coord a, coord b)
{
  return a.user <= b.user && a.truesize <= b.truesize;
}

bool operator >= (coord a, coord b)
{
  return a.user >= b.user && a.truesize >= b.truesize;
}

// Find the maximal elements of the input array, using the partial ordering
// given.
coord[] maxcoords(coord[] in, bool operator <= (coord,coord))
{
  // As operator <= is defined in the parameter list, it has a special
  // meaning in the body of the function.

  coord best;
  coord[] c;

  int n=in.length;
  
  // Find the first finite restriction.
  int first=0;
  for(first=0; first < n; ++first)
    if(finite(in[first].user)) break;
        
  if(first == n)
    return c;
  else {
    // Add the first coord without checking restrictions (as there are none).
    best=in[first];
    c.push(best);
  }

  static int NONE=-1;

  int dominator(coord x)
  {
    // This assumes it has already been checked against the best.
    for(int i=1; i < c.length; ++i)
      if(x <= c[i])
        return i;
    return NONE;
  }

  void promote(int i)
  {
    // Swap with the top
    coord x=c[i];
    c[i]=best;
    best=c[0]=x;
  }

  void addmaximal(coord x)
  {
    coord[] newc;

    // Check if it beats any others.
    for(int i=0; i < c.length; ++i) {
      coord y=c[i];
      if(!(y <= x))
        newc.push(y);
    }
    newc.push(x);
    c=newc;
    best=c[0];
  }

  void add(coord x)
  {
    if(x <= best || !finite(x.user))
      return;
    else {
      int i=dominator(x);
      if(i == NONE)
        addmaximal(x);
      else
        promote(i);
    }
  }

  for(int i=1; i < n; ++i)
    add(in[i]);

  return c;
}

typedef real scalefcn(real x);
                                              
struct scaleT {
  scalefcn T,Tinv;
  bool logarithmic;
  bool automin,automax;
  void operator init(scalefcn T, scalefcn Tinv, bool logarithmic=false,
                     bool automin=false, bool automax=false) {
    this.T=T;
    this.Tinv=Tinv;
    this.logarithmic=logarithmic;
    this.automin=automin;
    this.automax=automax;
  }
  scaleT copy() {
    scaleT dest=scaleT(T,Tinv,logarithmic,automin,automax);
    return dest;
  }
};

scaleT operator init()
{
  scaleT S=scaleT(identity,identity);
  return S;
}
                                  
typedef void boundRoutine();

struct autoscaleT {
  scaleT scale;
  scaleT postscale;
  real tickMin=-infinity, tickMax=infinity;
  boundRoutine[] bound; // Optional routines to recompute the bounding box.
  bool automin=false, automax=false;
  bool automin() {return automin && scale.automin;}
  bool automax() {return automax && scale.automax;}
  
  real T(real x) {return postscale.T(scale.T(x));}
  scalefcn T() {return scale.logarithmic ? postscale.T : T;}
  real Tinv(real x) {return scale.Tinv(postscale.Tinv(x));}
  
  autoscaleT copy() {
    autoscaleT dest=new autoscaleT;
    dest.scale=scale.copy();
    dest.postscale=postscale.copy();
    dest.tickMin=tickMin;
    dest.tickMax=tickMax;
    dest.bound=copy(bound);
    dest.automin=(bool) automin;
    dest.automax=(bool) automax;
    return dest;
  }
}

struct ScaleT {
  bool set;
  autoscaleT x;
  autoscaleT y;
  autoscaleT z;
  
  ScaleT copy() {
    ScaleT dest=new ScaleT;
    dest.set=set;
    dest.x=x.copy();
    dest.y=y.copy();
    dest.z=z.copy();
    return dest;
  }
};

struct Legend {
  string label;
  pen plabel;
  pen p;
  frame mark;
  bool above;
  void operator init(string label, pen plabel=currentpen, pen p=nullpen,
                     frame mark=newframe, bool above=true) {
    this.label=label;
    this.plabel=plabel;
    this.p=(p == nullpen) ? plabel : p;
    this.mark=mark;
    this.above=above;
  }
}

pair rectify(pair dir) 
{
  real scale=max(abs(dir.x),abs(dir.y));
  if(scale != 0) dir *= 0.5/scale;
  dir += (0.5,0.5);
  return dir;
}

pair point(frame f, pair dir)
{
  pair m=min(f);
  pair M=max(f);
  return m+realmult(rectify(dir),M-m);
}

path[] align(path[] g, transform t=identity(), pair position,
             pair align, pen p=currentpen)
{
  pair m=min(g);
  pair M=max(g);
  pair dir=rectify(inverse(t)*-align);
  if(basealign(p) == 1)
    dir -= (0,m.y/(M.y-m.y));
  pair a=m+realmult(dir,M-m);
  return shift(position+align*labelmargin(p))*t*shift(-a)*g;
}

// Returns a transform for aligning frame f in the direction align
transform shift(frame f, pair align) 
{
  return shift(align-point(f,-align));
}

// Returns a copy of frame f aligned in the direction align
frame align(frame f, pair align) 
{
  return shift(f,align)*f;
}

struct transformation {
  transform3 modelview;  // For orientation and positioning
  transform3 projection; // For 3D to 2D projection
  bool infinity;
  bool oblique;
  void operator init(transform3 modelview, bool oblique=false) {
    this.modelview=modelview;
    this.projection=identity4;
    infinity=true;
    this.oblique=oblique;
  }
  void operator init(transform3 modelview, transform3 projection) {
    this.modelview=modelview;
    this.projection=projection;
    infinity=false;
    oblique=false;
  }
  transform3 compute() {
    return infinity ? modelview : projection*modelview;
  }
  transformation copy() {
    transformation T=new transformation;
    T.modelview=copy(modelview);
    T.projection=copy(projection);
    T.infinity=infinity;
    T.oblique=oblique;
    return T;
  }
}

struct projection {
  transform3 t;         // projection*modelview (cached)
  bool infinity;
  bool oblique;
  bool absolute=false;
  triple camera;        // Position of camera.
  triple up;            // A vector that should be projected to direction (0,1).
  triple target;        // Point where camera is looking at.
  pair viewportshift;   // Fractional viewport shift.
  real zoom=1;          // Zoom factor.
  real angle;           // Lens angle (for perspective projection).
  bool showtarget=true; // Expand bounding volume to include target?
  typedef transformation projector(triple camera, triple up, triple target);
  projector projector;
  bool autoadjust=true; // Adjust camera to lie outside bounding volume?
  bool center=false;    // Center target within bounding volume?
  int ninterpolate;     // Used for projecting nurbs to 2D Bezier curves.
  transformation T;

  void calculate() {
    T=projector(camera,up,target);
    t=T.compute();
    infinity=T.infinity;
    oblique=T.oblique;
    ninterpolate=infinity ? 1 : 16;
  }

  triple vector() {
    return camera-target;
  }

  void operator init(triple camera, triple up=(0,0,1), triple target=(0,0,0),
                     real zoom=1, real angle=0, pair viewportshift=0,
                     bool showtarget=true, bool autoadjust=true,
                     bool center=false, projector projector) {
    this.camera=camera;
    this.up=up;
    this.target=target;
    this.zoom=zoom;
    this.angle=angle;
    this.viewportshift=viewportshift;
    this.showtarget=showtarget;
    this.autoadjust=autoadjust;
    this.center=center;
    this.projector=projector;
    calculate();
  }

  projection copy() {
    projection P=new projection;
    P.t=t;
    P.infinity=infinity;
    P.absolute=absolute;
    P.oblique=oblique;
    P.camera=camera;
    P.up=up;
    P.target=target;
    P.zoom=zoom;
    P.angle=angle;
    P.viewportshift=viewportshift;
    P.showtarget=showtarget;
    P.autoadjust=autoadjust;
    P.center=center;
    P.projector=projector;
    P.ninterpolate=ninterpolate;
    P.T=T.copy();
    return P;
  }

