private import math; if(inXasyMode) settings.render=0; if(prc0()) { if(settings.tex == "context") settings.prc=false; else { access embed; Embed=embed.embedplayer; } } // Useful lossy compression values. restricted real Zero=0; restricted real Low=0.0001; restricted real Medium=0.001; restricted real High=0.01; restricted int PRCsphere=0; // Renders slowly but produces smaller PRC files. restricted int NURBSsphere=1; // Renders fast but produces larger PRC files. struct render { // PRC parameters: real compression; // lossy compression parameter (0=no compression) real granularity; // PRC rendering granularity bool closed; // use one-sided rendering? bool tessellate; // use tessellated mesh to store straight patches? bool3 merge; // merge nodes before rendering, for faster but // lower quality PRC rendering (the value default means // merge opaque patches only). int sphere; // PRC sphere type (PRCsphere or NURBSsphere). // General parameters: real margin; // shrink amount for rendered openGL viewport, in bp. real tubegranularity; // granularity for rendering tubes bool labelfill; // fill subdivision cracks in unlighted labels bool partnames; // assign part name indices to compound objects bool defaultnames; // assign default names to unnamed objects static render defaultrender; void operator init(real compression=defaultrender.compression, real granularity=defaultrender.granularity, bool closed=defaultrender.closed, bool tessellate=defaultrender.tessellate, bool3 merge=defaultrender.merge, int sphere=defaultrender.sphere, real margin=defaultrender.margin, real tubegranularity=defaultrender.tubegranularity, bool labelfill=defaultrender.labelfill, bool partnames=defaultrender.partnames, bool defaultnames=defaultrender.defaultnames) { this.compression=compression; this.granularity=granularity; this.closed=closed; this.tessellate=tessellate; this.merge=merge; this.sphere=sphere; this.margin=margin; this.tubegranularity=tubegranularity; this.labelfill=labelfill; this.partnames=partnames; this.defaultnames=defaultnames; } } render operator init() {return render();} render defaultrender=render.defaultrender=new render; defaultrender.compression=High; defaultrender.granularity=Medium; defaultrender.closed=false; defaultrender.tessellate=false; defaultrender.merge=false; defaultrender.margin=0.02; defaultrender.tubegranularity=0.005; defaultrender.sphere=NURBSsphere; defaultrender.labelfill=true; defaultrender.partnames=false; defaultrender.defaultnames=true; real defaultshininess=0.25; real angleprecision=1e-5; // Precision for centering perspective projections. int maxangleiterations=25; string defaultembed3Doptions="3Dmenu"; string defaultembed3Dscript; real defaulteyetoview=63mm/1000mm; string partname(int i, render render=defaultrender) { return render.partnames ? string(i+1) : ""; } triple O=(0,0,0); triple X=(1,0,0), Y=(0,1,0), Z=(0,0,1); // A translation in 3D space. transform3 shift(triple v) { transform3 t=identity(4); t[0][3]=v.x; t[1][3]=v.y; t[2][3]=v.z; return t; } // Avoid two parentheses. transform3 shift(real x, real y, real z) { return shift((x,y,z)); } transform3 shift(transform3 t) { transform3 T=identity(4); T[0][3]=t[0][3]; T[1][3]=t[1][3]; T[2][3]=t[2][3]; return T; } // A 3D scaling in the x direction. transform3 xscale3(real x) { transform3 t=identity(4); t[0][0]=x; return t; } // A 3D scaling in the y direction. transform3 yscale3(real y) { transform3 t=identity(4); t[1][1]=y; return t; } // A 3D scaling in the z direction. transform3 zscale3(real z) { transform3 t=identity(4); t[2][2]=z; return t; } // A 3D scaling by s in the v direction. transform3 scale(triple v, real s) { v=unit(v); s -= 1; return new real[][] { {1+s*v.x^2, s*v.x*v.y, s*v.x*v.z, 0}, {s*v.x*v.y, 1+s*v.y^2, s*v.y*v.z, 0}, {s*v.x*v.z, s*v.y*v.z, 1+s*v.z^2, 0}, {0, 0, 0, 1}}; } // A transformation representing rotation by an angle in degrees about // an axis v through the origin (in the right-handed direction). transform3 rotate(real angle, triple v) { if(v == O) abort("cannot rotate about the zero vector"); v=unit(v); real x=v.x, y=v.y, z=v.z; real s=Sin(angle), c=Cos(angle), t=1-c; return new real[][] { {t*x^2+c, t*x*y-s*z, t*x*z+s*y, 0}, {t*x*y+s*z, t*y^2+c, t*y*z-s*x, 0}, {t*x*z-s*y, t*y*z+s*x, t*z^2+c, 0}, {0, 0, 0, 1}}; } // A transformation representing rotation by an angle in degrees about // the line u--v (in the right-handed direction). transform3 rotate(real angle, triple u, triple v) { return shift(u)*rotate(angle,v-u)*shift(-u); } // Reflects about the plane through u, v, and w. transform3 reflect(triple u, triple v, triple w) { triple n=unit(cross(v-u,w-u)); if(n == O) abort("points determining reflection plane cannot be colinear"); return new real[][] { {1-2*n.x^2, -2*n.x*n.y, -2*n.x*n.z, u.x}, {-2*n.x*n.y, 1-2*n.y^2, -2*n.y*n.z, u.y}, {-2*n.x*n.z, -2*n.y*n.z, 1-2*n.z^2, u.z}, {0, 0, 0, 1} }*shift(-u); } // Project u onto v. triple project(triple u, triple v) { v=unit(v); return dot(u,v)*v; } // Return a unit vector perpendicular to a given unit vector v. triple perp(triple v) { triple u=cross(v,Y); if(abs(u) > sqrtEpsilon) return unit(u); u=cross(v,Z); return (abs(u) > sqrtEpsilon) ? unit(u) : X; } // Return the transformation corresponding to moving the camera from the target // (looking in the negative z direction) to the point 'eye' (looking at target), // orienting the camera so that direction 'up' points upwards. // Since, in actuality, we are transforming the points instead of the camera, // we calculate the inverse matrix. // Based on the gluLookAt implementation in the OpenGL manual. transform3 look(triple eye, triple up=Z, triple target=O) { triple f=unit(target-eye); if(f == O) f=-Z; // The eye is already at the origin: look down. triple s=cross(f,up); // If the eye is pointing either directly up or down, there is no // preferred "up" direction. Pick one arbitrarily. s=s != O ? unit(s) : perp(f); triple u=cross(s,f); transform3 M={{ s.x, s.y, s.z, 0}, { u.x, u.y, u.z, 0}, {-f.x, -f.y, -f.z, 0}, { 0, 0, 0, 1}}; return M*shift(-eye); } // Return a matrix to do perspective distortion based on a triple v. transform3 distort(triple v) { transform3 t=identity(4); real d=length(v); if(d == 0) return t; t[3][2]=-1/d; t[3][3]=0; return t; } projection operator * (transform3 t, projection P) { projection P=P.copy(); if(!P.absolute) { P.camera=t*P.camera; triple target=P.target; P.target=t*P.target; if(P.infinity) P.normal=t*(target+P.normal)-P.target; else P.normal=P.vector(); P.calculate(); } return P; } // With this, save() and restore() in plain also save and restore the // currentprojection. addSaveFunction(new restoreThunk() { projection P=currentprojection.copy(); return new void() { currentprojection=P; }; }); pair project(triple v, projection P=currentprojection) { return project(v,P.t); } pair dir(triple v, triple dir, projection P) { return unit(project(v+0.5dir,P)-project(v-0.5*dir,P)); } // Uses the homogenous coordinate to perform perspective distortion. // When combined with a projection to the XY plane, this effectively maps // points in three space to a plane through target and // perpendicular to the vector camera-target. projection perspective(triple camera, triple up=Z, triple target=O, real zoom=1, real angle=0, pair viewportshift=0, bool showtarget=true, bool autoadjust=true, bool center=autoadjust) { if(camera == target) abort("camera cannot be at target"); return projection(camera,up,target,zoom,angle,viewportshift, showtarget,autoadjust,center, new transformation(triple camera, triple up, triple target) {return transformation(look(camera,up,target), distort(camera-target));}); } projection perspective(real x, real y, real z, triple up=Z, triple target=O, real zoom=1, real angle=0, pair viewportshift=0, bool showtarget=true, bool autoadjust=true, bool center=autoadjust) { return perspective((x,y,z),up,target,zoom,angle,viewportshift,showtarget, autoadjust,center); } projection orthographic(triple camera, triple up=Z, triple target=O, real zoom=1, pair viewportshift=0, bool showtarget=true, bool center=false) { return projection(camera,up,target,zoom,viewportshift,showtarget, center=center,new transformation(triple camera, triple up, triple target) { return transformation(look(camera,up,target));}); } projection orthographic(real x, real y, real z, triple up=Z, triple target=O, real zoom=1, pair viewportshift=0, bool showtarget=true, bool center=false) { return orthographic((x,y,z),up,target,zoom,viewportshift,showtarget, center=center); } // Compute camera position with x axis below the horizontal at angle alpha, // y axis below the horizontal at angle beta, and z axis up. triple camera(real alpha, real beta) { real denom=Tan(alpha+beta); real Tanalpha=Tan(alpha); real Tanbeta=Tan(beta); return (sqrt(Tanalpha/denom),sqrt(Tanbeta/denom),sqrt(Tanalpha*Tanbeta)); } projection oblique(real angle=45) { transform3 t=identity(4); real c2=Cos(angle)^2; real s2=1-c2; t[0][2]=-c2; t[1][2]=-s2; t[2][2]=1; t[2][3]=-1; return projection((c2,s2,1),up=Y,normal=(0,0,1), new transformation(triple,triple,triple) { return transformation(t);}); } projection obliqueZ(real angle=45) {return oblique(angle);} projection obliqueX(real angle=45) { transform3 t=identity(4); real c2=Cos(angle)^2; real s2=1-c2; t[0][0]=-c2; t[1][0]=-s2; t[1][1]=0; t[0][1]=1; t[1][2]=1; t[2][2]=0; t[2][0]=1; t[2][3]=-1; return projection((1,c2,s2),normal=(1,0,0), new