import bezulate; private import interpolate; int nslice=12; real camerafactor=1.2; string meshname(string name) {return name+" mesh";} private real Fuzz=10.0*realEpsilon; private real nineth=1/9; // Return the default Coons interior control point for a Bezier triangle // based on the cyclic path3 external. triple coons3(path3 external) { return 0.25*(precontrol(external,0)+postcontrol(external,0)+ precontrol(external,1)+postcontrol(external,1)+ precontrol(external,2)+postcontrol(external,2))- (point(external,0)+point(external,1)+point(external,2))/6; } struct patch { triple[][] P; pen[] colors; // Optionally specify 4 corner colors. bool straight; // Patch is based on a piecewise straight external path. bool3 planar; // Patch is planar. bool triangular; // Patch is a Bezier triangle. path3 external() { return straight ? P[0][0]--P[3][0]--P[3][3]--P[0][3]--cycle : P[0][0]..controls P[1][0] and P[2][0].. P[3][0]..controls P[3][1] and P[3][2].. P[3][3]..controls P[2][3] and P[1][3].. P[0][3]..controls P[0][2] and P[0][1]..cycle; } path3 externaltriangular() { return P[0][0]..controls P[1][0] and P[2][0].. P[3][0]..controls P[3][1] and P[3][2].. P[3][3]..controls P[2][2] and P[1][1]..cycle; } triple[] internal() { return new triple[] {P[1][1],P[2][1],P[2][2],P[1][2]}; } triple[] internaltriangular() { return new triple[] {P[2][1]}; } triple cornermean() { return 0.25*(P[0][0]+P[0][3]+P[3][0]+P[3][3]); } triple cornermeantriangular() { return (P[0][0]+P[3][0]+P[3][3])/3; } triple[] corners() {return new triple[] {P[0][0],P[3][0],P[3][3],P[0][3]};} triple[] cornerstriangular() {return new triple[] {P[0][0],P[3][0],P[3][3]};} real[] map(real f(triple)) { return new real[] {f(P[0][0]),f(P[3][0]),f(P[3][3]),f(P[0][3])}; } real[] maptriangular(real f(triple)) { return new real[] {f(P[0][0]),f(P[3][0]),f(P[3][3])}; } triple Bu(int j, real u) {return bezier(P[0][j],P[1][j],P[2][j],P[3][j],u);} triple BuP(int j, real u) { return bezierP(P[0][j],P[1][j],P[2][j],P[3][j],u); } path3 uequals(real u) { triple z0=Bu(0,u); triple z1=Bu(3,u); return path3(new triple[] {z0,Bu(2,u)},new triple[] {z0,z1}, new triple[] {Bu(1,u),z1},new bool[] {straight,false},false); } triple Bv(int i, real v) {return bezier(P[i][0],P[i][1],P[i][2],P[i][3],v);} triple BvP(int i, real v) { return bezierP(P[i][0],P[i][1],P[i][2],P[i][3],v); } path3 vequals(real v) { triple z0=Bv(0,v); triple z1=Bv(3,v); return path3(new triple[] {z0,Bv(2,v)},new triple[] {z0,z1}, new triple[] {Bv(1,v),z1},new bool[] {straight,false},false); } triple point(real u, real v) { return bezier(Bu(0,u),Bu(1,u),Bu(2,u),Bu(3,u),v); } static real fuzz=1000*realEpsilon; triple normal(triple left3, triple left2, triple left1, triple middle, triple right1, triple right2, triple right3) { real epsilon=fuzz*change2(P); triple lp=3.0*(left1-middle); triple rp=3.0*(right1-middle); triple n=cross(rp,lp); if(abs(n) > epsilon) return n; // Return one-half of the second derivative of the Bezier curve defined // by a,b,c,d at 0. triple bezierPP(triple a, triple b, triple c) { return 3.0*(a+c-2.0*b); } triple lpp=bezierPP(middle,left1,left2); triple rpp=bezierPP(middle,right1,right2); n=cross(rpp,lp)+cross(rp,lpp); if(abs(n) > epsilon) return n; // Return one-sixth of the third derivative of the Bezier curve defined // by a,b,c,d at 0. triple bezierPPP(triple a, triple b, triple c, triple d) { return d-a+3.0*(b-c); } triple lppp=bezierPPP(middle,left1,left2,left3); triple rppp=bezierPPP(middle,right1,right2,right3); n=cross(rpp,lpp)+cross(rppp,lp)+cross(rp,lppp); if(abs(n) > epsilon) return n; n=cross(rppp,lpp)+cross(rpp,lppp); if(abs(n) > epsilon) return n; return cross(rppp,lppp); } triple partialu(real u, real v) { return bezier(BuP(0,u),BuP(1,u),BuP(2,u),BuP(3,u),v); } triple partialv(real u, real v) { return bezier(BvP(0,v),BvP(1,v),BvP(2,v),BvP(3,v),u); } triple normal00() { return normal(P[0][3],P[0][2],P[0][1],P[0][0],P[1][0],P[2][0],P[3][0]); } triple normal10() { return normal(P[0][0],P[1][0],P[2][0],P[3][0],P[3][1],P[3][2],P[3][3]); } triple normal11() { return normal(P[3][0],P[3][1],P[3][2],P[3][3],P[2][3],P[1][3],P[0][3]); } triple normal01() { return normal(P[3][3],P[2][3],P[1][3],P[0][3],P[0][2],P[0][1],P[0][0]); } triple normal(real u, real v) { if(u == 0) { if(v == 0) return normal00(); if(v == 1) return normal01(); } if(u == 1) { if(v == 0) return normal10(); if(v == 1) return normal11(); } return cross(partialu(u,v),partialv(u,v)); } triple pointtriangular(real u, real v) { real w=1-u-v; return w^2*(w*P[0][0]+3*(u*P[1][0]+v*P[1][1]))+ u^2*(3*(w*P[2][0]+v*P[3][1])+u*P[3][0])+ 6*u*v*w*P[2][1]+v^2*(3*(w*P[2][2]+u*P[3][2])+v*P[3][3]); } triple partialutriangular(real u, real v) { // Compute one-third of the directional derivative of a Bezier triangle // in the u direction at (u,v). real w=1-u-v; return -w^2*P[0][0]+w*(w-2*u)*P[1][0]-2*w*v*P[1][1]+u*(2*w-u)*P[2][0]+ 2*v*(w-u)*P[2][1]-v^2*P[2][2]+u^2*P[3][0]+2*u*v*P[3][1]+v^2*P[3][2]; } triple partialvtriangular(real u, real v) { // Compute one-third of the directional derivative of a Bezier triangle // in the v direction at (u,v). real w=1-u-v; return -w^2*P[0][0]-2*u*w*P[1][0]+w*(w-2*v)*P[1][1]-u^2*P[2][0]+ 2*u*(w-v)*P[2][1]+v*(2*w-v)*P[2][2]+u*u*P[3][1]+2*u*v*P[3][2]+ v^2*P[3][3]; } triple normal00triangular() { return normal(P[3][3],P[2][2],P[1][1],P[0][0],P[1][0],P[2][0],P[3][0]); } triple normal10triangular() { return normal(P[0][0],P[1][0],P[2][0],P[3][0],P[3][1],P[3][2],P[3][3]); } triple