  // Return the maximum distance of box(m,M) from target.
  real distance(triple m, triple M) {
    triple[] c={m,(m.x,m.y,M.z),(m.x,M.y,m.z),(m.x,M.y,M.z),
                (M.x,m.y,m.z),(M.x,m.y,M.z),(M.x,M.y,m.z),M};
    return max(abs(c-target));
  }
   
  // Move the camera so that the box(m,M) rotated about target will always
  // lie in front of the clipping plane.
  bool adjust(triple m, triple M) {
    triple v=camera-target;
    real d=distance(m,M);
    static real lambda=camerafactor*(1-sqrtEpsilon);
    if(lambda*d >= abs(v)) {
      camera=target+camerafactor*d*unit(v);
      calculate();
      return true;
    }
    return false;
  }
}

projection currentprojection;

struct light {
  real[][] diffuse;
  real[][] ambient;
  real[][] specular;
  pen background=nullpen; // Background color of the 3D canvas.
  real specularfactor;
  bool viewport; // Are the lights specified (and fixed) in the viewport frame?
  triple[] position; // Only directional lights are currently implemented.

  transform3 T=identity(4); // Transform to apply to normal vectors.

  bool on() {return position.length > 0;}
  
  void operator init(pen[] diffuse,
                     pen[] ambient=array(diffuse.length,black),
                     pen[] specular=diffuse, pen background=nullpen,
                     real specularfactor=1,
                     bool viewport=false, triple[] position) {
    int n=diffuse.length;
    assert(ambient.length == n && specular.length == n && position.length == n);
    
    this.diffuse=new real[n][];
    this.ambient=new real[n][];
    this.specular=new real[n][];
    this.background=background;
    this.position=new triple[n];
    for(int i=0; i < position.length; ++i) {
      this.diffuse[i]=rgba(diffuse[i]);
      this.ambient[i]=rgba(ambient[i]);
      this.specular[i]=rgba(specular[i]);
      this.position[i]=unit(position[i]);
    }
    this.specularfactor=specularfactor;
    this.viewport=viewport;
  }

  void operator init(pen diffuse=white, pen ambient=black, pen specular=diffuse,
                     pen background=nullpen, real specularfactor=1,
                     bool viewport=false...triple[] position) {
    int n=position.length;
    operator init(array(n,diffuse),array(n,ambient),array(n,specular),
                  background,specularfactor,viewport,position);
  }

  void operator init(pen diffuse=white, pen ambient=black, pen specular=diffuse,
                     pen background=nullpen, bool viewport=false,
                     real x, real y, real z) {
    operator init(diffuse,ambient,specular,background,viewport,(x,y,z));
  }

  void operator init(explicit light light) {
    diffuse=copy(light.diffuse);
    ambient=copy(light.ambient);
    specular=copy(light.specular);
    background=light.background;
    specularfactor=light.specularfactor;
    viewport=light.viewport;
    position=copy(light.position);
  }

  real[] background() {return rgba(background == nullpen ? white : background);}
}

light currentlight;

triple min3(pen p)
{
  return linewidth(p)*(-0.5,-0.5,-0.5);
}

triple max3(pen p)
{
  return linewidth(p)*(0.5,0.5,0.5);
}

// A function that draws an object to frame pic, given that the transform
// from user coordinates to true-size coordinates is t.
typedef void drawer(frame f, transform t);

// A generalization of drawer that includes the final frame's bounds.
typedef void drawerBound(frame f, transform t, transform T, pair lb, pair rt);

struct picture {
  // Three-dimensional version of drawer and drawerBound:
  typedef void drawer3(frame f, transform3 t, picture pic, projection P);
  typedef void drawerBound3(frame f, transform3 t, transform3 T,
                            picture pic, projection P, triple lb, triple rt);

  // The functions to do the deferred drawing.
  drawerBound[] nodes;
  drawerBound3[] nodes3;
  
  bool uptodate=true;

  // The coordinates in flex space to be used in sizing the picture.
  struct bounds {
    coords2 point,min,max;
    bool exact=true; // An accurate picture bounds is provided by the user.
    void erase() {
      point.erase();
      min.erase();
      max.erase();
    }
    bounds copy() {
      bounds b=new bounds;
      b.point=point.copy();
      b.min=min.copy();
      b.max=max.copy();
      b.exact=exact;
      return b;
    }
    void xclip(real Min, real Max) {
      point.xclip(Min,Max);
      min.xclip(Min,Max);
      max.xclip(Min,Max);
    }
    void yclip(real Min, real Max) {
      point.yclip(Min,Max);
      min.yclip(Min,Max);
      max.yclip(Min,Max);
    }
    void clip(triple Min, triple Max) {
      xclip(Min.x,Max.x);
      yclip(Min.y,Max.y);
    }
  }
  
  struct bounds3 {
    coords3 point,min,max;
    bool exact=true; // An accurate picture bounds is provided by the user.
    void erase() {
      point.erase();
      min.erase();
      max.erase();
    }
    bounds3 copy() {
      bounds3 b=new bounds3;
      b.point=point.copy();
      b.min=min.copy();
      b.max=max.copy();
      b.exact=exact;
      return b;
    }
  }
  
  bounds bounds;
  bounds3 bounds3;
    
  // Transform to be applied to this picture.
  transform T;
  transform3 T3;
  
  // Cached user-space bounding box
  triple userMin,userMax;
  bool userSetx,userSety,userSetz;
  
  ScaleT scale; // Needed by graph
  Legend[] legend;

  pair[] clipmax; // Used by beginclip/endclip
  pair[] clipmin;

  // The maximum sizes in the x, y, and z directions; zero means no restriction.
  real xsize=0, ysize=0;

  real xsize3=0, ysize3=0, zsize3=0;

  // Fixed unitsizes in the x y, and z directions; zero means use
  // xsize, ysize, and zsize.
  real xunitsize=0, yunitsize=0, zunitsize=0;
  
  // If true, the x and y directions must be scaled by the same amount.
  bool keepAspect=true;

  // A fixed scaling transform.
  bool fixed;
  transform fixedscaling;
  
  void init() {
    userMin=userMax=(0,0,0);
    userSetx=userSety=userSetz=false;
    T3=identity(4);
  }
  init();
  