transformation(triple,triple,triple) { return transformation(t);}); } projection obliqueY(real angle=45) { transform3 t=identity(4); real c2=Cos(angle)^2; real s2=1-c2; t[0][1]=c2; t[1][1]=s2; t[1][2]=1; t[2][1]=-1; t[2][2]=0; t[2][3]=-1; return projection((c2,-1,s2),normal=(0,-1,0), new transformation(triple,triple,triple) { return transformation(t);}); } projection oblique=oblique(); projection obliqueX=obliqueX(), obliqueY=obliqueY(), obliqueZ=obliqueZ(); projection LeftView=orthographic(-X,showtarget=true); projection RightView=orthographic(X,showtarget=true); projection FrontView=orthographic(-Y,showtarget=true); projection BackView=orthographic(Y,showtarget=true); projection BottomView=orthographic(-Z,up=-Y,showtarget=true); projection TopView=orthographic(Z,up=Y,showtarget=true); currentprojection=perspective(5,4,2); projection projection() { projection P; real[] a=_projection(); if(a.length == 0 || a[10] == 0.0) return currentprojection; int k=0; return a[0] == 1 ? orthographic((a[++k],a[++k],a[++k]),(a[++k],a[++k],a[++k]), (a[++k],a[++k],a[++k]),a[++k],(a[k += 2],a[++k])) : perspective((a[++k],a[++k],a[++k]),(a[++k],a[++k],a[++k]), (a[++k],a[++k],a[++k]),a[++k],a[++k],(a[++k],a[++k])); } // Map pair z to a triple by inverting the projection P onto the // plane perpendicular to normal and passing through point. triple invert(pair z, triple normal, triple point, projection P=currentprojection) { transform3 t=P.t; real[][] A={{t[0][0]-z.x*t[3][0],t[0][1]-z.x*t[3][1],t[0][2]-z.x*t[3][2]}, {t[1][0]-z.y*t[3][0],t[1][1]-z.y*t[3][1],t[1][2]-z.y*t[3][2]}, {normal.x,normal.y,normal.z}}; real[] b={z.x*t[3][3]-t[0][3],z.y*t[3][3]-t[1][3],dot(normal,point)}; real[] x=solve(A,b,warn=false); return x.length > 0 ? (x[0],x[1],x[2]) : P.camera; } // Map pair to a triple on the projection plane. triple invert(pair z, projection P=currentprojection) { return invert(z,P.normal,P.target,P); } // Map pair dir to a triple direction at point v on the projection plane. triple invert(pair dir, triple v, projection P=currentprojection) { return invert(project(v,P)+dir,P.normal,v,P)-v; } pair xypart(triple v) { return (v.x,v.y); } struct control { triple post,pre; bool active=false; bool straight=true; void operator init(triple post, triple pre, bool straight=false) { this.post=post; this.pre=pre; active=true; this.straight=straight; } } control nocontrol; control operator * (transform3 t, control c) { control C; C.post=t*c.post; C.pre=t*c.pre; C.active=c.active; C.straight=c.straight; return C; } void write(file file, control c) { write(file,".. controls "); write(file,c.post); write(file," and "); write(file,c.pre); } struct Tension { real out,in; bool atLeast; bool active; void operator init(real out=1, real in=1, bool atLeast=false, bool active=true) { real check(real val) { if(val < 0.75) abort("tension cannot be less than 3/4"); return val; } this.out=check(out); this.in=check(in); this.atLeast=atLeast; this.active=active; } } Tension operator init() { return Tension(); } Tension noTension; noTension.active=false; void write(file file, Tension t) { write(file,"..tension "); if(t.atLeast) write(file,"atleast "); write(file,t.out); write(file," and "); write(file,t.in); } struct dir { triple dir; real gamma=1; // endpoint curl bool Curl; // curl specified bool active() { return dir != O || Curl; } void init(triple v) { this.dir=v; } void init(real gamma) { if(gamma < 0) abort("curl cannot be less than 0"); this.gamma=gamma; this.Curl=true; } void init(dir d) { dir=d.dir; gamma=d.gamma; Curl=d.Curl; } void default(triple v) { if(!active()) init(v); } void default(dir d) { if(!active()) init(d); } dir copy() { dir d=new dir; d.init(this); return d; } } void write(file file, dir d) { if(d.dir != O) { write(file,"{"); write(file,unit(d.dir)); write(file,"}"); } else if(d.Curl) { write(file,"{curl "); write(file,d.gamma); write(file,"}"); } } dir operator * (transform3 t, dir d) { dir D=d.copy(); D.init(unit(shiftless(t)*d.dir)); return D; } void checkEmpty(int n) { if(n == 0) abort("nullpath3 has no points"); } int adjustedIndex(int i, int n, bool cycles) { checkEmpty(n); if(cycles) return i % n; else if(i < 0) return 0; else if(i >= n) return n-1; else return i; } struct flatguide3 { triple[] nodes; bool[] cyclic; // true if node is really a cycle control[] control; // control points for segment starting at node Tension[] Tension; // Tension parameters for segment starting at node dir[] in,out; // in and out directions for segment starting at node bool cyclic() {int n=cyclic.length; return n > 0 ? cyclic[n-1] : false;} bool precyclic() {int i=find(cyclic); return i >= 0 && i < cyclic.length-1;} int size() { return cyclic() ? nodes.length-1 : nodes.length; } void node(triple v, bool b=false) { nodes.push(v); control.push(nocontrol); Tension.push(noTension); in.push(new dir); out.push(new dir); cyclic.push(b); } void control(triple post, triple pre) { if(control.length > 0) { control c=control(post,pre,false); control[control.length-1]=c; } } void Tension(real out, real in, bool atLeast) { if(Tension.length > 0) Tension[Tension.length-1]=Tension(out,in,atLeast,true); } void in(triple v) { if(in.length > 0) { in[in.length-1].init(v); } } void out(triple v) { if(out.length > 0) { out[out.length-1].init(v); } } void in(real gamma) { if(in.length > 0) { in[in.length-1].init(gamma); } } void out(real gamma) { if(out.length > 0) { out[out.length-1].init(gamma); } } void cycleToken() { if(nodes.length > 0) node(nodes[0],true); } // Return true if outgoing direction at node i is known. bool solved(int i) { return out[i].active() || control[i].active; } } void write(file file, string s="", explicit flatguide3 x, suffix suffix=none) { write(file,s); if(x.size() == 0) write(file,""); else for(int i=0; i < x.nodes.length; ++i) { if(i > 0) write(file,endl); if(x.cyclic[i]) write(file,"cycle"); else write(file,x.nodes[i]); if(i < x.nodes.length-1) { // Explicit control points trump other specifiers if(x.control[i].active) write(file,x.control[i]); else { write(file,x.out[i]); if(x.Tension[i].active) write(file,x.Tension[i]); } write(file,".."); if(!x.control[i].active) write(file,x.in[i]); } } write(file,suffix); } void write(string s="", flatguide3 x, suffix suffix=endl) { write(stdout,s,x,suffix); } // A guide3 is most easily represented as something that modifies a flatguide3. typedef void guide3(flatguide3); restricted void nullpath3(flatguide3) {}; guide3 operator init() {return nullpath3;} guide3 operator cast(triple v) { return new void(flatguide3 f) { f.node(v); }; } guide3 operator cast(cycleToken) { return new void(flatguide3 f) { f.cycleToken(); }; } guide3 operator controls(triple post, triple pre) { return new void(flatguide3 f) { f.control(post,pre); }; }; guide3 operator controls(triple v) { return operator controls(v,v); } guide3 operator cast(tensionSpecifier t) { return new void(flatguide3 f) { f.Tension(t.out, t.in, t.atLeast); }; } guide3 operator cast(curlSpecifier spec) { return new void(flatguide3 f) { if(spec.side == JOIN_OUT) f.out(spec.value); else if(spec.side == JOIN_IN) f.in(spec.value); else abort("invalid curl specifier"); }; } guide3 operator spec(triple v, int side) { return new void(flatguide3 f) { if(side == JOIN_OUT) f.out(v); else if(side == JOIN_IN) f.in(v); else abort("invalid direction specifier"); }; } guide3 operator -- (... guide3[] g) { return new void(flatguide3 f) { if(g.length > 0) { for(int i=0; i < g.length-1; ++i) { g[i](f); f.out(1); f.in(1); } g[g.length-1](f); } }; } guide3 operator .. (... guide3[] g) { return new void(flatguide3 f) { for(int i=0; i < g.length; ++i) g[i](f); }; } typedef guide3 interpolate3(... guide3[]); interpolate3 join3(tensionSpecifier t) { return new guide3(... guide3[] a) { if(a.length == 0) return nullpath3; guide3 g=a[0]; for(int i=1; i < a.length; ++i) g=g..t..a[i]; return g; }; } interpolate3 operator ::=join3(operator tension(1,true)); interpolate3 operator ---=join3(operator tension(infinity,true)); flatguide3 operator cast(guide3 g) { flatguide3 f; g(f); return f; } flatguide3[] operator cast(guide3[] g) { flatguide3[] p=new flatguide3[g.length]; for(int i=0; i < g.length; ++i) { flatguide3 f; g[i](f); p[i]=f; } return p; } // A version of asin that tolerates numerical imprecision real asin1(real x) { return asin(min(max(x,-1),1)); } // A version of acos that tolerates numerical imprecision real acos1(real x) { return acos(min(max(x,-1),1)); } struct Controls { triple c0,c1; // 3D extension of John Hobby's control point formula // (cf. The MetaFont Book, page 131), // as described in John C. Bowman and A. Hammerlindl, // TUGBOAT: The Communications of the TeX Users Group 29:2 (2008). void operator init(triple v0, triple v1, triple d0, triple d1, real tout, real tin, bool atLeast) { triple v=v1-v0; triple u=unit(v); real L=length(v); d0=unit(d0); d1=unit(d1); real theta=acos1(dot(d0,u)); real phi=acos1(dot(d1,u)); if(dot(cross(d0,v),cross(v,d1)) < 0) phi=-phi; c0=v0+d0*L*relativedistance(theta,phi,tout,atLeast); c1=v1-d1*L*relativedistance(phi,theta,tin,atLeast); } } private triple cross(triple d0, triple d1, triple reference) { triple normal=cross(d0,d1); return normal == O ? reference : normal; } private triple dir(real theta, triple d0, triple d1, triple reference) { triple normal=cross(d0,d1,reference); if(normal == O) return d1; return rotate(degrees(theta),dot(normal,reference) >= 