normal01triangular() { return normal(P[3][0],P[3][1],P[3][2],P[3][3],P[2][2],P[1][1],P[0][0]); } // Compute the normal vector of a Bezier triangle at (u,v) triple normaltriangular(real u, real v) { if(u == 0) { if(v == 0) return normal00triangular(); if(v == 1) return normal01triangular(); } if(u == 1 && v == 0) return normal10triangular(); return cross(partialutriangular(u,v),partialvtriangular(u,v)); } pen[] colors(material m, light light=currentlight) { bool nocolors=colors.length == 0; if(planar) { triple normal=normal(0.5,0.5); return new pen[] {color(normal,nocolors ? m : colors[0],light), color(normal,nocolors ? m : colors[1],light), color(normal,nocolors ? m : colors[2],light), color(normal,nocolors ? m : colors[3],light)}; } return new pen[] {color(normal00(),nocolors ? m : colors[0],light), color(normal10(),nocolors ? m : colors[1],light), color(normal11(),nocolors ? m : colors[2],light), color(normal01(),nocolors ? m : colors[3],light)}; } pen[] colorstriangular(material m, light light=currentlight) { bool nocolors=colors.length == 0; if(planar) { triple normal=normal(1/3,1/3); return new pen[] {color(normal,nocolors ? m : colors[0],light), color(normal,nocolors ? m : colors[1],light), color(normal,nocolors ? m : colors[2],light)}; } return new pen[] {color(normal00(),nocolors ? m : colors[0],light), color(normal10(),nocolors ? m : colors[1],light), color(normal01(),nocolors ? m : colors[2],light)}; } triple min3,max3; bool havemin3,havemax3; void init() { havemin3=false; havemax3=false; if(triangular) { external=externaltriangular; internal=internaltriangular; cornermean=cornermeantriangular; corners=cornerstriangular; map=maptriangular; point=pointtriangular; normal=normaltriangular; normal00=normal00triangular; normal10=normal10triangular; normal01=normal01triangular; colors=colorstriangular; uequals=new path3(real u) {return nullpath3;}; vequals=new path3(real u) {return nullpath3;}; } } triple min(triple bound=P[0][0]) { if(havemin3) return minbound(min3,bound); havemin3=true; return min3=minbezier(P,bound); } triple max(triple bound=P[0][0]) { if(havemax3) return maxbound(max3,bound); havemax3=true; return max3=maxbezier(P,bound); } triple center() { return 0.5*(this.min()+this.max()); } pair min(projection P, pair bound=project(this.P[0][0],P.t)) { triple[][] Q=P.T.modelview*this.P; if(P.infinity) return xypart(minbezier(Q,(bound.x,bound.y,0))); real d=P.T.projection[3][2]; return maxratio(Q,d*bound)/d; // d is negative } pair max(projection P, pair bound=project(this.P[0][0],P.t)) { triple[][] Q=P.T.modelview*this.P; if(P.infinity) return xypart(maxbezier(Q,(bound.x,bound.y,0))); real d=P.T.projection[3][2]; return minratio(Q,d*bound)/d; // d is negative } void operator init(triple[][] P, pen[] colors=new pen[], bool straight=false, bool3 planar=default, bool triangular=false, bool copy=true) { this.P=copy ? copy(P) : P; if(colors.length != 0) this.colors=copy(colors); this.straight=straight; this.planar=planar; this.triangular=triangular; init(); } void operator init(pair[][] P, triple plane(pair)=XYplane, bool straight=false, bool triangular=false) { triple[][] Q=new triple[4][]; for(int i=0; i < 4; ++i) { pair[] Pi=P[i]; Q[i]=sequence(new triple(int j) {return plane(Pi[j]);},4); } operator init(Q,straight,planar=true,triangular); } void operator init(patch s) { operator init(s.P,s.colors,s.straight,s.planar,s.triangular); } // A constructor for a cyclic path3 of length 3 with a specified // internal point, corner normals, and pens (rendered as a Bezier triangle). void operator init(path3 external, triple internal, pen[] colors=new pen[], bool3 planar=default) { triangular=true; this.planar=planar; init(); if(colors.length != 0) this.colors=copy(colors); P=new triple[][] { {point(external,0)}, {postcontrol(external,0),precontrol(external,0)}, {precontrol(external,1),internal,postcontrol(external,2)}, {point(external,1),postcontrol(external,1),precontrol(external,2), point(external,2)} }; } // A constructor for a convex cyclic path3 of length <= 4 with optional // arrays of internal points (4 for a Bezier patch, 1 for a Bezier // triangle), and pens. void operator init(path3 external, triple[] internal=new triple[], pen[] colors=new pen[], bool3 planar=default) { if(internal.length == 0 && planar == default) this.planar=normal(external) != O; else this.planar=planar; int L=length(external); if(L == 3) { operator init(external,internal.length == 1 ? internal[0] : coons3(external),colors,this.planar); straight=piecewisestraight(external); return; } if(L > 4 || !cyclic(external)) abort("cyclic path3 of length <= 4 expected"); if(L == 1) { external=external--cycle--cycle--cycle; if(colors.length > 0) colors.append(array(3,colors[0])); } else if(L == 2) { external=external--cycle--cycle; if(colors.length > 0) colors.append(array(2,colors[0])); } init(); if(colors.length != 0) this.colors=copy(colors); if(internal.length == 0) { straight=piecewisestraight(external); internal=new triple[4]; for(int j=0; j < 4; ++j) internal[j]=nineth*(-4*point(external,j) +6*(precontrol(external,j)+postcontrol(external,j)) -2*(point(external,j-1)+point(external,j+1)) +3*(precontrol(external,j-1)+ postcontrol(external,j+1)) -point(external,j+2)); } P=new triple[][] { {point(external,0),precontrol(external,0),postcontrol(external,3), point(external,3)}, {postcontrol(external,0),internal[0],internal[3],precontrol(external,3)}, {precontrol(external,1),internal[1],internal[2],postcontrol(external,2)}, {point(external,1),postcontrol(external,1),precontrol(external,2), point(external,2)} }; } // A