  // Erase the current picture, retaining any size specification.
  void erase() {
    nodes.delete();
    nodes3.delete();
    bounds.erase();
    bounds3.erase();
    T=identity();
    scale=new ScaleT;
    legend.delete();
    init();
  }
  
  bool empty2() {
    return nodes.length == 0;
  }

  bool empty3() {
    return nodes3.length == 0;
  }

  bool empty() {
    return empty2() && empty3();
  }
  
  pair userMin() {return (userMin.x,userMin.y);}
  pair userMax() {return (userMax.x,userMax.y);}

  void userMinx(real x) {
    userMin=(x,userMin.y,userMin.z);
    userSetx=true;
  }
  
  void userMiny(real y) {
    userMin=(userMin.x,y,userMin.z);
    userSety=true;
  }
  
  void userMinz(real z) {
    userMin=(userMin.x,userMin.y,z);
    userSetz=true;
  }
  
  void userMaxx(real x) {
    userMax=(x,userMax.y,userMax.z);
    userSetx=true;
  }
  
  void userMaxy(real y) {
    userMax=(userMax.x,y,userMax.z);
    userSety=true;
  }
  
  void userMaxz(real z) {
    userMax=(userMax.x,userMax.y,z);
    userSetz=true;
  }
  
  void userCorners(pair c00, pair c01, pair c10, pair c11) {
    userMin=(min(c00.x,c01.x,c10.x,c11.x),min(c00.y,c01.y,c10.y,c11.y),
             userMin.z);
    userMax=(max(c00.x,c01.x,c10.x,c11.x),max(c00.y,c01.y,c10.y,c11.y),
             userMax.z);
  }
  
  void userCorners(triple c000, triple c001, triple c010, triple c011,
                   triple c100, triple c101, triple c110, triple c111) {
    userMin=(min(c000.x,c001.x,c010.x,c011.x,c100.x,c101.x,c110.x,c111.x),
             min(c000.y,c001.y,c010.y,c011.y,c100.y,c101.y,c110.y,c111.y),
             min(c000.z,c001.z,c010.z,c011.z,c100.z,c101.z,c110.z,c111.z));
    userMax=(max(c000.x,c001.x,c010.x,c011.x,c100.x,c101.x,c110.x,c111.x),
             max(c000.y,c001.y,c010.y,c011.y,c100.y,c101.y,c110.y,c111.y),
             max(c000.z,c001.z,c010.z,c011.z,c100.z,c101.z,c110.z,c111.z));
  }
  
  void userCopy(picture pic) {
    userMin=(triple) pic.userMin;
    userMax=(triple) pic.userMax;
    userSetx=pic.userSetx;
    userSety=pic.userSety;
    userSetz=pic.userSetz;
  }
  
  typedef real binop(real, real);

  // Cache the current user-space bounding box x coodinates
  void userBoxX(real min, real max, binop m=min, binop M=max) {
    if(userSetx) {
      userMin=(m(userMin.x,min),userMin.y,userMin.z);
      userMax=(M(userMax.x,max),userMax.y,userMax.z);
    } else {
      userMin=(min,userMin.y,userMin.z);
      userMax=(max,userMax.y,userMax.z);
      userSetx=true;
    }
  }
  
  // Cache the current user-space bounding box y coodinates
  void userBoxY(real min, real max, binop m=min, binop M=max) {
    if(userSety) {
      userMin=(userMin.x,m(userMin.y,min),userMin.z);
      userMax=(userMax.x,M(userMax.y,max),userMax.z);
    } else {
      userMin=(userMin.x,min,userMin.z);
      userMax=(userMax.x,max,userMax.z);
      userSety=true;
    }
  }
  
  // Cache the current user-space bounding box z coodinates
  void userBoxZ(real min, real max, binop m=min, binop M=max) {
    if(userSetz) {
      userMin=(userMin.x,userMin.y,m(userMin.z,min));
      userMax=(userMax.x,userMax.y,M(userMax.z,max));
    } else {
      userMin=(userMin.x,userMin.y,min);
      userMax=(userMax.x,userMax.y,max);
      userSetz=true;
    }
  }
  
  // Cache the current user-space bounding box
  void userBox(pair min, pair max) {
    userBoxX(min.x,max.x);
    userBoxY(min.y,max.y);
  }
  
  // Cache the current user-space bounding box
  void userBox(triple min, triple max) {
    userBoxX(min.x,max.x);
    userBoxY(min.y,max.y);
    userBoxZ(min.z,max.z);
  }
  
  // Clip the current user-space bounding box
  void userClip(pair min, pair max) {
    userBoxX(min.x,max.x,max,min);
    userBoxY(min.y,max.y,max,min);
  }
  
  void add(drawerBound d, bool exact=false) {
    uptodate=false;
    if(!exact) bounds.exact=false;
    nodes.push(d);
  }

  void add(drawer d, bool exact=false, bool above=true) {
    uptodate=false;
    if(!exact) bounds.exact=false;
    nodes.push(new void(frame f, transform t, transform T, pair, pair) {
        d(f,t*T);
      });
  }

  void add(drawerBound3 d, bool exact=false, bool above=true) {
    uptodate=false;
    if(!exact) bounds.exact=false;
    if(above)
      nodes3.push(d);
    else
      nodes3.insert(0,d);
  }

  void add(drawer3 d, bool exact=false, bool above=true) {
    add(new void(frame f, transform3 t, transform3 T, picture pic,
                 projection P, triple, triple) {
          d(f,t*T,pic,P);
        },exact,above);
  }

  void clip(drawer d, bool exact=false) {
    bounds.clip(userMin,userMax);
    this.add(d,exact);
  }

  void clip(drawerBound d, bool exact=false) {
    bounds.clip(userMin,userMax);
    this.add(d,exact);
  }

  // Add a point to the sizing.
  void addPoint(pair user, pair truesize=0) {
    bounds.point.push(user,truesize);
    userBox(user,user);
  }
  
  // Add a point to the sizing, accounting also for the size of the pen.
  void addPoint(pair user, pair truesize=0, pen p) {
    addPoint(user,truesize+min(p));
    addPoint(user,truesize+max(p));
  }
  
  void addPoint(triple user, triple truesize=(0,0,0)) {
    bounds3.point.push(user,truesize);
    userBox(user,user);
  }

  void addPoint(triple user, triple truesize=(0,0,0), pen p) {
    addPoint(user,truesize+min3(p));
    addPoint(user,truesize+max3(p));
  }

  // Add a box to the sizing.
  void addBox(pair userMin, pair userMax, pair trueMin=0, pair trueMax=0) {
    bounds.min.push(userMin,trueMin);
    bounds.max.push(userMax,trueMax);
    userBox(userMin,userMax);
  }

  void addBox(triple userMin, triple userMax, triple trueMin=(0,0,0),
              triple trueMax=(0,0,0)) {
    bounds3.min.push(userMin,trueMin);
    bounds3.max.push(userMax,trueMax);
    userBox(userMin,userMax);
  }

  // Add a (user space) path to the sizing.
  void addPath(path g) {
    if(size(g) > 0)
      addBox(min(g),max(g));
  }
  void addPath(path[] g) {
    for(int i=0; i < g.length; ++i) 
      addPath(g[i]);
  }

  // Add a path to the sizing with the additional padding of a pen.
  void addPath(path g, pen p) {
    if(size(g) > 0)
      addBox(min(g),max(g),min(p),max(p));
  }

  void size(real x, real y=x, bool keepAspect=this.keepAspect) {
    if(!empty()) uptodate=false;
    xsize=x;
    ysize=y;
    this.keepAspect=keepAspect;
  }

  void size3(real x, real y=x, real z=y, bool keepAspect=this.keepAspect) {
    if(!empty3()) uptodate=false;
    xsize3=x;
    ysize3=y;
    zsize3=z;
    this.keepAspect=keepAspect;
  }

  void unitsize(real x, real y=x, real z=y) {
    uptodate=false;
    xunitsize=x;
    yunitsize=y;
    zunitsize=z;
  }