0 ? normal : -normal)* d1; } private real angle(triple d0, triple d1, triple reference) { real theta=acos1(dot(unit(d0),unit(d1))); return dot(cross(d0,d1,reference),reference) >= 0 ? theta : -theta; } // 3D extension of John Hobby's angle formula (The MetaFont Book, page 131). // Notational differences: here psi[i] is the turning angle at z[i+1], // beta[i] is the tension for segment i, and in[i] is the incoming // direction for segment i (where segment i begins at node i). real[] theta(triple[] v, real[] alpha, real[] beta, triple dir0, triple dirn, real g0, real gn, triple reference) { real[] a,b,c,f,l,psi; int n=alpha.length; bool cyclic=v.cyclic; for(int i=0; i < n; ++i) l[i]=1/length(v[i+1]-v[i]); int i0,in; if(cyclic) {i0=0; in=n;} else {i0=1; in=n-1;} for(int i=0; i < in; ++i) psi[i]=angle(v[i+1]-v[i],v[i+2]-v[i+1],reference); if(cyclic) { l.cyclic=true; psi.cyclic=true; } else { psi[n-1]=0; if(dir0 == O) { real a0=alpha[0]; real b0=beta[0]; real chi=g0*(b0/a0)^2; a[0]=0; b[0]=3a0-a0/b0+chi; real C=chi*(3a0-1)+a0/b0; c[0]=C; f[0]=-C*psi[0]; } else { a[0]=c[0]=0; b[0]=1; f[0]=angle(v[1]-v[0],dir0,reference); } if(dirn == O) { real an=alpha[n-1]; real bn=beta[n-1]; real chi=gn*(an/bn)^2; a[n]=chi*(3bn-1)+bn/an; b[n]=3bn-bn/an+chi; c[n]=f[n]=0; } else { a[n]=c[n]=0; b[n]=1; f[n]=angle(v[n]-v[n-1],dirn,reference); } } for(int i=i0; i < n; ++i) { real in=beta[i-1]^2*l[i-1]; real A=in/alpha[i-1]; a[i]=A; real B=3*in-A; real out=alpha[i]^2*l[i]; real C=out/beta[i]; b[i]=B+3*out-C; c[i]=C; f[i]=-B*psi[i-1]-C*psi[i]; } return tridiagonal(a,b,c,f); } triple reference(triple[] v, int n, triple d0, triple d1) { triple[] V=sequence(new triple(int i) { return cross(v[i+1]-v[i],v[i+2]-v[i+1]); },n-1); if(n > 0) { V.push(cross(d0,v[1]-v[0])); V.push(cross(v[n]-v[n-1],d1)); } triple max=V[0]; real M=abs(max); for(int i=1; i < V.length; ++i) { triple vi=V[i]; real a=abs(vi); if(a > M) { M=a; max=vi; } } triple reference; for(int i=0; i < V.length; ++i) { triple u=unit(V[i]); reference += dot(u,max) < 0 ? -u : u; } return reference; } // Fill in missing directions for n cyclic nodes. void aim(flatguide3 g, int N) { bool cyclic=true; int start=0, end=0; // If the cycle contains one or more direction specifiers, break the loop. for(int k=0; k < N; ++k) if(g.solved(k)) {cyclic=false; end=k; break;} for(int k=N-1; k >= 0; --k) if(g.solved(k)) {cyclic=false; start=k; break;} while(start < N && g.control[start].active) ++start; int n=N-(start-end); if(n <= 1 || (cyclic && n <= 2)) return; triple[] v=new triple[cyclic ? n : n+1]; real[] alpha=new real[n]; real[] beta=new real[n]; for(int k=0; k < n; ++k) { int K=(start+k) % N; v[k]=g.nodes[K]; alpha[k]=g.Tension[K].out; beta[k]=g.Tension[K].in; } if(cyclic) { v.cyclic=true; alpha.cyclic=true; beta.cyclic=true; } else v[n]=g.nodes[(start+n) % N]; int final=(end-1) % N; triple d0=g.out[start].dir; triple d1=g.in[final].dir; triple reference=reference(v,n,d0,d1); real[] theta=theta(v,alpha,beta,d0,d1,g.out[start].gamma,g.in[final].gamma, reference); v.cyclic=true; theta.cyclic=true; for(int k=1; k < (cyclic ? n+1 : n); ++k) { triple w=dir(theta[k],v[k]-v[k-1],v[k+1]-v[k],reference); g.in[(start+k-1) % N].init(w); g.out[(start+k) % N].init(w); } if(g.out[start].dir == O) g.out[start].init(dir(theta[0],v[0]-g.nodes[(start-1) % N],v[1]-v[0], reference)); if(g.in[final].dir == O) g.in[final].init(dir(theta[n],v[n-1]-v[n-2],v[n]-v[n-1],reference)); } // Fill in missing directions for the sequence of nodes i...n. void aim(flatguide3 g, int i, int n) { int j=n-i; if(j > 1 || g.out[i].dir != O || g.in[i].dir != O) { triple[] v=new triple[j+1]; real[] alpha=new real[j]; real[] beta=new real[j]; for(int k=0; k < j; ++k) { v[k]=g.nodes[i+k]; alpha[k]=g.Tension[i+k].out; beta[k]=g.Tension[i+k].in; } v[j]=g.nodes[n]; triple d0=g.out[i].dir; triple d1=g.in[n-1].dir; triple reference=reference(v,j,d0,d1); real[] theta=theta(v,alpha,beta,d0,d1,g.out[i].gamma,g.in[n-1].gamma, reference); for(int k=1; k < j; ++k) { triple w=dir(theta[k],v[k]-v[k-1],v[k+1]-v[k],reference); g.in[i+k-1].init(w); g.out[i+k].init(w); } if(g.out[i].dir == O) { triple w=dir(theta[0],g.in[i].dir,v[1]-v[0],reference); if(i > 0) g.in[i-1].init(w); g.out[i].init(w); } if(g.in[n-1].dir == O) { triple w=dir(theta[j],g.out[n-1].dir,v[j]-v[j-1],reference); g.in[n-1].init(w); g.out[n].init(w); } } } private real Fuzz=10*realEpsilon; triple XYplane(pair z) {return (z.x,z.y,0);} triple YZplane(pair z) {return (0,z.x,z.y);} triple ZXplane(pair z) {return (z.y,0,z.x);} bool cyclic(guide3 g) {flatguide3 f; g(f); return f.cyclic();} int size(guide3 g) {flatguide3 f; g(f); return f.size();} int length(guide3 g) {flatguide3 f; g(f); return f.nodes.length-1;} triple dir(path3 p) { return dir(p,length(p)); } triple dir(path3 p, path3 h) { return 0.5*(dir(p)+dir(h)); } // return the point on path3 p at arclength L triple arcpoint(path3 p, real L) { return point(p,arctime(p,L)); } // return the direction on path3 p at arclength L triple arcdir(path3 p, real L) { return dir(p,arctime(p,L)); } // return the time on path3 p at the relative fraction l of its arclength real reltime(path3 p, real l) { return arctime(p,l*arclength(p)); } // return the point on path3 p at the relative fraction l of its arclength triple relpoint(path3 p, real l) { return point(p,reltime(p,l)); } // return the direction of path3 p at the relative fraction l of its arclength triple reldir(path3 p, real l) { return dir(p,reltime(p,l)); } // return the initial point of path3 p triple beginpoint(path3 p) { return point(p,0); } // return the point on path3 p at half of its arclength triple midpoint(path3 p) { return relpoint(p,0.5); } // return the final point of path3 p triple endpoint(path3 p) { return point(p,length(p)); } path3 path3(triple v) { triple[] point={v}; return path3(point,point,point,new bool[] {false},false); } path3 path3(path p, triple plane(pair)=XYplane) { int n=size(p); return path3(sequence(new triple(int i) {return plane(precontrol(p,i));},n), sequence(new triple(int i) {return plane(point(p,i));},n), sequence(new triple(int i) {return plane(postcontrol(p,i));},n), sequence(new bool(int i) {return straight(p,i);},n), cyclic(p)); } path3[] path3(explicit path[] g, triple plane(pair)=XYplane) { return sequence(new path3(int i) {return path3(g[i],plane);},g.length); } path3 invert(path p, triple normal, triple point, projection P=currentprojection) { return path3(p,new triple(pair z) {return invert(z,normal,point,P);}); } path3 invert(path p, triple point, projection P=currentprojection) { return path3(p,new triple(pair z) {return invert(z,P.normal,point,P);}); } path3 invert(path p, projection P=currentprojection) { return path3(p,new triple(pair z) {return invert(z,P.normal,P.target,P);}); } // Construct a path from a path3 by applying P to each control point. path path(path3 p, pair P(triple)=xypart) { int n=length(p); if(n < 0) return nullpath; guide g=P(point(p,0)); if(n == 0) return g; for(int i=1; i < n; ++i) g=straight(p,i-1) ? g--P(point(p,i)) : g..controls P(postcontrol(p,i-1)) and P(precontrol(p,i))..P(point(p,i)); if(straight(p,n-1)) return cyclic(p) ? g--cycle : g--P(point(p,n)); pair post=P(postcontrol(p,n-1)); pair pre=P(precontrol(p,n)); return cyclic(p) ? g..controls post and pre..cycle : g..controls post and pre..P(point(p,n)); } void write(file file, string s="", explicit path3 x, suffix suffix=none) { write(file,s); int n=length(x); if(n < 0) write(""); else { for(int i=0; i < n; ++i) { write(file,point(x,i)); if(i < length(x)) { if(straight(x,i)) write(file,"--"); else { write(file,".. controls "); write(file,postcontrol(x,i)); write(file," and "); write(file,precontrol(x,i+1),newl); write(file," .."); } } } if(cyclic(x)) write(file,"cycle",suffix); else write(file,point(x,n),suffix); } } void write(string s="", explicit path3 x, suffix suffix=endl) { write(stdout,s,x,suffix); } void write(file file, string s="", explicit path3[] x, suffix suffix=none) { write(file,s); if(x.length > 0) write(file,x[0]); for(int i=1; i < x.length; ++i) { write(file,endl); write(file," ^^"); write(file,x[i]); } write(file,suffix); } void write(string s="", explicit path3[] x, suffix suffix=endl) { write(stdout,s,x,suffix); } path3 solve(flatguide3 g) { int n=g.nodes.length-1; // If duplicate points occur consecutively, add dummy controls (if absent). for(int i=0; i < n; ++i) { if(g.nodes[i] == g.nodes[i+1] && !g.control[i].active) g.control[i]=control(g.nodes[i],g.nodes[i],straight=true); } // Fill in empty direction specifiers inherited from explicit control points. for(int i=0; i < n; ++i) { if(g.control[i].active) { g.out[i].init(g.control[i].post-g.nodes[i]); g.in[i].init(g.nodes[i+1]-g.control[i].pre); } } // Propagate directions across nodes. for(int i=0; i < n; ++i) { int next=g.cyclic[i+1] ? 0 : i+1; if(g.out[next].active()) g.in[i].default(g.out[next]); if(g.in[i].active()) { g.out[next].default(g.in[i]); g.out[i+1].default(g.in[i]); } } // Compute missing 3D directions. // First, resolve cycles int i=find(g.cyclic); if(i > 0) { aim(g,i); // All other cycles can now be reduced to sequences. triple v=g.out[0].dir; for(int j=i; j <= n; ++j) { if(g.cyclic[j]) { g.in[j-1].default(v); g.out[j].default(v); if(g.nodes[j-1] == g.nodes[j] && !g.control[j-1].active) g.control[j-1]=control(g.nodes[j-1],g.nodes[j-1]); } } } // Next, resolve sequences. int i=0; int start=0; while(i < n) { // Look for a missing outgoing direction. while(i <= n && g.solved(i)) {start=i; ++i;} if(i > n) break; // Look for the end of the sequence. while(i < n && !g.solved(i)) ++i; while(start < i && g.control[start].active) ++start; if(start < i) aim(g,start,i); } // Compute missing 3D control points. for(int i=0; i < n; ++i) { int next=g.cyclic[i+1] ? 