constructor for a convex quadrilateral. void operator init(triple[] external, triple[] internal=new triple[], pen[] colors=new pen[], bool3 planar=default) { init(); if(internal.length == 0 && planar == default) this.planar=normal(external) != O; else this.planar=planar; if(colors.length != 0) this.colors=copy(colors); if(internal.length == 0) { internal=new triple[4]; for(int j=0; j < 4; ++j) internal[j]=nineth*(4*external[j]+2*external[(j+1)%4]+ external[(j+2)%4]+2*external[(j+3)%4]); } straight=true; triple delta[]=new triple[4]; for(int j=0; j < 4; ++j) delta[j]=(external[(j+1)% 4]-external[j])/3; P=new triple[][] { {external[0],external[0]-delta[3],external[3]+delta[3],external[3]}, {external[0]+delta[0],internal[0],internal[3],external[3]-delta[2]}, {external[1]-delta[0],internal[1],internal[2],external[2]+delta[2]}, {external[1],external[1]+delta[1],external[2]-delta[1],external[2]} }; } } patch operator * (transform3 t, patch s) { patch S; S.P=new triple[s.P.length][]; for(int i=0; i < s.P.length; ++i) { triple[] si=s.P[i]; triple[] Si=S.P[i]; for(int j=0; j < si.length; ++j) Si[j]=t*si[j]; } S.colors=copy(s.colors); S.planar=s.planar; S.straight=s.straight; S.triangular=s.triangular; S.init(); return S; } patch reverse(patch s) { assert(!s.triangular); patch S; S.P=transpose(s.P); if(s.colors.length > 0) S.colors=new pen[] {s.colors[0],s.colors[3],s.colors[2],s.colors[1]}; S.straight=s.straight; S.planar=s.planar; return S; } // Return a degenerate tensor patch representation of a Bezier triangle. patch tensor(patch s) { if(!s.triangular) return patch(s); triple[][] P=s.P; return patch(new triple[][] {{P[0][0],P[0][0],P[0][0],P[0][0]}, {P[1][0],P[1][0]*2/3+P[1][1]/3,P[1][0]/3+P[1][1]*2/3,P[1][1]}, {P[2][0],P[2][0]/3+P[2][1]*2/3,P[2][1]*2/3+P[2][2]/3,P[2][2]}, {P[3][0],P[3][1],P[3][2],P[3][3]}}, s.colors.length > 0 ? new pen[] {s.colors[0],s.colors[1],s.colors[2],s.colors[0]} : new pen[], s.straight,s.planar,false,false); } // Return the tensor product patch control points corresponding to path p // and points internal. pair[][] tensor(path p, pair[] internal) { return new pair[][] { {point(p,0),precontrol(p,0),postcontrol(p,3),point(p,3)}, {postcontrol(p,0),internal[0],internal[3],precontrol(p,3)}, {precontrol(p,1),internal[1],internal[2],postcontrol(p,2)}, {point(p,1),postcontrol(p,1),precontrol(p,2),point(p,2)} }; } // Return the Coons patch control points corresponding to path p. pair[][] coons(path p) { int L=length(p); if(L == 1) p=p--cycle--cycle--cycle; else if(L == 2) p=p--cycle--cycle; else if(L == 3) p=p--cycle; pair[] internal=new pair[4]; for(int j=0; j < 4; ++j) { internal[j]=nineth*(-4*point(p,j) +6*(precontrol(p,j)+postcontrol(p,j)) -2*(point(p,j-1)+point(p,j+1)) +3*(precontrol(p,j-1)+postcontrol(p,j+1)) -point(p,j+2)); } return tensor(p,internal); } // Decompose a possibly nonconvex cyclic path into an array of paths that // yield nondegenerate Coons patches. path[] regularize(path p, bool checkboundary=true) { path[] s; if(!cyclic(p)) abort("cyclic path expected"); int L=length(p); if(L > 4) { for(path g : bezulate(p)) s.append(regularize(g,checkboundary)); return s; } bool straight=piecewisestraight(p); if(L <= 3 && straight) { return new path[] {p}; } // Split p along the angle bisector at t. bool split(path p, real t) { pair dir=dir(p,t); if(dir != 0) { path g=subpath(p,t,t+length(p)); int L=length(g); pair z=point(g,0); real[] T=intersections(g,z,z+I*abs(z)*dir); for(int i=0; i < T.length; ++i) { real cut=T[i]; if(cut > sqrtEpsilon && cut < L-sqrtEpsilon) { pair w=point(g,cut); if(!inside(p,0.5*(z+w),zerowinding)) continue; pair delta=sqrtEpsilon*(w-z); if(intersections(g,z-delta--w+delta).length != 2) continue; s.append(regularize(subpath(g,0,cut)--cycle,checkboundary)); s.append(regularize(subpath(g,cut,L)--cycle,checkboundary)); return true; } } } return false; } // Ensure that all interior angles are less than 180 degrees. real fuzz=1e-4; int sign=sgn(windingnumber(p,inside(p,zerowinding))); for(int i=0; i < L; ++i) { if(sign*(conj(dir(p,i,-1))*dir(p,i,1)).y < -fuzz) { if(split(p,i)) return s; } } if(straight) return new path[] {p}; pair[][] P=coons(p); // Check for degeneracy. pair[][] U=new pair[3][4]; pair[][] V=new pair[4][3]; for(int i=0; i < 3; ++i) { for(int j=0; j < 4; ++j) U[i][j]=P[i+1][j]-P[i][j]; } for(int i=0; i < 4; ++i) { for(int j=0; j < 3; ++j) V[i][j]=P[i][j+1]-P[i][j]; } int[] choose2={1,2,1}; int[] choose3={1,3,3,1}; real T[][]=new real[6][6]; for(int p=0; p < 6; ++p) { int kstart=max(p-2,0); int kstop=min(p,3); real[] Tp=T[p]; for(int q=0; q < 6; ++q) { real Tpq; int jstop=min(q,3); int jstart=max(q-2,0); for(int k=kstart; k <= kstop; ++k) { int choose3k=choose3[k]; for(int j=jstart; j <= jstop; ++j) { int i=p-k; int l=q-j; Tpq += (conj(U[i][j])*V[k][l]).y* choose2[i]*choose3k*choose3[j]*choose2[l]; } } Tp[q]=Tpq; } } bool3 aligned=default; bool degenerate=false; for(int p=0; p < 6; ++p) { for(int q=0; q < 6; ++q) { if(aligned == default) { if(T[p][q] > sqrtEpsilon) aligned=true; if(T[p][q] < -sqrtEpsilon) aligned=false; } else { if((T[p][q] > sqrtEpsilon && aligned == false) || (T[p][q] < -sqrtEpsilon && aligned == true)) degenerate=true; } } } if(!degenerate) { if(aligned == (sign >= 0)) return new path[] {p}; return s; } if(checkboundary) { // Polynomial