private import math;

private import math;

if(settings.v3d) settings.prc=false;

if(prc0() || settings.v3d) {
if(!latex()) {
settings.prc=false;
settings.v3d=false;
}
else {
access embed;
Embed=embed.embedplayer;
}
}

// Useful lossy PRC compression values.
restricted real Zero=0;
restricted real Single=0.000001;
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 interaction
{
int type;
triple center; // position to rotate billboard objects about
bool targetsize;
static interaction defaultinteraction;

void operator init(interaction interaction=defaultinteraction,
int type=interaction.type,
triple center=interaction.center, bool targetsize=interaction.targetsize) {
this.type=type;
this.center=center;
this.targetsize=targetsize;
}
}

interaction.defaultinteraction=new interaction;

restricted interaction Embedded=interaction();
restricted interaction Billboard=interaction(1);

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

bool closed; // use one-sided PRC rendering?

bool tessellate; // use tessellated mesh to store straight patches

bool3 merge; // merge PRC nodes before rendering, for faster but
// lower quality 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.
bool partnames; // assign part name indices to compound objects
bool defaultnames; // assign default names to unnamed objects
interaction interaction; // billboard interaction mode

static render defaultrender;

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

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.sphere=NURBSsphere;
defaultrender.partnames=false;
defaultrender.defaultnames=true;
defaultrender.interaction=Embedded;

real defaultshininess=0.7;
real defaultmetallic=0.0;
real defaultfresnel0=0.04;

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);
real norm=sqrtEpsilon*abs(v);
if(abs(u) > norm) return unit(u);
u=cross(v,Z);
return (abs(u) > norm) ? 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=true)
{
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=true)
{
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 unit(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 interp(path3 a, path3 b, real t)
{
int n=size(a);
return path3(sequence(new triple(int i) {
return interp(precontrol(a,i),precontrol(b,i),t);},n),
sequence(new triple(int i) {return interp(point(a,i),point(b,i),t);},n),
sequence(new triple(int i) {return interp(postcontrol(a,i),
postcontrol(b,i),t);},n),
sequence(new bool(int i) {return straight(a,i) && straight(b,i);},n),
cyclic(a) && cyclic(b));
}

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;
real[] tO=t*new real[] {O.x,O.y,O.z,1};
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);
}

// Align Label with normal in direction dir.
Label align(Label L, triple dir)
{
Label L=L.copy();
L.transform3(align(unit(dir)));
return L;
}

// 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;
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)
{
_begingroup3(f,name,render.compression,render.granularity,render.closed,
render.tessellate,render.merge == false,
render.merge == true,render.interaction.center,render.interaction.type);
}

void begingroup3(picture pic=currentpicture, string name="",
render render=defaultrender)
{
pic.add(new void(frame f, transform3, picture pic, projection) {
if(is3D())
begingroup3(f,name,render);
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()) {
real width=linewidth(q);
void drawthick(path3 g) {
if(settings.thick && width > 0) {
bool prc=prc();
bool primitive=primitive();
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);
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.push(t0*scale2r*unitdisk);
if(L > 0) {
T.s.push(tL*scale2r*unitdisk);
}
} else if(linecap == 1) {
transform3 scale3r=scale3(r);
T.s.push(t0*scale3r*(straight(c,0) ?
unithemisphere : unitsphere));
if(L > 0)
T.s.push(tL*scale3r*(straight(c,Lc-1) ?
unithemisphere : unitsphere));
}
}
// Draw central core for better small-scale rendering.
if((!prc || piecewisestraight(g)) && !primitive && opacity(q) == 1)
_draw(f,c,p,light);
}
for(surface s : T.s)
draw(f,s,p,light,render);
} else _draw(f,g,p,light);
}
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]);
}

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";

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() ? "Headlamp" : "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);
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;
bool keepAspect=true;

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;
this.keepAspect=keepAspect;

if(xsize3 == 0 && ysize3 == 0 && zsize3 == 0) {
xsize3=ysize3=zsize3=max(xsize,ysize);
warn=false;
}

this.P=P.copy();

if(!P.absolute && P.showtarget && !pic.empty3())
draw(pic,this.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*this.P;
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;
bool prc=prc(format);
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(Q.t[2][3] == -1) // PRC can't handle oblique projections
Q=orthographic(P.camera,P.up,P.target,P.zoom,P.viewportshift,
P.showtarget,P.center);

if(P.infinity) {
triple m=min3(S.f);
triple M=max3(S.f);
triple lambda=M-m;
if(S.keepAspect || prc) {
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 preview=settings.render > 0 && !prconly() && !settings.v3d;
if(prc || settings.v3d) {
// 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) || settings.v3d) {
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 && !S.pic2.empty()) abort("camera too close");

if(primitive())
format=settings.v3d ? "v3d" : settings.outformat;

transform3 s=inv*shift(0,0,zcenter);

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,S.viewportmargin,
tinv*s,s,Light.background(),Light.position,
Light.diffuse,Light.specular,
view && !preview);
if(settings.v3d) {
string content=prefix+".v3d";
F.L=Embed(content,S.width,S.height);
if(!settings.inlinetex) file3.push(content);
return F;
}
if(!preview) return F;
}

string image;
if((preview || (prc && settings.render == 0)) && settings.embed) {
image=prefix;
if(settings.inlinetex) image += "_0";
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 && !(settings.xasy && format == ""))
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) || settings.v3d;
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,settings.inlinetex ?
"\input{"+prefix+"}" : 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 || pic.queueErase3) {
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);
pic.queueErase3=false;
}

if(is3D(format) || pic.queueErase || settings.render == 0)
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);