0 : i+1; if(!g.control[i].active) { control c; if((g.out[i].Curl && g.in[i].Curl) || (g.out[i].dir == O && g.in[i].dir == O)) { // fill in straight control points for path3 functions triple delta=(g.nodes[i+1]-g.nodes[i])/3; c=control(g.nodes[i]+delta,g.nodes[i+1]-delta,straight=true); } else { Controls C=Controls(g.nodes[i],g.nodes[next],g.out[i].dir,g.in[i].dir, g.Tension[i].out,g.Tension[i].in, g.Tension[i].atLeast); c=control(C.c0,C.c1); } g.control[i]=c; } } // Convert to Knuth's format (control points stored with nodes) int n=g.nodes.length; bool cyclic; if(n > 0) { cyclic=g.cyclic[n-1]; if(cyclic) --n; } triple[] pre=new triple[n]; triple[] point=new triple[n]; triple[] post=new triple[n]; bool[] straight=new bool[n]; if(n > 0) { for(int i=0; i < n-1; ++i) { point[i]=g.nodes[i]; post[i]=g.control[i].post; pre[i+1]=g.control[i].pre; straight[i]=g.control[i].straight; } point[n-1]=g.nodes[n-1]; if(cyclic) { pre[0]=g.control[n-1].pre; post[n-1]=g.control[n-1].post; straight[n-1]=g.control[n-1].straight; } else { pre[0]=point[0]; post[n-1]=point[n-1]; straight[n-1]=false; } } return path3(pre,point,post,straight,cyclic); } path nurb(path3 p, projection P, int ninterpolate=P.ninterpolate) { triple f=P.camera; triple u=unit(P.normal); transform3 t=P.t; path nurb(triple v0, triple v1, triple v2, triple v3) { return nurb(project(v0,t),project(v1,t),project(v2,t),project(v3,t), dot(u,f-v0),dot(u,f-v1),dot(u,f-v2),dot(u,f-v3),ninterpolate); } path g; if(straight(p,0)) g=project(point(p,0),t); int last=length(p); for(int i=0; i < last; ++i) { if(straight(p,i)) g=g--project(point(p,i+1),t); else g=g&nurb(point(p,i),postcontrol(p,i),precontrol(p,i+1),point(p,i+1)); } int n=length(g); if(cyclic(p)) g=g&cycle; return g; } path project(path3 p, projection P=currentprojection, int ninterpolate=P.ninterpolate) { guide g; int last=length(p); if(last < 0) return g; transform3 t=P.t; if(ninterpolate == 1 || piecewisestraight(p)) { g=project(point(p,0),t); // Construct the path. int stop=cyclic(p) ? last-1 : last; for(int i=0; i < stop; ++i) { if(straight(p,i)) g=g--project(point(p,i+1),t); else { g=g..controls project(postcontrol(p,i),t) and project(precontrol(p,i+1),t)..project(point(p,i+1),t); } } } else return nurb(p,P); if(cyclic(p)) g=straight(p,last-1) ? g--cycle : g..controls project(postcontrol(p,last-1),t) and project(precontrol(p,last),t)..cycle; return g; } pair[] project(triple[] v, projection P=currentprojection) { return sequence(new pair(int i) {return project(v[i],P.t);},v.length); } path[] project(explicit path3[] g, projection P=currentprojection) { return sequence(new path(int i) {return project(g[i],P);},g.length); } guide3 operator cast(path3 p) { int last=length(p); bool cyclic=cyclic(p); int stop=cyclic ? last-1 : last; return new void(flatguide3 f) { if(last >= 0) { f.node(point(p,0)); for(int i=0; i < stop; ++i) { if(straight(p,i)) { f.out(1); f.in(1); } else f.control(postcontrol(p,i),precontrol(p,i+1)); f.node(point(p,i+1)); } if(cyclic) { if(straight(p,stop)) { f.out(1); f.in(1); } else f.control(postcontrol(p,stop),precontrol(p,last)); f.cycleToken(); } } }; } // Return a unit normal vector to a planar path p (or O if the path is // nonplanar). triple normal(path3 p) { triple normal; real fuzz=sqrtEpsilon*abs(max(p)-min(p)); real absnormal; real theta; bool Cross(triple a, triple b) { if(abs(a) >= fuzz && abs(b) >= fuzz) { triple n=cross(unit(a),unit(b)); real absn=abs(n); if(absn < sqrtEpsilon) return false; n=unit(n); if(absnormal > 0 && abs(normal-n) > sqrtEpsilon && abs(normal+n) > sqrtEpsilon) return true; else { int sign=dot(n,normal) >= 0 ? 1 : -1; theta += sign*asin1(absn); if(absn > absnormal) { absnormal=absn; normal=n; theta=sign*theta; } } } return false; } int L=length(p); if(L <= 0) return O; triple zi=point(p,0); triple v0=zi-precontrol(p,0); for(int i=0; i < L; ++i) { triple c0=postcontrol(p,i); triple c1=precontrol(p,i+1); triple zp=point(p,i+1); triple v1=c0-zi; triple v2=c1-c0; triple v3=zp-c1; if(Cross(v0,v1) || Cross(v1,v2) || Cross(v2,v3)) return O; v0=v3; zi=zp; } return theta >= 0 ? normal : -normal; } // Return a unit normal vector to a polygon with vertices in p. triple normal(triple[] p) { triple normal; real fuzz=sqrtEpsilon*abs(maxbound(p)-minbound(p)); real absnormal; real theta; bool Cross(triple a, triple b) { if(abs(a) >= fuzz && abs(b) >= fuzz) { triple n=cross(unit(a),unit(b)); real absn=abs(n); n=unit(n); if(absnormal > 0 && absn > sqrtEpsilon && abs(normal-n) > sqrtEpsilon && abs(normal+n) > sqrtEpsilon) return true; else { int sign=dot(n,normal) >= 0 ? 1 : -1; theta += sign*asin1(absn); if(absn > absnormal) { absnormal=absn; normal=n; theta=sign*theta; } } } return false; } if(p.length <= 0) return O; triple zi=p[0]; triple v0=zi-p[p.length-1]; for(int i=0; i < p.length-1; ++i) { triple zp=p[i+1]; triple v1=zp-zi; if(Cross(v0,v1)) return O; v0=v1; zi=zp; } return theta >= 0 ? normal : -normal; } // Transforms that map XY plane to YX, YZ, ZY, ZX, and XZ planes. restricted transform3 XY=identity4; restricted transform3 YX=rotate(-90,O,Z); restricted transform3 YZ=rotate(90,O,Z)*rotate(90,O,X); restricted transform3 ZY=rotate(-90,O,X)*YZ; restricted transform3 ZX=rotate(-90,O,Z)*rotate(-90,O,Y); restricted transform3 XZ=rotate(-90,O,Y)*ZX; private transform3 flip(transform3 t, triple X, triple Y, triple Z, projection P) { static transform3 flip(triple v) { static real s(real x) {return x > 0 ? -1 : 1;} return scale(s(v.x),s(v.y),s(v.z)); } triple u=unit(P.normal); triple up=unit(perp(P.up,u)); bool upright=dot(Z,u) >= 0; if(dot(Y,up) < 0) { t=flip(Y)*t; upright=!upright; } return upright ? t : flip(X)*t; } restricted transform3 XY(projection P=currentprojection) { return flip(XY,X,Y,Z,P); } restricted transform3 YX(projection P=currentprojection) { return flip(YX,Y,X,Z,P); } restricted transform3 YZ(projection P=currentprojection) { return flip(YZ,Y,Z,X,P); } restricted transform3 ZY(projection P=currentprojection) { return flip(ZY,Z,Y,X,P); } restricted transform3 ZX(projection P=currentprojection) { return flip(ZX,Z,X,Y,P); } restricted transform3 XZ(projection P=currentprojection) { return flip(XZ,X,Z,Y,P); } // Transform3 that projects in direction dir onto plane with normal n // through point O. transform3 planeproject(triple n, triple O=O, triple dir=n) { real a=n.x, b=n.y, c=n.z; real u=dir.x, v=dir.y, w=dir.z; real delta=1.0/(a*u+b*v+c*w); real d=-(a*O.x+b*O.y+c*O.z)*delta; return new real[][] { {(b*v+c*w)*delta,-b*u*delta,-c*u*delta,-d*u}, {-a*v*delta,(a*u+c*w)*delta,-c*v*delta,-d*v}, {-a*w*delta,-b*w*delta,(a*u+b*v)*delta,-d*w}, {0,0,0,1} }; } // Transform3 that projects in direction dir onto plane defined by p. transform3 planeproject(path3 p, triple dir=O) { triple n=normal(p); return planeproject(n,point(p,0),dir == O ? n : dir); } // Transform for projecting onto plane through point O with normal cross(u,v). transform transform(triple u, triple v, triple O=O, projection P=currentprojection) { transform3 t=P.t; static real[] O={0,0,0,1}; real[] tO=t*O; real tO3=tO[3]; real factor=1/tO3^2; real[] x=(tO3*t[0]-tO[0]*t[3])*factor; real[] y=(tO3*t[1]-tO[1]*t[3])*factor; triple x=(x[0],x[1],x[2]); triple y=(y[0],y[1],y[2]); u=unit(u); v=unit(v); return (0,0,dot(u,x),dot(v,x),dot(u,y),dot(v,y)); } // Project Label onto plane through point O with normal cross(u,v). Label project(Label L, triple u, triple v, triple O=O, projection P=currentprojection) { Label L=L.copy(); L.position=project(O,P.t); L.transform(transform(u,v,O,P)); return L; } path3 operator cast(guide3 g) {return solve(g);} path3 operator cast(triple v) {return path3(v);} guide3[] operator cast(triple[] v) { return sequence(new guide3(int i) {return v[i];},v.length); } path3[] operator cast(triple[] v) { return sequence(new path3(int i) {return v[i];},v.length); } path3[] operator cast(guide3[] g) { return sequence(new path3(int i) {return solve(g[i]);},g.length); } guide3[] operator cast(path3[] g) { return sequence(new guide3(int i) {return g[i];},g.length); } void write(file file, string s="", explicit guide3[] x, suffix suffix=none) { write(file,s,(path3[]) x,suffix); } void write(string s="", explicit guide3[] x, suffix suffix=endl) { write(stdout,s,(path3[]) x,suffix); } triple point(explicit guide3 g, int t) { flatguide3 f; g(f); int n=f.size(); return f.nodes[adjustedIndex(t,n,f.cyclic())]; } triple[] dirSpecifier(guide3 g, int t) { flatguide3 f; g(f); int n=f.size(); checkEmpty(n); if(f.cyclic()) t=t % n; else if(t < 0 || t >= n-1) return new triple[]; return new triple[] {f.out[t].dir,f.in[t].dir}; } triple[] controlSpecifier(guide3 g, int t) { flatguide3 f; g(f); int n=f.size(); checkEmpty(n); if(f.cyclic()) t=t % n; else if(t < 0 || t >= n-1) return new triple[]; control c=f.control[t]; if(c.active) return new triple[] {c.post,c.pre}; else return new triple[]; } tensionSpecifier tensionSpecifier(guide3 g, int t) { flatguide3 f; g(f); int n=f.size(); checkEmpty(n); if(f.cyclic()) t=t % n; else if(t < 0 || t >= n-1) return