coefficients of (B_i'' B_j + B_i' B_j')/3. static real[][][] fpv0={ {{5, -20, 30, -20, 5}, {-3, 24, -54, 48, -15}, {0, -6, 27, -36, 15}, {0, 0, -3, 8, -5}}, {{-7, 36, -66, 52, -15}, {3, -36, 108, -120, 45}, {0, 6, -45, 84, -45}, {0, 0, 3, -16, 15}}, {{2, -18, 45, -44, 15}, {0, 12, -63, 96, -45}, {0, 0, 18, -60, 45}, {0, 0, 0, 8, -15}}, {{0, 2, -9, 12, -5}, {0, 0, 9, -24, 15}, {0, 0, 0, 12, -15}, {0, 0, 0, 0, 5}} }; // Compute one-ninth of the derivative of the Jacobian along the boundary. real[][] c=array(4,array(5,0.0)); for(int i=0; i < 4; ++i) { real[][] fpv0i=fpv0[i]; for(int j=0; j < 4; ++j) { real[] w=fpv0i[j]; c[0] += w*(conj(P[i][0])*(P[j][1]-P[j][0])).y; // v=0 c[1] += w*(conj(P[3][j]-P[2][j])*P[3][i]).y; // u=1 c[2] += w*(conj(P[i][3])*(P[j][3]-P[j][2])).y; // v=1 c[3] += w*(conj(P[0][j]-P[1][j])*P[0][i]).y; // u=0 } } pair BuP(int j, real u) { return bezierP(P[0][j],P[1][j],P[2][j],P[3][j],u); } pair BvP(int i, real v) { return bezierP(P[i][0],P[i][1],P[i][2],P[i][3],v); } real normal(real u, real v) { return (conj(bezier(BuP(0,u),BuP(1,u),BuP(2,u),BuP(3,u),v))* bezier(BvP(0,v),BvP(1,v),BvP(2,v),BvP(3,v),u)).y; } // Use Rolle's theorem to check for degeneracy on the boundary. real M=0; real cut; for(int i=0; i < 4; ++i) { if(!straight(p,i)) { real[] ci=c[i]; pair[] R=quarticroots(ci[4],ci[3],ci[2],ci[1],ci[0]); for(pair r : R) { if(fabs(r.y) < sqrtEpsilon) { real t=r.x; if(0 <= t && t <= 1) { real[] U={t,1,t,0}; real[] V={0,t,1,t}; real[] T={t,t,1-t,1-t}; real N=sign*normal(U[i],V[i]); if(N < M) { M=N; cut=i+T[i]; } } } } } } // Split at the worst boundary degeneracy. if(M < 0 && split(p,cut)) return s; } // Split arbitrarily to resolve any remaining (internal) degeneracy. checkboundary=false; for(int i=0; i < L; ++i) if(!straight(p,i) && split(p,i+0.5)) return s; while(true) for(int i=0; i < L; ++i) if(!straight(p,i) && split(p,i+unitrand())) return s; return s; } typedef void drawfcn(frame f, transform3 t=identity4, material[] m, light light=currentlight, render render=defaultrender); struct surface { patch[] s; int index[][];// Position of patch corresponding to major U,V parameter in s. bool vcyclic; transform3 T=identity4; drawfcn draw; bool PRCprimitive=true; // True unless no PRC primitive is available. bool empty() { return s.length == 0; } void operator init(int n) { s=new patch[n]; } void operator init(... patch[] s) { this.s=s; } void operator init(surface s) { this.s=new patch[s.s.length]; for(int i=0; i < s.s.length; ++i) this.s[i]=patch(s.s[i]); this.index=copy(s.index); this.vcyclic=s.vcyclic; } void operator init(triple[][][] P, pen[][] colors=new pen[][], bool3 planar=default, bool triangular=false) { s=sequence(new patch(int i) { return patch(P[i],colors.length == 0 ? new pen[] : colors[i],planar, triangular); },P.length); } void colors(pen[][] palette) { for(int i=0; i < s.length; ++i) s[i].colors=copy(palette[i]); } triple[][] corners() { triple[][] a=new triple[s.length][]; for(int i=0; i < s.length; ++i) a[i]=s[i].corners(); return a; } real[][] map(real f(triple)) { real[][] a=new real[s.length][]; for(int i=0; i < s.length; ++i) a[i]=s[i].map(f); return a; } triple[] cornermean() { return sequence(new triple(int i) {return s[i].cornermean();},s.length); } triple point(real u, real v) { int U=floor(u); int V=floor(v); int index=index.length == 0 ? U+V : index[U][V]; return s[index].point(u-U,v-V); } triple normal(real u, real v) { int U=floor(u); int V=floor(v); int index=index.length == 0 ? U+V : index[U][V]; return s[index].normal(u-U,v-V); } void ucyclic(bool f) { index.cyclic=f; } void vcyclic(bool f) { for(int[] i : index) i.cyclic=f; vcyclic=f; } bool ucyclic() { return index.cyclic; } bool vcyclic() { return vcyclic; } path3 uequals(real u) { if(index.length == 0) return nullpath3; int U=floor(u); int[] index=index[U]; path3 g; for(int i : index) g=g&s[i].uequals(u-U); return vcyclic() ? g&cycle : g; } path3 vequals(real v) { if(index.length == 0) return nullpath3; int V=floor(v); path3 g; for(int[] i : index) g=g&s[i[V]].vequals(v-V); return ucyclic() ? g&cycle : g; } // A constructor for a possibly nonconvex simple cyclic path in a given // plane. void operator init(path p, triple plane(pair)=XYplane) { for(path g : regularize(p)) { if(length(g) == 3) { path3 G=path3(g,plane); s.push(patch(G,coons3(G),planar=true)); } else s.push(patch(coons(g),plane,piecewisestraight(g))); } } void operator init(explicit path[] g, triple plane(pair)=XYplane) { for(path p : bezulate(g)) s.append(surface(p,plane).s); } // A general surface constructor for both planar and nonplanar 3D paths. void construct(path3 external, triple[] internal=new triple[], pen[] colors=new pen[], bool3 planar=default) { int L=length(external); if(!cyclic(external)) abort("cyclic path expected"); if(L <= 3 && piecewisestraight(external)) { s.push(patch(external,internal,colors,planar)); return; } // Construct a surface from a possibly nonconvex planar cyclic path3. if(planar != false && internal.length == 0 && colors.length == 0) { triple n=normal(external); if(n != O) { transform3 T=align(n); external=transpose(T)*external; T *= shift(0,0,point(external,0).z); for(patch p : surface(path(external)).s) s.push(T*p); return; } } if(L <= 4 || internal.length > 0) { s.push(patch(external,internal,colors,planar)); return; } // Path is not planar; split into patches. real factor=1/L; pen[] p; triple[] n; bool nocolors=colors.length == 0; triple center; for(int i=0; i < L; ++i) center += point(external,i); center *= factor; if(!nocolors) p=new pen[] {mean(colors)}; // Use triangles for nonplanar surfaces. int step=normal(external) == O ? 