  // The scaling in one dimension:  x --> a*x + b
  struct scaling {
    real a,b;
    static scaling build(real a, real b) {
      scaling s=new scaling;
      s.a=a; s.b=b;
      return s;
    }
    real scale(real x) {
      return a*x+b;
    }
    real scale(coord c) {
      return scale(c.user) + c.truesize;
    }
  }

  // Calculate the minimum point in scaling the coords.
  real min(real m, scaling s, coord[] c) {
    for(int i=0; i < c.length; ++i)
      if(finite(c[i].user) && s.scale(c[i]) < m)
        m=s.scale(c[i]);
    return m;
  }
 
  // Calculate the maximum point in scaling the coords.
  real max(real M, scaling s, coord[] c) {
    for(int i=0; i < c.length; ++i)
      if(finite(c[i].user) && s.scale(c[i]) > M)
        M=s.scale(c[i]);
    return M;
  }

  // Calculate the min for the final frame, given the coordinate transform.
  pair min(transform t) {
    if(bounds.min.x.length == 0 && bounds.point.x.length == 0 &&
       bounds.max.x.length == 0) return 0;
    pair a=t*(1,1)-t*(0,0), b=t*(0,0);
    scaling xs=scaling.build(a.x,b.x);
    scaling ys=scaling.build(a.y,b.y);
    return (min(min(min(infinity,xs,bounds.point.x),xs,bounds.min.x),
                xs,bounds.max.x),
            min(min(min(infinity,ys,bounds.point.y),ys,bounds.min.y),
                ys,bounds.max.y));
  }

  // Calculate the max for the final frame, given the coordinate transform.
  pair max(transform t) {
    if(bounds.min.x.length == 0 && bounds.point.x.length == 0 &&
       bounds.max.x.length == 0) return 0;
    pair a=t*(1,1)-t*(0,0), b=t*(0,0);
    scaling xs=scaling.build(a.x,b.x);
    scaling ys=scaling.build(a.y,b.y);
    return (max(max(max(-infinity,xs,bounds.point.x),xs,bounds.min.x),
                xs,bounds.max.x),
            max(max(max(-infinity,ys,bounds.point.y),ys,bounds.min.y),
                ys,bounds.max.y));
  }

  // Calculate the min for the final frame, given the coordinate transform.
  triple min(transform3 t) {
    if(bounds3.min.x.length == 0 && bounds3.point.x.length == 0 &&
       bounds3.max.x.length == 0) return (0,0,0);
    triple a=t*(1,1,1)-t*(0,0,0), b=t*(0,0,0);
    scaling xs=scaling.build(a.x,b.x);
    scaling ys=scaling.build(a.y,b.y);
    scaling zs=scaling.build(a.z,b.z);
    return (min(min(min(infinity,xs,bounds3.point.x),xs,bounds3.min.x),
                xs,bounds3.max.x),
            min(min(min(infinity,ys,bounds3.point.y),ys,bounds3.min.y),
                ys,bounds3.max.y),
            min(min(min(infinity,zs,bounds3.point.z),zs,bounds3.min.z),
                zs,bounds3.max.z));
  }

  // Calculate the max for the final frame, given the coordinate transform.
  triple max(transform3 t) {
    if(bounds3.min.x.length == 0 && bounds3.point.x.length == 0 &&
       bounds3.max.x.length == 0) return (0,0,0);
    triple a=t*(1,1,1)-t*(0,0,0), b=t*(0,0,0);
    scaling xs=scaling.build(a.x,b.x);
    scaling ys=scaling.build(a.y,b.y);
    scaling zs=scaling.build(a.z,b.z);
    return (max(max(max(-infinity,xs,bounds3.point.x),xs,bounds3.min.x),
                xs,bounds3.max.x),
            max(max(max(-infinity,ys,bounds3.point.y),ys,bounds3.min.y),
                ys,bounds3.max.y),
            max(max(max(-infinity,zs,bounds3.point.z),zs,bounds3.min.z),
                zs,bounds3.max.z));
  }

  // Calculate the sizing constants for the given array and maximum size.
  real calculateScaling(string dir, coord[] coords, real size,
                        bool warn=true) {
    access simplex;
    simplex.problem p=new simplex.problem;
   
    void addMinCoord(coord c) {
      // (a*user + b) + truesize >= 0:
      p.addRestriction(c.user,1,c.truesize);
    }
    void addMaxCoord(coord c) {
      // (a*user + b) + truesize <= size:
      p.addRestriction(-c.user,-1,size-c.truesize);
    }

    coord[] m=maxcoords(coords,operator >=);
    coord[] M=maxcoords(coords,operator <=);
    
    for(int i=0; i < m.length; ++i)
      addMinCoord(m[i]);
    for(int i=0; i < M.length; ++i)
      addMaxCoord(M[i]);

    int status=p.optimize();
    if(status == simplex.problem.OPTIMAL) {
      return scaling.build(p.a(),p.b()).a;
    } else if(status == simplex.problem.UNBOUNDED) {
      if(warn) warning("unbounded",dir+" scaling in picture unbounded");
      return 0;
    } else {
      if(!warn) return 1;
      bool userzero=true;
      for(int i=0; i < coords.length; ++i) {
        if(coords[i].user != 0) userzero=false;
        if(!finite(coords[i].user) || !finite(coords[i].truesize))
          abort("unbounded picture");
      }
      if(userzero) return 1;
      warning("cannotfit","cannot fit picture to "+dir+"size "+(string) size
              +"...enlarging...");
      return calculateScaling(dir,coords,sqrt(2)*size,warn);
    }
  }

  void append(coords2 point, coords2 min, coords2 max, transform t,
              bounds bounds) 
  {
    // Add the coord info to this picture.
    if(t == identity()) {
      point.append(bounds.point);
      min.append(bounds.min);
      max.append(bounds.max);
    } else {
      point.push(t,bounds.point,bounds.point);
      // Add in all 4 corner points, to properly size rectangular pictures.
      point.push(t,bounds.min,bounds.min);
      point.push(t,bounds.min,bounds.max);
      point.push(t,bounds.max,bounds.min);
      point.push(t,bounds.max,bounds.max);
    }
  }
  
  void append(coords3 point, coords3 min, coords3 max, transform3 t,
              bounds3 bounds) 
  {
    // Add the coord info to this picture.
    if(t == identity4) {
      point.append(bounds.point);
      min.append(bounds.min);
      max.append(bounds.max);
    } else {
      point.push(t,bounds.point,bounds.point,bounds.point);
      // Add in all 8 corner points, to properly size cuboid pictures.
      point.push(t,bounds.min,bounds.min,bounds.min);
      point.push(t,bounds.min,bounds.min,bounds.max);
      point.push(t,bounds.min,bounds.max,bounds.min);
      point.push(t,bounds.min,bounds.max,bounds.max);
      point.push(t,bounds.max,bounds.min,bounds.min);
      point.push(t,bounds.max,bounds.min,bounds.max);
      point.push(t,bounds.max,bounds.max,bounds.min);
      point.push(t,bounds.max,bounds.max,bounds.max);
    }
  }
  