operator tension(1,1,false); Tension T=f.Tension[t]; return operator tension(T.out,T.in,T.atLeast); } real[] curlSpecifier(guide3 g, int t) { flatguide3 f; g(f); int n=f.size(); checkEmpty(n); if(f.cyclic()) t=t % n; else if(t < 0 || t >= n-1) return new real[]; return new real[] {f.out[t].gamma,f.in[t].gamma}; } guide3 reverse(guide3 g) { flatguide3 f; bool cyclic=cyclic(g); g(f); if(f.precyclic()) return reverse(solve(g)); int n=f.size(); checkEmpty(n); guide3 G; if(n >= 0) { int start=cyclic ? n : n-1; for(int i=start; i > 0; --i) { G=G..f.nodes[i]; control c=f.control[i-1]; if(c.active) G=G..operator controls(c.pre,c.post); else { dir in=f.in[i-1]; triple d=in.dir; if(d != O) G=G..operator spec(-d,JOIN_OUT); else if(in.Curl) G=G..operator curl(in.gamma,JOIN_OUT); dir out=f.out[i-1]; triple d=out.dir; if(d != O) G=G..operator spec(-d,JOIN_IN); else if(out.Curl) G=G..operator curl(out.gamma,JOIN_IN); } } if(cyclic) G=G..cycle; else G=G..f.nodes[0]; } return G; } triple intersectionpoint(path3 p, path3 q, real fuzz=-1) { real[] t=intersect(p,q,fuzz); if(t.length == 0) abort("paths do not intersect"); return point(p,t[0]); } // return an array containing all intersection points of p and q triple[] intersectionpoints(path3 p, path3 q, real fuzz=-1) { real[][] t=intersections(p,q,fuzz); return sequence(new triple(int i) {return point(p,t[i][0]);},t.length); } triple[] intersectionpoints(explicit path3[] p, explicit path3[] q, real fuzz=-1) { triple[] v; for(int i=0; i < p.length; ++i) for(int j=0; j < q.length; ++j) v.append(intersectionpoints(p[i],q[j],fuzz)); return v; } path3 operator &(path3 p, cycleToken tok) { int n=length(p); if(n < 0) return nullpath3; triple a=point(p,0); triple b=point(p,n); return subpath(p,0,n-1)..controls postcontrol(p,n-1) and precontrol(p,n).. cycle; } // return the point on path3 p at arclength L triple arcpoint(path3 p, real L) { return point(p,arctime(p,L)); } // return the point on path3 p at arclength L triple arcpoint(path3 p, real L) { return point(p,arctime(p,L)); } // return the direction on path3 p at arclength L triple arcdir(path3 p, real L) { return dir(p,arctime(p,L)); } // return the time on path3 p at the relative fraction l of its arclength real reltime(path3 p, real l) { return arctime(p,l*arclength(p)); } // return the point on path3 p at the relative fraction l of its arclength triple relpoint(path3 p, real l) { return point(p,reltime(p,l)); } // return the direction of path3 p at the relative fraction l of its arclength triple reldir(path3 p, real l) { return dir(p,reltime(p,l)); } // return the point on path3 p at half of its arclength triple midpoint(path3 p) { return relpoint(p,0.5); } real relative(Label L, path3 g) { return L.position.relative ? reltime(g,L.relative()) : L.relative(); } // return the linear transformation that maps X,Y,Z to u,v,w. transform3 transform3(triple u, triple v, triple w=cross(u,v)) { return new real[][] { {u.x,v.x,w.x,0}, {u.y,v.y,w.y,0}, {u.z,v.z,w.z,0}, {0,0,0,1} }; } // return the rotation that maps Z to a unit vector u about cross(u,Z), transform3 align(triple u) { real a=u.x; real b=u.y; real c=u.z; real d=a^2+b^2; if(d != 0) { d=sqrt(d); real e=1/d; return new real[][] { {-b*e,-a*c*e,a,0}, {a*e,-b*c*e,b,0}, {0,d,c,0}, {0,0,0,1}}; } return c >= 0 ? identity(4) : diagonal(1,-1,-1,1); } // return a rotation that maps X,Y to the projection plane. transform3 transform3(projection P=currentprojection) { triple w=unit(P.normal); triple v=unit(perp(P.up,w)); if(v == O) v=cross(perp(w),w); triple u=cross(v,w); return u != O ? transform3(u,v,w) : identity(4); } triple[] triples(real[] x, real[] y, real[] z) { if(x.length != y.length || x.length != z.length) abort("arrays have different lengths"); return sequence(new triple(int i) {return (x[i],y[i],z[i]);},x.length); } path3[] operator cast(path3 p) { return new path3[] {p}; } path3[] operator cast(guide3 g) { return new path3[] {(path3) g}; } path3[] operator ^^ (path3 p, path3 q) { return new path3[] {p,q}; } path3[] operator ^^ (path3 p, explicit path3[] q) { return concat(new path3[] {p},q); } path3[] operator ^^ (explicit path3[] p, path3 q) { return concat(p,new path3[] {q}); } path3[] operator ^^ (explicit path3[] p, explicit path3[] q) { return concat(p,q); } path3[] operator * (transform3 t, explicit path3[] p) { return sequence(new path3(int i) {return t*p[i];},p.length); } triple[] operator * (transform3 t, triple[] v) { return sequence(new triple(int i) {return t*v[i];},v.length); } triple[][] operator * (transform3 t, triple[][] v) { triple[][] V=new triple[v.length][]; for(int i=0; i < v.length; ++i) { triple[] vi=v[i]; V[i]=sequence(new triple(int j) {return t*vi[j];},vi.length); } return V; } triple min(explicit path3[] p) { checkEmpty(p.length); triple minp=min(p[0]); for(int i=1; i < p.length; ++i) minp=minbound(minp,min(p[i])); return minp; } triple max(explicit path3[] p) { checkEmpty(p.length); triple maxp=max(p[0]); for(int i=1; i < p.length; ++i) maxp=maxbound(maxp,max(p[i])); return maxp; } path3 randompath3(int n, bool cumulate=true, interpolate3 join=operator ..) { guide3 g; triple w; for(int i=0; i <= n; ++i) { triple z=(unitrand()-0.5,unitrand()-0.5,unitrand()-0.5); if(cumulate) w += z; else w=z; g=join(g,w); } return g; } path3[] box(triple v1, triple v2) { return (v1.x,v1.y,v1.z)-- (v1.x,v1.y,v2.z)-- (v1.x,v2.y,v2.z)-- (v1.x,v2.y,v1.z)-- (v1.x,v1.y,v1.z)-- (v2.x,v1.y,v1.z)-- (v2.x,v1.y,v2.z)-- (v2.x,v2.y,v2.z)-- (v2.x,v2.y,v1.z)-- (v2.x,v1.y,v1.z)^^ (v2.x,v2.y,v1.z)-- (v1.x,v2.y,v1.z)^^ (v1.x,v2.y,v2.z)-- (v2.x,v2.y,v2.z)^^ (v2.x,v1.y,v2.z)-- (v1.x,v1.y,v2.z); } restricted path3[] unitbox=box(O,(1,1,1)); restricted path3 unitcircle3=X..Y..-X..-Y..cycle; restricted path3 unitsquare3=O--X--X+Y--Y--cycle; path3 circle(triple c, real r, triple normal=Z) { path3 p=normal == Z ? unitcircle3 : align(unit(normal))*unitcircle3; return shift(c)*scale3(r)*p; } // return an arc centered at c from triple v1 to v2 (assuming |v2-c|=|v1-c|), // drawing in the given direction. // The normal must be explicitly specified if c and the endpoints are colinear. path3 arc(triple c, triple v1, triple v2, triple normal=O, bool direction=CCW) { v1 -= c; real r=abs(v1); v1=unit(v1); v2=unit(v2-c); if(normal == O) { normal=cross(v1,v2); if(normal == O) abort("explicit normal required for these endpoints"); } transform3 T; bool align=normal != Z; if(align) { T=align(unit(normal)); transform3 Tinv=transpose(T); v1=Tinv*v1; v2=Tinv*v2; } string invalidnormal="invalid normal vector"; real fuzz=sqrtEpsilon*max(abs(v1),abs(v2)); if(abs(v1.z) > fuzz || abs(v2.z) > fuzz) abort(invalidnormal); real[] t1=intersect(unitcircle3,O--2*(v1.x,v1.y,0)); real[] t2=intersect(unitcircle3,O--2*(v2.x,v2.y,0)); if(t1.length == 0 || t2.length == 0) abort(invalidnormal); real t1=t1[0]; real t2=t2[0]; int n=length(unitcircle3); if(direction) { if(t1 >= t2) t1 -= n; } else if(t2 >= t1) t2 -= n; path3 p=subpath(unitcircle3,t1,t2); if(align) p=T*p; return shift(c)*scale3(r)*p; } // return an arc centered at c with radius r from c+r*dir(theta1,phi1) to // c+r*dir(theta2,phi2) in degrees, drawing in the given direction // relative to the normal vector cross(dir(theta1,phi1),dir(theta2,phi2)). // The normal must be explicitly specified if c and the endpoints are colinear. path3 arc(triple c, real r, real theta1, real phi1, real theta2, real phi2, triple normal=O, bool direction) { return arc(c,c+r*dir(theta1,phi1),c+r*dir(theta2,phi2),normal,direction); } // return an arc centered at c with radius r from c+r*dir(theta1,phi1) to // c+r*dir(theta2,phi2) in degrees, drawing drawing counterclockwise // relative to the normal vector cross(dir(theta1,phi1),dir(theta2,phi2)) // iff theta2 > theta1 or (theta2 == theta1 and phi2 >= phi1). // The normal must be explicitly specified if c and the endpoints are colinear. path3 arc(triple c, real r, real theta1, real phi1, real theta2, real phi2, triple normal=O) { return arc(c,r,theta1,phi1,theta2,phi2,normal, theta2 > theta1 || (theta2 == theta1 && phi2 >= phi1) ? CCW : CW); } private real epsilon=1000*realEpsilon; // Return a representation of the plane through point O with normal cross(u,v). path3 plane(triple u, triple v, triple O=O) { return O--O+u--O+u+v--O+v--cycle; } // PRC/OpenGL support include three_light; void draw(frame f, path3 g, material p=currentpen, light light=nolight, string name="", render render=defaultrender, projection P=currentprojection); void begingroup3(frame f, string name="", render render=defaultrender, triple center=O, int interaction=0) { _begingroup3(f,name,render.compression,render.granularity,render.closed, render.tessellate,render.merge == false, render.merge == true,center,interaction); } void begingroup3(picture pic=currentpicture, string name="", render render=defaultrender, triple center=O, int interaction=0) { pic.add(new void(frame f, transform3, picture pic, projection) { if(is3D()) begingroup3(f,name,render,center,interaction); if(pic != null) begingroup(pic); },true); } void endgroup3(picture pic=currentpicture) { pic.add(new void(frame f, transform3, picture pic, projection) { if(is3D()) endgroup3(f); if(pic != null) endgroup(pic); },true); } void addPath(picture pic, path3 g, pen p) { if(size(g) > 0) pic.addBox(min(g),max(g),min3(p),max3(p)); } include three_surface; include three_margins; pair min(path3 p, projection P) { path3 q=P.T.modelview*p; if(P.infinity) return