1 : 2; int i=0; int end; while((end=i+step) < L) { s.push(patch(subpath(external,i,end)--center--cycle, nocolors ? p : concat(colors[i:end+1],p),planar)); i=end; } s.push(patch(subpath(external,i,L)--center--cycle, nocolors ? p : concat(colors[i:],colors[0:1],p),planar)); } void operator init(path3 external, triple[] internal=new triple[], pen[] colors=new pen[], bool3 planar=default) { s=new patch[]; construct(external,internal,colors,planar); } void operator init(explicit path3[] external, triple[][] internal=new triple[][], pen[][] colors=new pen[][], bool3 planar=default) { s=new patch[]; if(planar == true) {// Assume all path3 elements share a common normal. if(external.length != 0) { triple n=normal(external[0]); if(n != O) { transform3 T=align(n); external=transpose(T)*external; T *= shift(0,0,point(external[0],0).z); path[] g=sequence(new path(int i) {return path(external[i]);}, external.length); for(patch p : surface(g).s) s.push(T*p); return; } } } for(int i=0; i < external.length; ++i) construct(external[i], internal.length == 0 ? new triple[] : internal[i], colors.length == 0 ? new pen[] : colors[i],planar); } void push(path3 external, triple[] internal=new triple[], pen[] colors=new pen[], bool3 planar=default) { s.push(patch(external,internal,colors,planar)); } // Construct the surface of rotation generated by rotating g // from angle1 to angle2 sampled n times about the line c--c+axis. // An optional surface pen color(int i, real j) may be specified // to override the color at vertex(i,j). void operator init(triple c, path3 g, triple axis, int n=nslice, real angle1=0, real angle2=360, pen color(int i, real j)=null) { axis=unit(axis); real w=(angle2-angle1)/n; int L=length(g); s=new patch[L*n]; index=new int[n][L]; int m=-1; transform3[] T=new transform3[n+1]; transform3 t=rotate(w,c,c+axis); T[0]=rotate(angle1,c,c+axis); for(int k=1; k <= n; ++k) T[k]=T[k-1]*t; typedef pen colorfcn(int i, real j); bool defaultcolors=(colorfcn) color == null; for(int i=0; i < L; ++i) { path3 h=subpath(g,i,i+1); path3 r=reverse(h); path3 H=shift(-c)*h; real M=0; triple perp; void test(real[] t) { for(int i=0; i < 3; ++i) { triple v=point(H,t[i]); triple V=v-dot(v,axis)*axis; real a=abs(V); if(a > M) {M=a; perp=V;} } } test(maxtimes(H)); test(mintimes(H)); perp=unit(perp); triple normal=unit(cross(axis,perp)); triple dir(real j) {return Cos(j)*normal-Sin(j)*perp;} real j=angle1; transform3 Tk=T[0]; triple dirj=dir(j); for(int k=0; k < n; ++k, j += w) { transform3 Tp=T[k+1]; triple dirp=dir(j+w); path3 G=reverse(Tk*h{dirj}..{dirp}Tp*r{-dirp}..{-dirj}cycle); Tk=Tp; dirj=dirp; s[++m]=defaultcolors ? patch(G) : patch(G,new pen[] {color(i,j),color(i,j+w),color(i+1,j+w), color(i+1,j)}); index[k][i]=m; } ucyclic((angle2-angle1) % 360 == 0); vcyclic(cyclic(g)); } } void push(patch s) { this.s.push(s); } void append(surface s) { this.s.append(s.s); } void operator init(... surface[] s) { for(surface S : s) this.s.append(S.s); } } surface operator * (transform3 t, surface s) { surface S; S.s=new patch[s.s.length]; for(int i=0; i < s.s.length; ++i) S.s[i]=t*s.s[i]; S.index=copy(s.index); S.vcyclic=(bool) s.vcyclic; S.T=t*s.T; S.draw=s.draw; S.PRCprimitive=s.PRCprimitive; return S; } private string nullsurface="null surface"; triple min(surface s) { if(s.s.length == 0) abort(nullsurface); triple bound=s.s[0].min(); for(int i=1; i < s.s.length; ++i) bound=s.s[i].min(bound); return bound; } triple max(surface s) { if(s.s.length == 0) abort(nullsurface); triple bound=s.s[0].max(); for(int i=1; i < s.s.length; ++i) bound=s.s[i].max(bound); return bound; } pair min(surface s, projection P) { if(s.s.length == 0) abort(nullsurface); pair bound=s.s[0].min(P); for(int i=1; i < s.s.length; ++i) bound=s.s[i].min(P,bound); return bound; } pair max(surface s, projection P) { if(s.s.length == 0) abort(nullsurface); pair bound=s.s[0].max(P); for(int i=1; i < s.s.length; ++i) bound=s.s[i].max(P,bound); return bound; } private triple[] split(triple z0, triple c0, triple c1, triple z1, real t=0.5) { triple m0=interp(z0,c0,t); triple m1=interp(c0,c1,t); triple m2=interp(c1,z1,t); triple m3=interp(m0,m1,t); triple m4=interp(m1,m2,t); triple m5=interp(m3,m4,t); return new triple[] {m0,m3,m5,m4,m2}; } // Return the control points of the subpatches // produced by a horizontal split of P triple[][][] hsplit(triple[][] P, real v=0.5) { // get control points in rows triple[] P0=P[0]; triple[] P1=P[1]; triple[] P2=P[2]; triple[] P3=P[3]; triple[] c0=split(P0[0],P0[1],P0[2],P0[3],v); triple[] c1=split(P1[0],P1[1],P1[2],P1[3],v); triple[] c2=split(P2[0],P2[1],P2[2],P2[3],v); triple[] c3=split(P3[0],P3[1],P3[2],P3[3],v); // bottom, top return new triple[][][] { {{P0[0],c0[0],c0[1],c0[2]}, {P1[0],c1[0],c1[1],c1[2]}, {P2[0],c2[0],c2[1],c2[2]}, {P3[0],c3[0],c3[1],c3[2]}}, {{c0[2],c0[3],c0[4],P0[3]}, {c1[2],c1[3],c1[4],P1[3]}, {c2[2],c2[3],c2[4],P2[3]}, {c3[2],c3[3],c3[4],P3[3]}} }; } // Return the control points of the subpatches // produced by a vertical split of P triple[][][] vsplit(triple[][] P, real u=0.5) { // get control points in rows triple[] P0=P[0]; triple[] P1=P[1]; triple[] P2=P[2]; triple[] P3=P[3]; triple[] c0=split(P0[0],P1[0],P2[0],P3[0],u); triple[] c1=split(P0[1],P1[1],P2[1],P3[1],u); triple[] c2=split(P0[2],P1[2],P2[2],P3[2],u); triple[] c3=split(P0[3],P1[3],P2[3],P3[3],u); // left, right