  // Returns the transform for turning user-space pairs into true-space pairs.
  transform scaling(real xsize, real ysize, bool keepAspect=true,
                    bool warn=true) {
    if(xsize == 0 && xunitsize == 0 && ysize == 0 && yunitsize == 0)
      return identity();

    coords2 Coords;
    
    append(Coords,Coords,Coords,T,bounds);
    
    real sx;
    if(xunitsize == 0) {
      if(xsize != 0) sx=calculateScaling("x",Coords.x,xsize,warn);
    } else sx=xunitsize;

    real sy;
    if(yunitsize == 0) {
      if(ysize != 0) sy=calculateScaling("y",Coords.y,ysize,warn);
    } else sy=yunitsize;

    if(sx == 0) {
      sx=sy;
      if(sx == 0)
        return identity();
    } else if(sy == 0) sy=sx;


    if(keepAspect && (xunitsize == 0 || yunitsize == 0))
      return scale(min(sx,sy));
    else
      return scale(sx,sy);
  }

  transform scaling(bool warn=true) {
    return scaling(xsize,ysize,keepAspect,warn);
  }

  // Returns the transform for turning user-space pairs into true-space triples.
  transform3 scaling(real xsize, real ysize, real zsize, bool keepAspect=true,
                     bool warn=true) {
    if(xsize == 0 && xunitsize == 0 && ysize == 0 && yunitsize == 0
       && zsize == 0 && zunitsize == 0)
      return identity(4);

    coords3 Coords;
    
    append(Coords,Coords,Coords,T3,bounds3);
    
    real sx;
    if(xunitsize == 0) {
      if(xsize != 0) sx=calculateScaling("x",Coords.x,xsize,warn);
    } else sx=xunitsize;

    real sy;
    if(yunitsize == 0) {
      if(ysize != 0) sy=calculateScaling("y",Coords.y,ysize,warn);
    } else sy=yunitsize;

    real sz;
    if(zunitsize == 0) {
      if(zsize != 0) sz=calculateScaling("z",Coords.z,zsize,warn);
    } else sz=zunitsize;

    if(sx == 0) {
      sx=max(sy,sz);
      if(sx == 0)
        return identity(4);
    }
    if(sy == 0) sy=max(sz,sx);
    if(sz == 0) sz=max(sx,sy);

    if(keepAspect && (xunitsize == 0 || yunitsize == 0 || zunitsize == 0))
      return scale3(min(sx,sy,sz));
    else
      return scale(sx,sy,sz);
  }

  transform3 scaling3(bool warn=true) {
    return scaling(xsize3,ysize3,zsize3,keepAspect,warn);
  }

  frame fit(transform t, transform T0=T, pair m, pair M) {
    frame f;
    for(int i=0; i < nodes.length; ++i)
      nodes[i](f,t,T0,m,M);
    return f;
  }

  frame fit3(transform3 t, transform3 T0=T3, picture pic, projection P,
             triple m, triple M) {
    frame f;
    for(int i=0; i < nodes3.length; ++i)
      nodes3[i](f,t,T0,pic,P,m,M);
    return f;
  }

  // Returns a rigid version of the picture using t to transform user coords
  // into truesize coords.
  frame fit(transform t) {
    return fit(t,min(t),max(t));
  }

  frame fit3(transform3 t, picture pic, projection P) {
    return fit3(t,pic,P,min(t),max(t));
  }

  void add(void d(picture, transform), bool exact=false) {
    add(new void(frame f, transform t) {
        picture opic=new picture;
        d(opic,t);
        add(f,opic.fit(identity()));
      },exact);
  }

  void add(void d(picture, transform3), bool exact=false, bool above=true) {
    add(new void(frame f, transform3 t, picture pic2, projection P) {
        picture opic=new picture;
        d(opic,t);
        add(f,opic.fit3(identity4,pic2,P));
      },exact,above);
  }

  void add(void d(picture, transform3, transform3, triple, triple),
           bool exact=false, bool above=true) {
    add(new void(frame f, transform3 t, transform3 T, picture pic2,
                 projection P, triple lb, triple rt) {
          picture opic=new picture;
          d(opic,t,T,lb,rt);
          add(f,opic.fit3(identity4,pic2,P));
        },exact,above);
  }

  frame scaled() {
    frame f=fit(fixedscaling);
    pair d=size(f);
    static real epsilon=100*realEpsilon;
    if(d.x > xsize*(1+epsilon)) 
      warning("xlimit","frame exceeds xlimit: "+(string) d.x+" > "+
              (string) xsize);
    if(d.y > ysize*(1+epsilon))
      warning("ylimit","frame exceeds ylimit: "+(string) d.y+" > "+
              (string) ysize);
    return f;
  }
  
  // Calculate additional scaling required if only an approximate picture
  // size estimate is available.
  transform scale(frame f, real xsize=this.xsize, real ysize=this.ysize,
                  bool keepaspect=this.keepAspect) {
    if(bounds.exact) return identity();
    pair m=min(f);
    pair M=max(f);
    real width=M.x-m.x;
    real height=M.y-m.y;
    real xgrow=xsize == 0 || width == 0 ? 1 : xsize/width;
    real ygrow=ysize == 0 || height == 0 ? 1 : ysize/height;
    if(keepAspect) {
      real[] grow;
      if(xsize > 0) grow.push(xgrow);
      if(ysize > 0) grow.push(ygrow);
      return scale(grow.length == 0 ? 1 : min(grow));
    } else return scale(xgrow,ygrow);

  }

  // Calculate additional scaling required if only an approximate thisture
  // size estimate is available.
  transform3 scale3(frame f, real xsize3=this.xsize3,
                    real ysize3=this.ysize3, real zsize3=this.zsize3,
                    bool keepaspect=this.keepAspect) {
    if(bounds3.exact) return identity(4);
    triple m=min3(f);
    triple M=max3(f);
    real width=M.x-m.x;
    real height=M.y-m.y;
    real depth=M.z-m.z;
    real xgrow=xsize3 == 0 || width == 0 ? 1 : xsize3/width;
    real ygrow=ysize3 == 0 || height == 0 ? 1 : ysize3/height;
    real zgrow=zsize3 == 0 || depth == 0 ? 1 : zsize3/depth;
    if(keepAspect) {
      real[] grow;
      if(xsize3 > 0) grow.push(xgrow);
      if(ysize3 > 0) grow.push(ygrow);
      if(zsize3 > 0) grow.push(zgrow);
      return scale3(grow.length == 0 ? 1 : min(grow));
    } else return scale(xgrow,ygrow,zgrow);
  }