xypart(min(q)); return maxratio(q)/P.T.projection[3][2]; } pair max(path3 p, projection P) { path3 q=P.T.modelview*p; if(P.infinity) return xypart(max(q)); return minratio(q)/P.T.projection[3][2]; } pair min(frame f, projection P) { frame g=P.T.modelview*f; if(P.infinity) return xypart(min3(g)); return maxratio(g)/P.T.projection[3][2]; } pair max(frame f, projection P) { frame g=P.T.modelview*f; if(P.infinity) return xypart(max3(g)); return minratio(g)/P.T.projection[3][2]; } void draw(picture pic=currentpicture, Label L="", path3 g, align align=NoAlign, material p=currentpen, margin3 margin=NoMargin3, light light=nolight, string name="", render render=defaultrender) { pen q=(pen) p; pic.add(new void(frame f, transform3 t, picture pic, projection P) { path3 G=margin(t*g,q).g; if(is3D()) { draw(f,G,p,light,name,render,null); if(pic != null && size(G) > 0) pic.addBox(min(G,P),max(G,P),min(q),max(q)); } if(pic != null) draw(pic,project(G,P),q); },true); Label L=L.copy(); L.align(align); if(L.s != "") { L.p(q); label(pic,L,g); } addPath(pic,g,q); } include three_tube; draw=new void(frame f, path3 g, material p=currentpen, light light=nolight, string name="", render render=defaultrender, projection P=currentprojection) { pen q=(pen) p; if(is3D()) { p=material(p); real width=linewidth(q); void drawthick(path3 g) { if(settings.thick) { if(width > 0) { bool prc=prc(); void cylinder(transform3) {}; void sphere(transform3, bool half) {}; void disk(transform3) {}; void pipe(path3, path3); if(prc) { cylinder=new void(transform3 t) {drawPRCcylinder(f,t,p,light);}; sphere=new void(transform3 t, bool half) {drawPRCsphere(f,t,half,p,light,render);}; disk=new void(transform3 t) {draw(f,t*unitdisk,p,light,render);}; pipe=new void(path3 center, path3 g) {drawPRCtube(f,center,g,p,light);}; } real linecap=linecap(q); real r=0.5*width; bool open=!cyclic(g); int L=length(g); triple g0=point(g,0); triple gL=point(g,L); if(open && L > 0) { if(linecap == 2) { g0 -= r*dir(g,0); gL += r*dir(g,L); g=g0..g..gL; L += 2; } } tube T=tube(g,width,render,cylinder,sphere,pipe); path3 c=T.center; if(L >= 0) { if(open) { int Lc=length(c); triple c0=point(c,0); triple cL=point(c,Lc); triple dir0=dir(g,0); triple dirL=dir(g,L); triple dirc0=dir(c,0); triple dircL=dir(c,Lc); transform3 t0=shift(g0)*align(-dir0); transform3 tL=shift(gL)*align(dirL); transform3 tc0=shift(c0)*align(-dirc0); transform3 tcL=shift(cL)*align(dircL); if(linecap == 0 || linecap == 2) { transform3 scale2r=scale(r,r,1); T.s.append(t0*scale2r*unitdisk); disk(tc0*scale2r); if(L > 0) { T.s.append(tL*scale2r*unitdisk); disk(tcL*scale2r); } } else if(linecap == 1) { transform3 scale3r=scale3(r); T.s.append(t0*scale3r* (dir0 != O ? unithemisphere : unitsphere)); sphere(tc0*scale3r,half=straight(c,0)); if(L > 0) { T.s.append(tL*scale3r* (dirL != O ? unithemisphere : unitsphere)); sphere(tcL*scale3r,half=straight(c,Lc-1)); } } } if(opacity(q) == 1) _draw(f,c,q); } for(patch s : T.s.s) draw3D(f,s,p,light,prc=false); } else _draw(f,g,q); } else _draw(f,g,q); } bool group=q != nullpen && (name != "" || render.defaultnames); if(group) begingroup3(f,name == "" ? "curve" : name,render); if(linetype(q).length == 0) drawthick(g); else { real[] dash=linetype(adjust(q,arclength(g),cyclic(g))); if(sum(dash) > 0) { dash.cyclic=true; real offset=offset(q); real L=arclength(g); int i=0; real l=offset; while(l <= L) { real t1=arctime(g,l); l += dash[i]; real t2=arctime(g,min(l,L)); drawthick(subpath(g,t1,t2)); ++i; l += dash[i]; ++i; } } } if(group) endgroup3(f); } else draw(f,project(g,P),q); }; void draw(frame f, explicit path3[] g, material p=currentpen, light light=nolight, string name="", render render=defaultrender, projection P=currentprojection) { bool group=g.length > 1 && (name != "" || render.defaultnames); if(group) begingroup3(f,name == "" ? "curves" : name,render); for(int i=0; i < g.length; ++i) draw(f,g[i],p,light,partname(i,render),render,P); if(group) endgroup3(f); } void draw(picture pic=currentpicture, explicit path3[] g, material p=currentpen, margin3 margin=NoMargin3, light light=nolight, string name="", render render=defaultrender) { bool group=g.length > 1 && (name != "" || render.defaultnames); if(group) begingroup3(pic,name == "" ? "curves" : name,render); for(int i=0; i < g.length; ++i) draw(pic,g[i],p,margin,light,partname(i,render),render); if(group) endgroup3(pic); } include three_arrows; void draw(picture pic=currentpicture, Label L="", path3 g, align align=NoAlign, material p=currentpen, arrowbar3 arrow, arrowbar3 bar=None, margin3 margin=NoMargin3, light light=nolight, light arrowheadlight=currentlight, string name="", render render=defaultrender) { bool group=arrow != None || bar != None; if(group) begingroup3(pic,name,render); bool drawpath=arrow(pic,g,p,margin,light,arrowheadlight); if(bar(pic,g,p,margin,light,arrowheadlight) && drawpath) draw(pic,L,g,align,p,margin,light,render); if(group) endgroup3(pic); if(L.s != "") label(pic,L,g,align,(pen) p); } void draw(frame f, path3 g, material p=currentpen, arrowbar3 arrow, light light=nolight, light arrowheadlight=currentlight, string name="", render render=defaultrender, projection P=currentprojection) { picture pic; bool group=arrow != None; if(group) begingroup3(f,name,render); if(arrow(pic,g,p,NoMargin3,light,arrowheadlight)) draw(f,g,p,light,render,P); add(f,pic.fit()); if(group) endgroup3(f); } void add(picture pic=currentpicture, void d(picture,transform3), bool exact=false) { pic.add(d,exact); } // Fit the picture src using the identity transformation (so user // coordinates and truesize coordinates agree) and add it about the point // position to picture dest. void add(picture dest, picture src, triple position, bool group=true, bool above=true) { dest.add(new void(picture f, transform3 t) { f.add(shift(t*position)*src,group,above); }); } void add(picture src, triple position, bool group=true, bool above=true) { add(currentpicture,src,position,group,above); } // Align an arrow pointing to b from the direction dir. The arrow is // 'length' PostScript units long. void arrow(picture pic=currentpicture, Label L="", triple b, triple dir, real length=arrowlength, align align=NoAlign, pen p=currentpen, arrowbar3 arrow=Arrow3, margin3 margin=EndMargin3, light light=nolight, light arrowheadlight=currentlight, string name="", render render=defaultrender) { Label L=L.copy(); if(L.defaultposition) L.position(0); L.align(L.align,dir); L.p(p); picture opic; marginT3 margin=margin(b--b,p); // Extract margin.begin and margin.end triple a=(margin.begin+length+margin.end)*unit(dir); draw(opic,L,a--O,align,p,arrow,margin,light,arrowheadlight,name,render); add(pic,opic,b); } void arrow(picture pic=currentpicture, Label L="", triple b, pair dir, real length=arrowlength, align align=NoAlign, pen p=currentpen, arrowbar3 arrow=Arrow3, margin3 margin=EndMargin3, light light=nolight, light arrowheadlight=currentlight, string name="", render render=defaultrender, projection P=currentprojection) { arrow(pic,L,b,invert(dir,b,P),length,align,p,arrow,margin,light, arrowheadlight,name,render); } triple min3(picture pic, projection P=currentprojection) { return pic.min3(P); } triple max3(picture pic, projection P=currentprojection) { return pic.max3(P); } triple size3(picture pic, bool user=false, projection P=currentprojection) { transform3 t=pic.calculateTransform3(P); triple M=pic.max(t); triple m=pic.min(t); if(!user) return M-m; t=inverse(t); return t*M-t*m; } triple point(frame f, triple dir) { triple m=min3(f); triple M=max3(f); return m+realmult(rectify(dir),M-m); } triple point(picture pic=currentpicture, triple dir, bool user=true, projection P=currentprojection) { triple min = pic.userMin(), max = pic.userMax(); triple v=min+realmult(rectify(dir),max-min); return user ? v : pic.calculateTransform3(P)*v; } triple truepoint(picture pic=currentpicture, triple dir, bool user=true, projection P=currentprojection) { transform3 t=pic.calculateTransform3(P); triple m=pic.min(t); triple M=pic.max(t); triple v=m+realmult(rectify(dir),M-m); return user ? inverse(t)*v : v; } void add(picture dest=currentpicture, object src, pair position=0, pair align=0, bool group=true, filltype filltype=NoFill, bool above=true) { if(prc()) label(dest,src,position,align); else if(settings.render == 0) plain.add(dest,src,position,align,group,filltype,above); } private struct viewpoint { triple target,camera,up; real angle; void operator init(string s) { s=replace(s,'\n'," "); string[] S=split(s); int pos(string s, string key) { int pos=find(s,key); if(pos < 0) return -1; pos += length(key); while(substr(s,pos,1) == " ") ++pos; if(substr(s,pos,1) == "=") return pos+1; return -1; } triple C2C=X; real ROO=1; real ROLL=0; angle=30; int pos; for(int k=0; k < S.length; ++k) { if((pos=pos(S[k],"COO")) >= 0) target=((real) substr(S[k],pos),(real) S[++k],(real) S[++k]); else if((pos=pos(S[k],"C2C")) >= 0) C2C=((real) substr(S[k],pos),(real) S[++k],(real) S[++k]); else if((pos=pos(S[k],"ROO")) >= 0) ROO=(real) substr(S[k],pos); else if((pos=pos(S[k],"ROLL")) >= 0) ROLL=(real) substr(S[k],pos); else if((pos=pos(S[k],"AAC")) >= 0) angle=(real) substr(S[k],pos); } camera=target+ROO*C2C; triple u=unit(target-camera); triple w=unit(Z-u.z*u); up=rotate(ROLL,O,u)*w; } } projection perspective(string s) { viewpoint v=viewpoint(s); projection