return new triple[][][] { {{P0[0],P0[1],P0[2],P0[3]}, {c0[0],c1[0],c2[0],c3[0]}, {c0[1],c1[1],c2[1],c3[1]}, {c0[2],c1[2],c2[2],c3[2]}}, {{c0[2],c1[2],c2[2],c3[2]}, {c0[3],c1[3],c2[3],c3[3]}, {c0[4],c1[4],c2[4],c3[4]}, {P3[0],P3[1],P3[2],P3[3]}} }; } // Return a 2D array of the control point arrays of the subpatches // produced by horizontal and vertical splits of P at u and v triple[][][][] split(triple[][] P, real u=0.5, real v=0.5) { triple[] P0=P[0]; triple[] P1=P[1]; triple[] P2=P[2]; triple[] P3=P[3]; // slice horizontally triple[] c0=split(P0[0],P0[1],P0[2],P0[3],v); triple[] c1=split(P1[0],P1[1],P1[2],P1[3],v); triple[] c2=split(P2[0],P2[1],P2[2],P2[3],v); triple[] c3=split(P3[0],P3[1],P3[2],P3[3],v); // bottom patch triple[] c4=split(P0[0],P1[0],P2[0],P3[0],u); triple[] c5=split(c0[0],c1[0],c2[0],c3[0],u); triple[] c6=split(c0[1],c1[1],c2[1],c3[1],u); triple[] c7=split(c0[2],c1[2],c2[2],c3[2],u); // top patch triple[] c8=split(c0[3],c1[3],c2[3],c3[3],u); triple[] c9=split(c0[4],c1[4],c2[4],c3[4],u); triple[] cA=split(P0[3],P1[3],P2[3],P3[3],u); // {{bottom-left, top-left}, {bottom-right, top-right}} return new triple[][][][] { {{{P0[0],c0[0],c0[1],c0[2]}, {c4[0],c5[0],c6[0],c7[0]}, {c4[1],c5[1],c6[1],c7[1]}, {c4[2],c5[2],c6[2],c7[2]}}, {{c0[2],c0[3],c0[4],P0[3]}, {c7[0],c8[0],c9[0],cA[0]}, {c7[1],c8[1],c9[1],cA[1]}, {c7[2],c8[2],c9[2],cA[2]}}}, {{{c4[2],c5[2],c6[2],c7[2]}, {c4[3],c5[3],c6[3],c7[3]}, {c4[4],c5[4],c6[4],c7[4]}, {P3[0],c3[0],c3[1],c3[2]}}, {{c7[2],c8[2],c9[2],cA[2]}, {c7[3],c8[3],c9[3],cA[3]}, {c7[4],c8[4],c9[4],cA[4]}, {c3[2],c3[3],c3[4],P3[3]}}} }; } // Return the control points for a subpatch of P on [u,1] x [v,1]. triple[][] subpatchend(triple[][] P, real u, real v) { triple[] P0=P[0]; triple[] P1=P[1]; triple[] P2=P[2]; triple[] P3=P[3]; triple[] c0=split(P0[0],P0[1],P0[2],P0[3],v); triple[] c1=split(P1[0],P1[1],P1[2],P1[3],v); triple[] c2=split(P2[0],P2[1],P2[2],P2[3],v); triple[] c3=split(P3[0],P3[1],P3[2],P3[3],v); triple[] c7=split(c0[2],c1[2],c2[2],c3[2],u); triple[] c8=split(c0[3],c1[3],c2[3],c3[3],u); triple[] c9=split(c0[4],c1[4],c2[4],c3[4],u); triple[] cA=split(P0[3],P1[3],P2[3],P3[3],u); return new triple[][] { {c7[2],c8[2],c9[2],cA[2]}, {c7[3],c8[3],c9[3],cA[3]}, {c7[4],c8[4],c9[4],cA[4]}, {c3[2],c3[3],c3[4],P3[3]}}; } // Return the control points for a subpatch of P on [0,u] x [0,v]. triple[][] subpatchbegin(triple[][] P, real u, real v) { triple[] P0=P[0]; triple[] P1=P[1]; triple[] P2=P[2]; triple[] P3=P[3]; triple[] c0=split(P0[0],P0[1],P0[2],P0[3],v); triple[] c1=split(P1[0],P1[1],P1[2],P1[3],v); triple[] c2=split(P2[0],P2[1],P2[2],P2[3],v); triple[] c3=split(P3[0],P3[1],P3[2],P3[3],v); triple[] c4=split(P0[0],P1[0],P2[0],P3[0],u); triple[] c5=split(c0[0],c1[0],c2[0],c3[0],u); triple[] c6=split(c0[1],c1[1],c2[1],c3[1],u); triple[] c7=split(c0[2],c1[2],c2[2],c3[2],u); return new triple[][] { {P0[0],c0[0],c0[1],c0[2]}, {c4[0],c5[0],c6[0],c7[0]}, {c4[1],c5[1],c6[1],c7[1]}, {c4[2],c5[2],c6[2],c7[2]}}; } triple[][] subpatch(triple[][] P, pair a, pair b) { return subpatchend(subpatchbegin(P,b.x,b.y),a.x/b.x,a.y/b.y); } patch subpatch(patch s, pair a, pair b) { assert(a.x >= 0 && a.y >= 0 && b.x <= 1 && b.y <= 1 && a.x < b.x && a.y < b.y && !s.triangular); return patch(subpatch(s.P,a,b),s.straight,s.planar); } private string triangular= "Intersection of path3 with Bezier triangle is not yet implemented"; // return an array containing the times for one intersection of path p and // patch s. real[] intersect(path3 p, patch s, real fuzz=-1) { if(s.triangular) abort(triangular); return intersect(p,s.P,fuzz); } // return an array containing the times for one intersection of path p and // surface s. real[] intersect(path3 p, surface s, real fuzz=-1) { for(int i=0; i < s.s.length; ++i) { real[] T=intersect(p,s.s[i],fuzz); if(T.length > 0) return T; } return new real[]; } // return an array containing all intersection times of path p and patch s. real[][] intersections(path3 p, patch s, real fuzz=-1) { if(s.triangular) abort(triangular); return sort(intersections(p,s.P,fuzz)); } // return an array containing all intersection times of path p and surface s. real[][] intersections(path3 p, surface s, real fuzz=-1) { real[][] T; if(length(p) < 0) return T; for(int i=0; i < s.s.length; ++i) for(real[] s: intersections(p,s.s[i],fuzz)) T.push(s); static real Fuzz=1000*realEpsilon; real fuzz=max(10*fuzz,Fuzz*max(abs(min(s)),abs(max(s)))); // Remove intrapatch duplicate points. for(int i=0; i < T.length; ++i) { triple v=point(p,T[i][0]); for(int j=i+1; j < T.length;) { if(abs(v-point(p,T[j][0])) < fuzz) T.delete(j); else ++j; } } return sort(T); } // return an array containing all intersection points of path p and surface s. triple[] intersectionpoints(path3 p, patch s, real fuzz=-1) { real[][] t=intersections(p,s,fuzz); return sequence(new triple(int i) {return point(p,t[i][0]);},t.length); } // return an array containing all intersection points of path p and surface s. triple[] intersectionpoints(path3 p, surface s, real fuzz=-1) { real[][] t=intersections(p,s,fuzz); return sequence(new triple(int i) {return point(p,t[i][0]);},t.length); } // Return true iff the control point bounding boxes of patches p and q overlap. bool overlap(triple[][] p, triple[][] q, real fuzz=-1) { triple pmin=minbound(p); triple pmax=maxbound(p); triple qmin=minbound(q); triple