  // Return the transform that would be used to fit the picture to a frame
  transform calculateTransform(real xsize, real ysize, bool keepAspect=true,
                               bool warn=true) {
    transform t=scaling(xsize,ysize,keepAspect,warn);
    return scale(fit(t),xsize,ysize,keepAspect)*t;
  }

  transform calculateTransform(bool warn=true) {
    if(fixed) return fixedscaling;
    return calculateTransform(xsize,ysize,keepAspect,warn);
  }

  transform3 calculateTransform3(real xsize=xsize3, real ysize=ysize3,
                                 real zsize=zsize3,
                                 bool keepAspect=true, bool warn=true,
                                 projection P=currentprojection) {
    transform3 t=scaling(xsize,ysize,zsize,keepAspect,warn);
    return scale3(fit3(t,null,P),keepAspect)*t;
  }

  pair min(real xsize=this.xsize, real ysize=this.ysize,
           bool keepAspect=this.keepAspect, bool warn=true) {
    return min(calculateTransform(xsize,ysize,keepAspect,warn));
  }
  
  pair max(real xsize=this.xsize, real ysize=this.ysize,
           bool keepAspect=this.keepAspect, bool warn=true) {
    return max(calculateTransform(xsize,ysize,keepAspect,warn));
  }
  
  triple min3(real xsize=this.xsize3, real ysize=this.ysize3,
              real zsize=this.zsize3, bool keepAspect=this.keepAspect,
              bool warn=true, projection P) {
    return min(calculateTransform3(xsize,ysize,zsize,keepAspect,warn,P));
  }
  
  triple max3(real xsize=this.xsize3, real ysize=this.ysize3,
              real zsize=this.zsize3, bool keepAspect=this.keepAspect,
              bool warn=true, projection P) {
    return max(calculateTransform3(xsize,ysize,zsize,keepAspect,warn,P));
  }
  
  // Returns the 2D picture fit to the requested size.
  frame fit2(real xsize=this.xsize, real ysize=this.ysize,
             bool keepAspect=this.keepAspect) {
    if(fixed) return scaled();
    if(empty2()) return newframe;
    transform t=scaling(xsize,ysize,keepAspect);
    frame f=fit(t);
    transform s=scale(f,xsize,ysize,keepAspect);
    if(s == identity()) return f;
    return fit(s*t);
  }

  static frame fitter(string,picture,string,real,real,bool,bool,string,string,
                      light,projection);
  frame fit(string prefix="", string format="",
            real xsize=this.xsize, real ysize=this.ysize,
            bool keepAspect=this.keepAspect, bool view=false,
            string options="", string script="", light light=currentlight,
            projection P=currentprojection) {
    return fitter == null ? fit2(xsize,ysize,keepAspect) :
      fitter(prefix,this,format,xsize,ysize,keepAspect,view,options,script,
             light,P);
  }
  
  // In case only an approximate picture size estimate is available, return the
  // fitted frame slightly scaled (including labels and true size distances)
  // so that it precisely meets the given size specification. 
  frame scale(real xsize=this.xsize, real ysize=this.ysize,
              bool keepAspect=this.keepAspect) {
    frame f=fit(xsize,ysize,keepAspect);
    transform s=scale(f,xsize,ysize,keepAspect);
    if(s == identity()) return f;
    return s*f;
  }

  // Copies the drawing information, but not the sizing information into a new
  // picture. Fitting this picture will not scale as the original picture would.
  picture drawcopy() {
    picture dest=new picture;
    dest.nodes=copy(nodes);
    dest.nodes3=copy(nodes3);
    dest.T=T;
    dest.T3=T3;
    dest.userCopy(this);
    dest.scale=scale.copy();
    dest.legend=copy(legend);

    return dest;
  }

  // A deep copy of this picture.  Modifying the copied picture will not affect
  // the original.
  picture copy() {
    picture dest=drawcopy();

    dest.uptodate=uptodate;
    dest.bounds=bounds.copy();
    dest.bounds3=bounds3.copy();
    
    dest.xsize=xsize; dest.ysize=ysize;
    dest.xsize3=xsize; dest.ysize3=ysize3; dest.zsize3=zsize3;
    dest.keepAspect=keepAspect;
    dest.xunitsize=xunitsize; dest.yunitsize=yunitsize;
    dest.zunitsize=zunitsize;
    dest.fixed=fixed; dest.fixedscaling=fixedscaling;
    
    return dest;
  }

  // Add a picture to this picture, such that the user coordinates will be
  // scaled identically when fitted
  void add(picture src, bool group=true, filltype filltype=NoFill,
           bool above=true) {
    // Copy the picture.  Only the drawing function closures are needed, so we
    // only copy them.  This needs to be a deep copy, as src could later have
    // objects added to it that should not be included in this picture.

    if(src == this) abort("cannot add picture to itself");
    
    uptodate=false;

    picture srcCopy=src.drawcopy();
    // Draw by drawing the copied picture.
    if(srcCopy.nodes.length > 0)
      nodes.push(new void(frame f, transform t, transform T, pair m, pair M) {
          add(f,srcCopy.fit(t,T*srcCopy.T,m,M),group,filltype,above);
        });
    
    if(srcCopy.nodes3.length > 0) {
      nodes3.push(new void(frame f, transform3 t, transform3 T3, picture pic,
                           projection P, triple m, triple M) {
                    add(f,srcCopy.fit3(t,T3*srcCopy.T3,pic,P,m,M),group,above);
                  });
    }
    
    legend.append(src.legend);
    
    if(src.userSetx) userBoxX(src.userMin.x,src.userMax.x);
    if(src.userSety) userBoxY(src.userMin.y,src.userMax.y);
    if(src.userSetz) userBoxZ(src.userMin.z,src.userMax.z);
    
    append(bounds.point,bounds.min,bounds.max,srcCopy.T,src.bounds);
    append(bounds3.point,bounds3.min,bounds3.max,srcCopy.T3,src.bounds3);

    if(!src.bounds.exact) bounds.exact=false;
    if(!src.bounds3.exact) bounds3.exact=false;
  }
}

picture operator * (transform t, picture orig)
{
  picture pic=orig.copy();
  pic.T=t*pic.T;
  pic.userCorners(t*(pic.userMin.x,pic.userMin.y),
                  t*(pic.userMin.x,pic.userMax.y),
                  t*(pic.userMax.x,pic.userMin.y),
                  t*(pic.userMax.x,pic.userMax.y));
  pic.bounds.exact=false;
  return pic;
}

picture operator * (transform3 t, picture orig)
{
  picture pic=orig.copy();
  pic.T3=t*pic.T3;
  pic.userCorners(t*pic.userMin,
                  t*(pic.userMin.x,pic.userMin.y,pic.userMax.z),
                  t*(pic.userMin.x,pic.userMax.y,pic.userMin.z),
                  t*(pic.userMin.x,pic.userMax.y,pic.userMax.z),
                  t*(pic.userMax.x,pic.userMin.y,pic.userMin.z),
                  t*(pic.userMax.x,pic.userMin.y,pic.userMax.z),
                  t*(pic.userMax.x,pic.userMax.y,pic.userMin.z),
                  t*pic.userMax);
  pic.bounds3.exact=false;
  return pic;
}

picture currentpicture;

void size(picture pic=currentpicture, real x, real y=x,
          bool keepAspect=pic.keepAspect)
{
  pic.size(x,y,keepAspect);
}

void size3(picture pic=currentpicture, real x, real y=x, real z=y,
           bool keepAspect=pic.keepAspect)
{
  pic.size3(x,y,z,keepAspect);
}

void unitsize(picture pic=currentpicture, real x, real y=x, real z=y) 
{
  pic.unitsize(x,y,z);
}

void size(picture pic=currentpicture, real xsize, real ysize,
          pair min, pair max)
{
  pair size=max-min;
  pic.unitsize(size.x != 0 ? xsize/size.x : 0,
               size.y != 0 ? ysize/size.y : 0);
}