P=perspective(v.camera,v.up,v.target); P.angle=v.angle; P.absolute=true; return P; } projection absorthographic(triple camera=Z, triple target=O, real roll=0) { triple u=unit(target-camera); triple w=unit(Z-u.z*u); triple up=rotate(roll,O,u)*w; projection P= projection(camera,up,target,1,0,false,false, new transformation(triple camera, triple up, triple target) {return transformation(look(camera,up,target));}); P.absolute=true; return P; } projection absperspective(triple camera=Z, triple target=O, real roll=0, real angle=30) { triple u=unit(target-camera); triple w=unit(Z-u.z*u); triple up=rotate(roll,O,u)*w; projection P=perspective(camera,up,target); P.angle=angle; P.absolute=true; return P; } private string Format(real x) { assert(abs(x) < 1e17,"Number too large: "+string(x)); return format("%.9f",x,"C"); } private string Format(triple v, string sep=" ") { return Format(v.x)+sep+Format(v.y)+sep+Format(v.z); } private string Format(real[] c) { return Format((c[0],c[1],c[2])); } private string format(triple v, string sep=" ") { return string(v.x)+sep+string(v.y)+sep+string(v.z); } private string Format(transform3 t, string sep=" ") { return Format(t[0][0])+sep+Format(t[1][0])+sep+Format(t[2][0])+sep+ Format(t[0][1])+sep+Format(t[1][1])+sep+Format(t[2][1])+sep+ Format(t[0][2])+sep+Format(t[1][2])+sep+Format(t[2][2])+sep+ Format(t[0][3])+sep+Format(t[1][3])+sep+Format(t[2][3]); } string lightscript(light light) { string script="for(var i=scene.lights.count-1; i >= 0; i--) scene.lights.removeByIndex(i);"+'\n\n'; for(int i=0; i < light.position.length; ++i) { string Li="L"+string(i); real[] diffuse=light.diffuse[i]; script += Li+"=scene.createLight();"+'\n'+ Li+".direction.set("+format(-light.position[i],",")+");"+'\n'+ Li+".color.set("+format((diffuse[0],diffuse[1],diffuse[2]),",")+");"+'\n'; } // Work around initialization bug in Adobe Reader 8.0: return script +" scene.lightScheme=scene.LIGHT_MODE_HEADLAMP; scene.lightScheme=scene.LIGHT_MODE_FILE; "; } void writeJavaScript(string name, string preamble, string script) { file out=output(name); write(out,preamble); if(script != "") { write(out,endl); file in=input(script); while(true) { string line=in; if(eof(in)) break; write(out,line,endl); } } close(out); if(settings.verbose > 1) write("Wrote "+name); if(!settings.inlinetex) file3.push(name); } pair viewportmargin(pair lambda) { return maxbound(0.5*(viewportsize-lambda),viewportmargin); } string embed3D(string prefix, string label=prefix, string text=label, frame f, string format="", real width=0, real height=0, string options="", string script="", light light=currentlight, projection P=currentprojection, real viewplanesize=0) { if(!prc(format) || Embed == null) return ""; if(width == 0) width=settings.paperwidth; if(height == 0) height=settings.paperheight; if(script == "") script=defaultembed3Dscript; if(P.infinity) { if(viewplanesize==0) { triple lambda=max3(f)-min3(f); pair margin=viewportmargin((lambda.x,lambda.y)); viewplanesize=(max(lambda.x+2*margin.x,lambda.y+2*margin.y))/P.zoom; } } else if(!P.absolute) P.angle=2*aTan(Tan(0.5*P.angle)); shipout3(prefix,f); string name=prefix+".js"; // Adobe Reader doesn't appear to support user-specified viewport lights. bool lightscript=light.on() && !light.viewport; if(lightscript) writeJavaScript(name,lightscript(light),script); if(!settings.inlinetex && !prconly()) file3.push(prefix+".prc"); static transform3 flipxz=xscale3(-1)*zscale3(-1); transform3 inv=inverse(flipxz*P.T.modelview); string options3="3Dlights="+ (light.on() ? (light.viewport ? "Headlamp" : "File") : "None"); if(defaultembed3Doptions != "") options3 += ","+defaultembed3Doptions; if((settings.render < 0 || !settings.embed) && settings.auto3D) options3 += ",activate=pagevisible"; options3 += ",3Dtoolbar="+(settings.toolbar ? "true" : "false")+ ",label="+label+ (P.infinity ? ",3Dortho="+Format(1/viewplanesize) : ",3Daac="+Format(P.angle))+ ",3Dc2w="+Format(inv)+ ",3Droo="+Format(abs(P.vector()))+ ",3Dpsob="+(P.infinity ? "Max" : "H")+ ",3Dbg="+Format(light.background()); if(options != "") options3 += ","+options; if(settings.inlinetex) prefix=jobname(prefix); if(lightscript) options3 += ",add3Djscript="+prefix+".js"; options3 += ",add3Djscript=asylabels.js"; return text == "" ? Embed(prefix+".prc","",options3,width,height) : "\hbox to 0pt{"+text+"\hss}"+Embed(prefix+".prc","\phantom{"+text+"}", options3); } struct scene { frame f; transform3 t; projection P; bool adjusted; real width,height; pair viewportmargin; transform3 T=identity4; picture pic2; void operator init(frame f, real width, real height, projection P=currentprojection) { this.f=f; this.t=identity4; this.P=P; this.width=width; this.height=height; } void operator init(picture pic, real xsize=pic.xsize, real ysize=pic.ysize, bool keepAspect=pic.keepAspect, bool is3D=true, projection P=currentprojection) { real xsize3=pic.xsize3, ysize3=pic.ysize3, zsize3=pic.zsize3; bool warn=true; if(xsize3 == 0 && ysize3 == 0 && zsize3 == 0) { xsize3=ysize3=zsize3=max(xsize,ysize); warn=false; } if(P.absolute) this.P=P.copy(); else if(P.showtarget) draw(pic,P.target,nullpen); t=pic.scaling(xsize3,ysize3,zsize3,keepAspect,warn); adjusted=false; triple m=pic.min(t); triple M=pic.max(t); if(!P.absolute) { this.P=t*P; if(this.P.center && settings.render != 0) { triple target=0.5*(m+M); this.P.target=target; this.P.calculate(); } if(this.P.autoadjust || this.P.infinity) adjusted=adjusted | this.P.adjust(m,M); } bool scale=xsize != 0 || ysize != 0; bool scaleAdjust=scale && this.P.autoadjust; bool noAdjust=(this.P.absolute || !scaleAdjust); if(pic.bounds3.exact && noAdjust) this.P.bboxonly=false; f=pic.fit3(t,pic.bounds3.exact ? pic2 : null,this.P); if(!pic.bounds3.exact) { if(noAdjust) this.P.bboxonly=false; transform3 s=pic.scale3(f,xsize3,ysize3,zsize3,keepAspect); t=s*t; this.P=s*this.P; f=pic.fit3(t,pic2,this.P); } if(is3D || scale) { pic2.bounds.exact=true; transform s=pic2.scaling(xsize,ysize,keepAspect); pair m2=pic2.min(s); pair M2=pic2.max(s); pair lambda=M2-m2; viewportmargin=viewportmargin(lambda); width=ceil(lambda.x+2*viewportmargin.x); height=ceil(lambda.y+2*viewportmargin.y); if(!this.P.absolute) { if(scaleAdjust) { pair v=(s.xx,s.yy); transform3 T=this.P.t; pair x=project(X,T); pair y=project(Y,T); pair z=project(Z,T); real f(pair a, pair b) { return b == 0 ? (0.5*(a.x+a.y)) : (b.x^2*a.x+b.y^2*a.y)/(b.x^2+b.y^2); } transform3 s=keepAspect ? scale3(min(f(v,x),f(v,y),f(v,z))) : xscale3(f(v,x))*yscale3(f(v,y))*zscale3(f(v,z)); s=shift(this.P.target)*s*shift(-this.P.target); t=s*t; this.P=s*this.P; this.P.bboxonly=false; if(!is3D) pic2.erase(); f=pic.fit3(t,is3D ? null : pic2,this.P); } if(this.P.autoadjust || this.P.infinity) adjusted=adjusted | this.P.adjust(min3(f),max3(f)); } } } // Choose the angle to be just large enough to view the entire image. real angle(projection P) { T=identity4; real h=-0.5*P.target.z; pair r,R; real diff=realMax; pair s; int i; do { r=minratio(f); R=maxratio(f); pair lasts=s; if(P.autoadjust) { s=r+R; if(s != 0) { transform3 t=shift(h*s.x,h*s.y,0); f=t*f; T=t*T; adjusted=true; } } diff=abs(s-lasts); ++i; } while (diff > angleprecision && i < maxangleiterations); real aspect=width > 0 ? height/width : 1; real rx=-r.x*aspect; real Rx=R.x*aspect; real ry=-r.y; real Ry=R.y; if(!P.autoadjust) { if(rx > Rx) Rx=rx; else rx=Rx; if(ry > Ry) Ry=ry; else ry=Ry; } return (1+angleprecision)*max(aTan(rx)+aTan(Rx),aTan(ry)+aTan(Ry)); } } object embed(string prefix=outprefix(), string label=prefix, string text=label, scene S, string format="", bool view=true, string options="", string script="", light light=currentlight) { object F; transform3 modelview; projection P=S.P; transform3 tinv=inverse(S.t); projection Q; triple orthoshift; modelview=P.T.modelview; transform3 inv; if(P.absolute) { Q=modelview*P; inv=inverse(modelview); } else { triple target=P.target; S.f=modelview*S.f; P=modelview*P; Q=P.copy(); if(P.infinity) { triple m=min3(S.f); triple M=max3(S.f); triple lambda=M-m; S.viewportmargin=viewportmargin((lambda.x,lambda.y)); S.width=ceil(lambda.x+2*S.viewportmargin.x); S.height=ceil(lambda.y+2*S.viewportmargin.y); orthoshift=(-0.5(m.x+M.x),-0.5*(m.y+M.y),0); S.f=shift(orthoshift)*S.f; // Eye will be at (0,0,0) inv=inverse(modelview); } else { if(P.angle == 0) { P.angle=S.angle(P); modelview=S.T*modelview; if(S.viewportmargin.y != 0) P.angle=2*aTan(Tan(0.5*P.angle)-S.viewportmargin.y/P.target.z); } inv=inverse(modelview); Q.angle=P.angle; if(settings.verbose > 0) { if(S.adjusted) write("adjusting camera to ",tinv*inv*P.camera); target=inv*P.target; } P=S.T*P; } if(settings.verbose > 0) { if((P.center && settings.render != 0) || (!P.infinity && P.autoadjust)) write("adjusting target to ",tinv*target); } } light Light=modelview*light; if(prefix == "") prefix=outprefix(); bool prc=prc(format); bool preview=settings.render > 0 && !prconly(); if(prc) { // The media9.sty package cannot handle spaces or dots in filenames. string dir=stripfile(prefix); prefix=dir+replace(stripdirectory(prefix), new string[][]{{" ","_"},{".","_"}}); if((settings.embed || nativeformat() == "pdf") && !prconly()) prefix += "+"+(string) file3.length; } else preview=false; if(preview || (!prc && settings.render != 0)) { frame f=S.f; triple m,M; real zcenter; real r; if(P.absolute) { f=modelview*f; m=min3(f); M=max3(f); r=0.5*abs(M-m); zcenter=0.5*(M.z+m.z); } else { m=min3(f); M=max3(f); zcenter=P.target.z; r=P.distance(m,M); } M=(M.x,M.y,zcenter+r); m=(m.x,m.y,zcenter-r); if(P.infinity) { triple margin=(S.viewportmargin.x,S.viewportmargin.y,0); M += margin; m -= margin; } else if(M.z >= 0) abort("camera too close"); shipout3(prefix,f,preview ? nativeformat() : format, S.width-defaultrender.margin,S.height-defaultrender.margin, P.infinity ? 