qmax=maxbound(q); if(fuzz == -1) fuzz=1000*realEpsilon*max(abs(pmin),abs(pmax),abs(qmin),abs(qmax)); return pmax.x+fuzz >= qmin.x && pmax.y+fuzz >= qmin.y && pmax.z+fuzz >= qmin.z && qmax.x+fuzz >= pmin.x && qmax.y+fuzz >= pmin.y && qmax.z+fuzz >= pmin.z; // Overlapping bounding boxes? } triple point(patch s, real u, real v) { return s.point(u,v); } struct interaction { int type; bool targetsize; void operator init(int type, bool targetsize=false) { this.type=type; this.targetsize=targetsize; } } restricted interaction Embedded=interaction(0); restricted interaction Billboard=interaction(1); interaction LabelInteraction() { return settings.autobillboard ? Billboard : Embedded; } material material(material m, light light, bool colors=false) { return light.on() || invisible((pen) m) ? m : emissive(m,colors); } void draw3D(frame f, patch s, triple center=O, material m, light light=currentlight, interaction interaction=Embedded, bool primitive=false) { bool straight=s.straight && s.planar; // Planar Bezier surfaces require extra precision in WebGL int digits=s.planar && !straight ? 12 : settings.digits; if(s.colors.length > 0) { if(prc() && light.on()) straight=false; // PRC vertex colors (for quads only) ignore lighting m.diffuse(mean(s.colors)); } m=material(m,light,s.colors.length > 0); (s.triangular ? drawbeziertriangle : draw) (f,s.P,center,straight,m.p,m.opacity,m.shininess, m.metallic,m.fresnel0,s.colors,interaction.type,digits,primitive); } void _draw(frame f, path3 g, triple center=O, material m, light light=currentlight, interaction interaction=Embedded) { if(!prc()) m=material(m,light); _draw(f,g,center,m.p,m.opacity,m.shininess,m.metallic,m.fresnel0, interaction.type); } int computeNormals(triple[] v, int[][] vi, triple[] n, int[][] ni) { triple lastnormal=O; for(int i=0; i < vi.length; ++i) { int[] vii=vi[i]; int[] nii=ni[i]; triple normal=normal(new triple[] {v[vii[0]],v[vii[1]],v[vii[2]]}); if(normal != lastnormal || n.length == 0) { n.push(normal); lastnormal=normal; } nii[0]=nii[1]=nii[2]=n.length-1; } return ni.length; } // Draw triangles on a frame. void draw(frame f, triple[] v, int[][] vi, triple[] n={}, int[][] ni={}, material m=currentpen, pen[] p={}, int[][] pi={}, light light=currentlight) { bool normals=n.length > 0; if(!normals) { ni=new int[vi.length][3]; normals=computeNormals(v,vi,n,ni) > 0; } if(p.length > 0) m=mean(p); m=material(m,light); draw(f,v,vi,n,ni,m.p,m.opacity,m.shininess,m.metallic,m.fresnel0,p,pi); } // Draw triangles on a picture. void draw(picture pic=currentpicture, triple[] v, int[][] vi, triple[] n={}, int[][] ni={}, material m=currentpen, pen[] p={}, int[][] pi={}, light light=currentlight) { bool prc=prc(); bool normals=n.length > 0; if(!normals) { ni=new int[vi.length][3]; normals=computeNormals(v,vi,n,ni) > 0; } bool colors=pi.length > 0; pic.add(new void(frame f, transform3 t, picture pic, projection P) { triple[] v=t*v; triple[] n=t*n; if(is3D()) { draw(f,v,vi,n,ni,m,p,pi,light); if(pic != null) { for(int[] vii : vi) for(int viij : vii) pic.addPoint(project(v[viij],P)); } } else if(pic != null) { static int[] edges={0,0,1}; if(colors) { for(int i=0; i < vi.length; ++i) { int[] vii=vi[i]; int[] pii=pi[i]; gouraudshade(pic,project(v[vii[0]],P)--project(v[vii[1]],P)-- project(v[vii[2]],P)--cycle, new pen[] {p[pii[0]],p[pii[1]],p[pii[2]]},edges); } } else { if(normals) { for(int i=0; i < vi.length; ++i) { int[] vii=vi[i]; int[] nii=ni[i]; gouraudshade(pic,project(v[vii[0]],P)--project(v[vii[1]],P)-- project(v[vii[2]],P)--cycle, new pen[] {color(n[nii[0]],m,light), color(n[nii[1]],m,light), color(n[nii[2]],m,light)},edges); } } else { for(int i=0; i < vi.length; ++i) { int[] vii=vi[i]; path g=project(v[vii[0]],P)--project(v[vii[1]],P)-- project(v[vii[2]],P)--cycle; pen p=color(n[ni[i][0]],m,light); fill(pic,g,p); if(prc && opacity(m.diffuse()) == 1) // Fill subdivision cracks draw(pic,g,p); } } } } },true); for(int[] vii : vi) for(int viij : vii) pic.addPoint(v[viij]); } void tensorshade(transform t=identity(), frame f, patch s, material m, light light=currentlight, projection P) { pen[] p; if(s.triangular) { p=s.colorstriangular(m,light); p.push(p[0]); s=tensor(s); } else p=s.colors(m,light); path g=t*project(s.external(),P,1); pair[] internal=t*project(s.internal(),P); pen fillrule=m.diffuse(); if(inside(g,internal[0],fillrule) && inside(g,internal[1],fillrule) && inside(g,internal[2],fillrule) && inside(g,internal[3],fillrule)) { if(p[0] == p[1] && p[1] == p[2] && p[2] == p[3]) fill(f,g,fillrule+p[0]); else tensorshade(f,g,fillrule,p,internal); } else { tensorshade(f,box(t*s.min(P),t*s.max(P)),fillrule,p,g,internal); } } restricted pen[] nullpens={nullpen}; nullpens.cyclic=true; void draw(transform t=identity(), frame f, surface s, int nu=1, int nv=1, material[] surfacepen, pen[] meshpen=nullpens, light light=currentlight, light meshlight=nolight, string name="", render render=defaultrender, projection P=currentprojection) { bool is3D=is3D(); if(is3D) { bool prc=prc(); if(s.draw != null && (settings.outformat == "html" || (prc && s.PRCprimitive))) { for(int k=0; k < s.s.length; ++k) draw3D(f,s.s[k],surfacepen[k],light,primitive=true); s.draw(f,s.T,surfacepen,light,render); } else { bool group=name != "" || render.defaultnames; if(group) begingroup3(f,name == "" ? "surface" : name,render); // Sort patches by mean distance from camera triple camera=P.camera; if(P.infinity) { triple m=min(s); triple M=max(s); camera=P.target+camerafactor*(abs(M-m)+abs(m-P.target))* unit(P.vector()); } real[][] depth=new real[s.s.length][]; for(int i=0; i < depth.length; ++i) depth[i]=new real[] {dot(P.normal,camera-s.s[i].cornermean()),i}; depth=sort(depth); for(int p=depth.length-1; p >= 0; --p) { real[] a=depth[p]; int k=round(a[1]); draw3D(f,s.s[k],surfacepen[k],light); } if(group) endgroup3(f); pen modifiers=thin()+squarecap; for(int p=depth.length-1; p >= 0; --p) { real[] a=depth[p]; int k=round(a[1]); patch S=s.s[k]; pen meshpen=meshpen[k]; if(!invisible(meshpen) && !S.triangular) { if(group) begingroup3(f,meshname(name),render); meshpen=modifiers+meshpen; real step=nu == 0 ? 