void size(picture dest, picture src)
{
  dest.size(src.xsize,src.ysize,src.keepAspect);
  dest.size3(src.xsize3,src.ysize3,src.zsize3,src.keepAspect);
  dest.unitsize(src.xunitsize,src.yunitsize,src.zunitsize);
}

pair min(picture pic, bool user=false)
{
  transform t=pic.calculateTransform();
  pair z=pic.min(t);
  return user ? inverse(t)*z : z;
}
  
pair max(picture pic, bool user=false)
{
  transform t=pic.calculateTransform();
  pair z=pic.max(t);
  return user ? inverse(t)*z : z;
}
  
pair size(picture pic, bool user=false)
{
  transform t=pic.calculateTransform();
  pair M=pic.max(t);
  pair m=pic.min(t);
  if(!user) return M-m;
  t=inverse(t);
  return t*M-t*m;
}

pair point(picture pic=currentpicture, pair dir, bool user=true)
{
  pair z=pic.userMin()+realmult(rectify(dir),pic.userMax()-pic.userMin());
  return user ? z : pic.calculateTransform()*z;
}

pair truepoint(picture pic=currentpicture, pair dir, bool user=true)
{
  transform t=pic.calculateTransform();
  pair m=pic.min(t);
  pair M=pic.max(t);
  pair z=m+realmult(rectify(dir),M-m);
  return user ? inverse(t)*z : z;
}

// Transform coordinate in [0,1]x[0,1] to current user coordinates.
pair relative(picture pic=currentpicture, pair z)
{
  return pic.userMin()+realmult(z,pic.userMax()-pic.userMin());
}

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

typedef void drawer3(frame f, transform3 t, picture pic, projection P);
void add(picture pic=currentpicture, drawer3 d, bool exact=false)
{
  pic.add(d,exact);
}

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

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

void begingroup(picture pic=currentpicture)
{
  pic.add(new void(frame f, transform) {
      begingroup(f);
    },true);
}

void endgroup(picture pic=currentpicture)
{
  pic.add(new void(frame f, transform) {
      endgroup(f);
    },true);
}

void Draw(picture pic=currentpicture, path g, pen p=currentpen)
{
  pic.add(new void(frame f, transform t) {
      draw(f,t*g,p);
    },true);
  pic.addPath(g,p);
}

void _draw(picture pic=currentpicture, path g, pen p=currentpen,
           margin margin=NoMargin)
{
  pic.add(new void(frame f, transform t) {
      draw(f,margin(t*g,p).g,p);
    },true);
  pic.addPath(g,p);
}

void Draw(picture pic=currentpicture, explicit path[] g, pen p=currentpen)
{
  for(int i=0; i < g.length; ++i) Draw(pic,g[i],p);
}

void fill(picture pic=currentpicture, path[] g, pen p=currentpen,
          bool copy=true)
{
  if(copy)
    g=copy(g);
  pic.add(new void(frame f, transform t) {
      fill(f,t*g,p,false);
    },true);
  pic.addPath(g);
}

void latticeshade(picture pic=currentpicture, path[] g, bool stroke=false,
                  pen fillrule=currentpen, pen[][] p, bool copy=true)
{
  if(copy) {
    g=copy(g);
    p=copy(p);
  }
  pic.add(new void(frame f, transform t) {
      latticeshade(f,t*g,stroke,fillrule,p,t,false);
    },true);
  pic.addPath(g);
}

void axialshade(picture pic=currentpicture, path[] g, bool stroke=false,
                pen pena, pair a, pen penb, pair b, bool copy=true)
{
  if(copy)
    g=copy(g);
  pic.add(new void(frame f, transform t) {
      axialshade(f,t*g,stroke,pena,t*a,penb,t*b,false);
    },true);
  pic.addPath(g);
}

void radialshade(picture pic=currentpicture, path[] g, bool stroke=false,
                 pen pena, pair a, real ra, pen penb, pair b, real rb,
                 bool copy=true)
{
  if(copy)
    g=copy(g);
  pic.add(new void(frame f, transform t) {
      pair A=t*a, B=t*b;
      real RA=abs(t*(a+ra)-A);
      real RB=abs(t*(b+rb)-B);
      radialshade(f,t*g,stroke,pena,A,RA,penb,B,RB,false);
    },true);
  pic.addPath(g);
}

void gouraudshade(picture pic=currentpicture, path[] g, bool stroke=false,
                  pen fillrule=currentpen, pen[] p, pair[] z, int[] edges,
                  bool copy=true)
{
  if(copy) {
    g=copy(g);
    p=copy(p);
    z=copy(z);
    edges=copy(edges);
  }
  pic.add(new void(frame f, transform t) {
      gouraudshade(f,t*g,stroke,fillrule,p,t*z,edges,false);
    },true);
  pic.addPath(g);
}

void gouraudshade(picture pic=currentpicture, path[] g, bool stroke=false,
                  pen fillrule=currentpen, pen[] p, int[] edges, bool copy=true)
{
  if(copy) {
    g=copy(g);
    p=copy(p);
    edges=copy(edges);
  }
  pic.add(new void(frame f, transform t) {
      gouraudshade(f,t*g,stroke,fillrule,p,edges,false);
    },true);
  pic.addPath(g);
}

void tensorshade(picture pic=currentpicture, path[] g, bool stroke=false,
                 pen fillrule=currentpen, pen[][] p, path[] b=g,
                 pair[][] z=new pair[][], bool copy=true)
{
  if(copy) {
    g=copy(g);
    p=copy(p);
    b=copy(b);
    z=copy(z);
  }
  pic.add(new void(frame f, transform t) {
      pair[][] Z=new pair[z.length][];
      for(int i=0; i < z.length; ++i)
        Z[i]=t*z[i];
      tensorshade(f,t*g,stroke,fillrule,p,t*b,Z,false);
    },true);
  pic.addPath(g);
}

void tensorshade(frame f, path[] g, bool stroke=false,
                 pen fillrule=currentpen, pen[] p,
                 path b=g.length > 0 ? g[0] : nullpath)
{
  tensorshade(f,g,stroke,fillrule,new pen[][] {p},b);
}

void tensorshade(frame f, path[] g, bool stroke=false,
                 pen fillrule=currentpen, pen[] p,
                 path b=g.length > 0 ? g[0] : nullpath, pair[] z)
{
  tensorshade(f,g,stroke,fillrule,new pen[][] {p},b,new pair[][] {z});
}

void tensorshade(picture pic=currentpicture, path[] g, bool stroke=false,
                 pen fillrule=currentpen, pen[] p,
                 path b=g.length > 0 ? g[0] : nullpath)
{
  tensorshade(pic,g,stroke,fillrule,new pen[][] {p},b);
}

void tensorshade(picture pic=currentpicture, path[] g, bool stroke=false,
                 pen fillrule=currentpen, pen[] p,
                 path b=g.length > 0 ? g[0] : nullpath, pair[] z)
{
  tensorshade(pic,g,stroke,fillrule,new pen[][] {p},b,new pair[][] {z});
}