0 : 2aTan(Tan(0.5*P.angle)*P.zoom), P.zoom,m,M,P.viewportshift, tinv*inv*shift(0,0,zcenter),Light.background(),Light.position, Light.diffuse,Light.ambient,Light.specular, Light.viewport,view && !preview); if(!preview) return F; } string image; if((preview || (prc && settings.render == 0)) && settings.embed) { image=prefix; if(settings.inlinetex) image += "_0"; if(!preview && !shipped && !S.pic2.empty2()) { transform T=S.pic2.scaling(S.width,S.height); shipout(image,S.pic2.fit(T),newframe,nativeformat(),false,false,null); } image += "."+nativeformat(); if(!settings.inlinetex) file3.push(image); image=graphic(image,"hiresbb"); } if(prc) { if(P.viewportshift != 0) { if(!P.infinity) warning("offaxis", "PRC does not support off-axis projections; use pan instead of shift"); triple lambda=max3(S.f)-min3(S.f); Q.target -= (P.viewportshift.x*lambda.x/P.zoom, P.viewportshift.y*lambda.y/P.zoom,0); } real viewplanesize=0; if(P.absolute) { if(P.infinity) { S.f=modelview*S.f; triple lambda=max3(S.f)-min3(S.f); pair margin=viewportmargin((lambda.x,lambda.y)); viewplanesize=(max(lambda.x+2*margin.x,lambda.y+2*margin.y))/Q.zoom; S.f=inv*S.f; } Q=inv*Q; } else { if(P.infinity) { triple lambda=max3(S.f)-min3(S.f); pair margin=viewportmargin((lambda.x,lambda.y)); viewplanesize=(max(lambda.x+2*margin.x,lambda.y+2*margin.y))/(Q.zoom); transform3 t=inv*shift(-orthoshift); Q=t*Q; S.f=t*S.f; } else { Q=inv*Q; S.f=inv*S.f; } } F.L=embed3D(prefix,label,text=image,S.f,format, S.width-2,S.height-2,options,script,light,Q,viewplanesize); } return F; } object embed(string prefix=outprefix(), string label=prefix, string text=label, picture pic, string format="", real xsize=pic.xsize, real ysize=pic.ysize, bool keepAspect=pic.keepAspect, bool view=true, string options="", string script="", light light=currentlight, projection P=currentprojection) { bool is3D=is3D(format); scene S=scene(pic,xsize,ysize,keepAspect,is3D,P); if(is3D) return embed(prefix,label,text,S,format,view,options,script,light); else { object F; transform T=S.pic2.scaling(xsize,ysize,keepAspect); F.f=pic.fit(scale(S.t[0][0])*T); add(F.f,S.pic2.fit()); return F; } } object embed(string prefix=outprefix(), string label=prefix, string text=label, frame f, string format="", real width=0, real height=0, bool view=true, string options="", string script="", light light=currentlight, projection P=currentprojection) { if(is3D(format)) return embed(label,text,prefix,scene(f,width,height,P),format,view,options, script,light); else { object F; F.f=f; return F; } } embed3=new object(string prefix, frame f, string format, string options, string script, light light, projection P) { return embed(prefix=prefix,f,format,options,script,light,P); }; frame embedder(object embedder(string prefix, string format), string prefix, string format, bool view, light light) { frame f; bool prc=prc(format); if(!prc && settings.render != 0 && !view) { static int previewcount=0; bool keep=prefix != ""; prefix=outprefix(prefix)+"+"+(string) previewcount; ++previewcount; format=nativeformat(); if(!keep) file3.push(prefix+"."+format); } object F=embedder(prefix,format); if(prc) label(f,F.L); else { if(settings.render == 0) { add(f,F.f); if(light.background != nullpen) box(f,light.background,Fill,above=false); } else if(!view) label(f,graphic(prefix,"hiresbb")); } return f; } currentpicture.fitter=new frame(string prefix, picture pic, string format, real xsize, real ysize, bool keepAspect, bool view, string options, string script, light light, projection P) { frame f; bool empty3=pic.empty3(); if(!empty3) f=embedder(new object(string prefix, string format) { return embed(prefix=prefix,pic,format,xsize,ysize,keepAspect,view, options,script,light,P); },prefix,format,view,light); if(is3D(format) || empty3) add(f,pic.fit2(xsize,ysize,keepAspect)); return f; }; frame embedder(string prefix, frame f, string format, real width, real height, bool view, string options, string script, light light, projection P) { return embedder(new object(string prefix, string format) { return embed(prefix=prefix,f,format,width,height,view,options,script, light,P); },prefix,format,view,light); } projection[][] ThreeViewsUS={{TopView}, {FrontView,RightView}}; projection[][] SixViewsUS={{null,TopView}, {LeftView,FrontView,RightView,BackView}, {null,BottomView}}; projection[][] ThreeViewsFR={{RightView,FrontView}, {null,TopView}}; projection[][] SixViewsFR={{null,BottomView}, {RightView,FrontView,LeftView,BackView}, {null,TopView}}; projection[][] ThreeViews={{FrontView,TopView,RightView}}; projection[][] SixViews={{FrontView,TopView,RightView}, {BackView,BottomView,LeftView}}; void addViews(picture dest, picture src, projection[][] views=SixViewsUS, bool group=true, filltype filltype=NoFill) { frame[][] F=array(views.length,new frame[]); pair[][] M=array(views.length,new pair[]); pair[][] m=array(views.length,new pair[]); for(int i=0; i < views.length; ++i) { projection[] viewsi=views[i]; frame[] Fi=F[i]; pair[] Mi=M[i]; pair[] mi=m[i]; for(projection P : viewsi) { if(P != null) { frame f=src.fit(P); mi.push(min(f)); Mi.push(max(f)); Fi.push(f); } else { pair Infinity=(infinity,infinity); mi.push(Infinity); Mi.push(-Infinity); Fi.push(newframe); } } } real[] my=new real[views.length]; real[] My=new real[views.length]; int Nj=0; for(int i=0; i < views.length; ++i) { my[i]=minbound(m[i]).y; My[i]=maxbound(M[i]).y; Nj=max(Nj,views[i].length); } real[] mx=array(Nj,infinity); real[] Mx=array(Nj,-infinity); for(int i=0; i < views.length; ++i) { pair[] mi=m[i]; pair[] Mi=M[i]; for(int j=0; j < views[i].length; ++j) { mx[j]=min(mx[j],mi[j].x); Mx[j]=max(Mx[j],Mi[j].x); } } if(group) begingroup(dest); real y; for(int i=0; i < views.length; ++i) { real x; pair[] mi=m[i]; for(int j=0; j < views[i].length; ++j) { if(size(F[i][j]) != 0) add(dest,shift(x-mx[j],y+my[i])*F[i][j],filltype); x += (Mx[j]-mx[j]); } y -= (My[i]-my[i]); } if(group) endgroup(dest); } void addViews(picture src, projection[][] views=SixViewsUS, bool group=true, filltype filltype=NoFill) { addViews(currentpicture,src,views,group,filltype); } void addStereoViews(picture dest, picture src, bool group=true, filltype filltype=NoFill, real eyetoview=defaulteyetoview, bool leftright=true, projection P=currentprojection) { triple v=P.vector(); triple h=0.5*abs(v)*eyetoview*unit(cross(P.up,v)); projection leftEye=P.copy(); leftEye.camera -= h; leftEye.calculate(); projection rightEye=P.copy(); rightEye.camera += h; rightEye.calculate(); addViews(dest,src,leftright ? new projection[][] {{leftEye,rightEye}} : new projection[][] {{rightEye,leftEye}},group,filltype); } void addStereoViews(picture src, bool group=true, filltype filltype=NoFill, real eyetoview=defaulteyetoview, bool leftright=true, projection P=currentprojection) { addStereoViews(currentpicture,src,group,filltype,eyetoview,leftright,P); } // Fit an array of 3D pictures simultaneously using the sizing of picture all. frame[] fit3(string prefix="", picture[] pictures, picture all, string format="", bool view=true, string options="", string script="", light light=currentlight, projection P=currentprojection) { frame[] out; scene S=scene(all,P); triple m=all.min(S.t); triple M=all.max(S.t); out=new frame[pictures.length]; int i=0; bool reverse=settings.reverse; settings.animating=true; for(picture pic : pictures) { picture pic2; frame f=pic.fit3(S.t,pic2,S.P); if(settings.interrupt) break; add(f,pic2.fit2()); draw(f,m,nullpen); draw(f,M,nullpen); out[i]=f; ++i; } while(!settings.interrupt) { for(int i=settings.reverse ? pictures.length-1 : 0; i >= 0 && i < pictures.length && !settings.interrupt; settings.reverse ? --i : ++i) { frame f=embedder(prefix,out[i],format,S.width,S.height,view,options, script,light,S.P); if(!settings.loop) out[i]=f; } if(!settings.loop) break; } settings.animating=false; settings.interrupt=false; settings.reverse=reverse; return out; } // Fit an array of pictures simultaneously using the size of the first picture. fit=new frame[](string prefix="", picture[] pictures, string format="", bool view=true, string options="", string script="", projection P=currentprojection) { if(pictures.length == 0) return new frame[]; picture all; size(all,pictures[0]); for(picture pic : pictures) add(all,pic); return all.empty3() ? fit2(pictures,all) : fit3(prefix,pictures,all,format,view,options,script,P); }; // Add frame src to picture dest about position. void add(picture dest=currentpicture, frame src, triple position) { if(is3D(src)) { dest.add(new void(frame f, transform3 t, picture, projection) { add(f,shift(t*position)*src); },true); } else { dest.add(new void(frame, transform3 t, picture pic, projection P) { if(pic != null) { pic.add(new void(frame f, transform T) { add(f,T*shift(project(t*position,P))*src); },true); } },true); } dest.addBox(position,position,min3(src),max3(src)); } exitfcn currentexitfunction=atexit(); void exitfunction() { if(currentexitfunction != null) currentexitfunction(); if(!settings.keep) for(int i=0; i < file3.length; ++i) delete(file3[i]); file3=new string[]; } atexit(exitfunction);