// Smoothly shade the regions between consecutive paths of a sequence using a
// given array of pens:
void draw(picture pic=currentpicture, path[] g, pen fillrule=currentpen,
          pen[] p)
{
  path[] G;
  pen[][] P;
  string differentlengths="arrays have different lengths";
  if(g.length != p.length) abort(differentlengths);
  for(int i=0; i < g.length-1; ++i) {
    path g0=g[i];
    path g1=g[i+1];
    if(length(g0) != length(g1)) abort(differentlengths);
    for(int j=0; j < length(g0); ++j) {
      G.push(subpath(g0,j,j+1)--reverse(subpath(g1,j,j+1))--cycle);
      P.push(new pen[] {p[i],p[i],p[i+1],p[i+1]});
    }
  }
  tensorshade(pic,G,fillrule,P);
}

void functionshade(picture pic=currentpicture, path[] g, bool stroke=false,
                   pen fillrule=currentpen, string shader, bool copy=true)
{
  if(copy)
    g=copy(g);
  pic.add(new void(frame f, transform t) {
      functionshade(f,t*g,stroke,fillrule,shader);
    },true);
  pic.addPath(g);
}

void filldraw(picture pic=currentpicture, path[] g, pen fillpen=currentpen,
              pen drawpen=currentpen)
{
  begingroup(pic);
  fill(pic,g,fillpen);
  Draw(pic,g,drawpen);
  endgroup(pic);
}

void clip(picture pic=currentpicture, path[] g, bool stroke=false,
          pen fillrule=currentpen, bool copy=true)
{
  if(copy)
    g=copy(g);
  pic.userClip(min(g),max(g));
  pic.clip(new void(frame f, transform t) {
      clip(f,t*g,stroke,fillrule,false);
    },true);
}

void beginclip(picture pic=currentpicture, path[] g, bool stroke=false,
               pen fillrule=currentpen, bool copy=true) 
{
  if(copy)
    g=copy(g);

  pic.clipmin.push(min(g));
  pic.clipmax.push(max(g));

  pic.add(new void(frame f, transform t) {
      beginclip(f,t*g,stroke,fillrule,false);
    },true);
}

void endclip(picture pic=currentpicture)
{
  if(pic.clipmin.length > 0 && pic.clipmax.length > 0)
    pic.userClip(pic.clipmin.pop(),pic.clipmax.pop());
  pic.clip(new void(frame f, transform) {
      endclip(f);
    },true);
}

void unfill(picture pic=currentpicture, path[] g, bool copy=true)
{
  if(copy)
    g=copy(g);
  pic.add(new void(frame f, transform t) {
      unfill(f,t*g,false);
    },true);
}

void filloutside(picture pic=currentpicture, path[] g, pen p=currentpen,
                 bool copy=true)
{
  if(copy)
    g=copy(g);
  pic.add(new void(frame f, transform t) {
      filloutside(f,t*g,p,false);
    },true);
  pic.addPath(g);
}

// Use a fixed scaling to map user coordinates in box(min,max) to the 
// desired picture size.
transform fixedscaling(picture pic=currentpicture, pair min, pair max,
                       pen p=nullpen, bool warn=false)
{
  Draw(pic,min,p+invisible);
  Draw(pic,max,p+invisible);
  pic.fixed=true;
  return pic.fixedscaling=pic.calculateTransform(pic.xsize,pic.ysize,
                                                 pic.keepAspect);
}

// Add frame src about position to frame dest with optional grouping.
void add(frame dest, frame src, pair position, bool group=false,
         filltype filltype=NoFill, bool above=true)
{
  add(dest,shift(position)*src,group,filltype,above);
}

// Add frame src about position to picture dest with optional grouping.
void add(picture dest=currentpicture, frame src, pair position=0,
         bool group=true, filltype filltype=NoFill, bool above=true)
{
  if(is3D(src)) {
    dest.add(new void(frame f, transform3, picture, projection) {
        add(f,src); // always add about 3D origin (ignore position)
      },true);
    dest.addBox((0,0,0),(0,0,0),min3(src),max3(src));
  } else {
    dest.add(new void(frame f, transform t) {
        add(f,shift(t*position)*src,group,filltype,above);
      },true);
    dest.addBox(position,position,min(src),max(src));
  }
}

// Like add(picture,frame,pair) but extend picture to accommodate frame.
void attach(picture dest=currentpicture, frame src, pair position=0,
            bool group=true, filltype filltype=NoFill, bool above=true)
{
  transform t=dest.calculateTransform();
  add(dest,src,position,group,filltype,above);
  pair s=size(dest.fit(t));
  size(dest,dest.xsize != 0 ? s.x : 0,dest.ysize != 0 ? s.y : 0);
}

// Like add(picture,frame,pair) but align frame in direction align.
void add(picture dest=currentpicture, frame src, pair position, pair align,
         bool group=true, filltype filltype=NoFill, bool above=true)
{
  add(dest,align(src,align),position,group,filltype,above);
}

// Like add(frame,frame,pair) but align frame in direction align.
void add(frame dest, frame src, pair position, pair align,
         bool group=true, filltype filltype=NoFill, bool above=true)
{
  add(dest,align(src,align),position,group,filltype,above);
}

// Like add(picture,frame,pair,pair) but extend picture to accommodate frame;
void attach(picture dest=currentpicture, frame src, pair position,
            pair align, bool group=true, filltype filltype=NoFill,
            bool above=true)
{
  attach(dest,align(src,align),position,group,filltype,above);
}

// Add a picture to another such that user coordinates in both will be scaled
// identically in the shipout.
void add(picture dest, picture src, bool group=true, filltype filltype=NoFill,
         bool above=true)
{
  dest.add(src,group,filltype,above);
}

void add(picture src, bool group=true, filltype filltype=NoFill,
         bool above=true)
{
  currentpicture.add(src,group,filltype,above);
}

// 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, pair position, bool group=true,
         filltype filltype=NoFill, bool above=true)
{
  add(dest,src.fit(identity()),position,group,filltype,above);
}

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

// Fill a region about the user-coordinate 'origin'.
void fill(pair origin, picture pic=currentpicture, path[] g, pen p=currentpen)
{
  picture opic;
  fill(opic,g,p);
  add(pic,opic,origin);
}

void postscript(picture pic=currentpicture, string s)
{
  pic.add(new void(frame f, transform) {
      postscript(f,s);
    },true);
}

void tex(picture pic=currentpicture, string s)
{
  pic.add(new void(frame f, transform) {
      tex(f,s);
    },true);
}

void postscript(picture pic=currentpicture, string s, pair min, pair max)
{
  pic.add(new void(frame f, transform t) {
      postscript(f,s,t*min,t*max);
    },true);
}

void tex(picture pic=currentpicture, string s, pair min, pair max)
{
  pic.add(new void(frame f, transform t) {
      tex(f,s,t*min,t*max);
    },true);
}

void layer(picture pic=currentpicture)
{
  pic.add(new void(frame f, transform) {
      layer(f);
    },true);
}

void erase(picture pic=currentpicture)
{
  pic.uptodate=false;
  pic.erase();
}

void begin(picture pic=currentpicture, string name, string id="",
           bool visible=true)
{
  if(!latex() || !pdf()) return;
  settings.twice=true;
  if(id == "") id=string(++ocgindex);
  tex(pic,"\begin{ocg}{"+name+"}{"+id+"}{"+(visible ? "1" : "0")+"}");
  layer(pic);
}

void end(picture pic=currentpicture)
{
  if(!latex() || !pdf()) return;
  tex(pic,"\end{ocg}");
  layer(pic);
}