0 : 1/nu; for(int i=0; i <= nu; ++i) draw(f,S.uequals(i*step),meshpen,meshlight,partname(i,render), render); step=nv == 0 ? 0 : 1/nv; for(int j=0; j <= nv; ++j) draw(f,S.vequals(j*step),meshpen,meshlight,partname(j,render), render); if(group) endgroup3(f); } } } } if(!is3D || settings.render == 0) { begingroup(f); // Sort patches by mean distance from camera triple camera=P.camera; if(P.infinity) { triple m=min(s); triple M=max(s); camera=P.target+camerafactor*(abs(M-m)+abs(m-P.target))*unit(P.vector()); } real[][] depth=new real[s.s.length][]; for(int i=0; i < depth.length; ++i) depth[i]=new real[] {dot(P.normal,camera-s.s[i].cornermean()),i}; depth=sort(depth); light.T=shiftless(P.T.modelview); // Draw from farthest to nearest for(int p=depth.length-1; p >= 0; --p) { real[] a=depth[p]; int k=round(a[1]); tensorshade(t,f,s.s[k],surfacepen[k],light,P); pen meshpen=meshpen[k]; if(!invisible(meshpen)) draw(f,t*project(s.s[k].external(),P),meshpen); } endgroup(f); } } void draw(transform t=identity(), frame f, surface s, int nu=1, int nv=1, material surfacepen=currentpen, pen meshpen=nullpen, light light=currentlight, light meshlight=nolight, string name="", render render=defaultrender, projection P=currentprojection) { material[] surfacepen={surfacepen}; pen[] meshpen={meshpen}; surfacepen.cyclic=true; meshpen.cyclic=true; draw(t,f,s,nu,nv,surfacepen,meshpen,light,meshlight,name,render,P); } void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1, material[] surfacepen, pen[] meshpen=nullpens, light light=currentlight, light meshlight=nolight, string name="", render render=defaultrender) { if(s.empty()) return; bool cyclic=surfacepen.cyclic; surfacepen=copy(surfacepen); surfacepen.cyclic=cyclic; cyclic=meshpen.cyclic; meshpen=copy(meshpen); meshpen.cyclic=cyclic; pic.add(new void(frame f, transform3 t, picture pic, projection P) { surface S=t*s; if(is3D()) draw(f,S,nu,nv,surfacepen,meshpen,light,meshlight,name,render); if(pic != null) { pic.add(new void(frame f, transform T) { draw(T,f,S,nu,nv,surfacepen,meshpen,light,meshlight,P); },true); pic.addPoint(min(S,P)); pic.addPoint(max(S,P)); } },true); pic.addPoint(min(s)); pic.addPoint(max(s)); pen modifiers; if(is3D()) modifiers=thin()+squarecap; for(int k=0; k < s.s.length; ++k) { patch S=s.s[k]; pen meshpen=meshpen[k]; if(!invisible(meshpen) && !S.triangular) { meshpen=modifiers+meshpen; real step=nu == 0 ? 0 : 1/nu; for(int i=0; i <= nu; ++i) addPath(pic,s.s[k].uequals(i*step),meshpen); step=nv == 0 ? 0 : 1/nv; for(int j=0; j <= nv; ++j) addPath(pic,s.s[k].vequals(j*step),meshpen); } } } void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1, material surfacepen=currentpen, pen meshpen=nullpen, light light=currentlight, light meshlight=nolight, string name="", render render=defaultrender) { material[] surfacepen={surfacepen}; pen[] meshpen={meshpen}; surfacepen.cyclic=true; meshpen.cyclic=true; draw(pic,s,nu,nv,surfacepen,meshpen,light,meshlight,name,render); } void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1, material[] surfacepen, pen meshpen, light light=currentlight, light meshlight=nolight, string name="", render render=defaultrender) { pen[] meshpen={meshpen}; meshpen.cyclic=true; draw(pic,s,nu,nv,surfacepen,meshpen,light,meshlight,name,render); } surface extrude(path3 p, path3 q) { static patch[] allocate; return surface(...sequence(new patch(int i) { return patch(subpath(p,i,i+1)--subpath(q,i+1,i)--cycle); },length(p))); } surface extrude(path3 p, triple axis=Z) { return extrude(p,shift(axis)*p); } surface extrude(path p, triple plane(pair)=XYplane, triple axis=Z) { return extrude(path3(p,plane),axis); } surface extrude(explicit path[] p, triple axis=Z) { surface s; for(path g:p) s.append(extrude(g,axis)); return s; } triple rectify(triple dir) { real scale=max(abs(dir.x),abs(dir.y),abs(dir.z)); if(scale != 0) dir *= 0.5/scale; dir += (0.5,0.5,0.5); return dir; } path3[] align(path3[] g, transform3 t=identity4, triple position, triple align, pen p=currentpen) { if(determinant(t) == 0 || g.length == 0) return g; triple m=min(g); triple dir=rectify(inverse(t)*-align); triple a=m+realmult(dir,max(g)-m); return shift(position+align*labelmargin(p))*t*shift(-a)*g; } surface align(surface s, transform3 t=identity4, triple position, triple align, pen p=currentpen) { if(determinant(t) == 0 || s.s.length == 0) return s; triple m=min(s); triple dir=rectify(inverse(t)*-align); triple a=m+realmult(dir,max(s)-m); return shift(position+align*labelmargin(p))*t*shift(-a)*s; } surface surface(Label L, triple position=O, bool bbox=false) { surface s=surface(texpath(L,bbox=bbox)); return L.align.is3D ? align(s,L.T3,position,L.align.dir3,L.p) : shift(position)*L.T3*s; } private path[] path(Label L, pair z=0, projection P) { path[] g=texpath(L,bbox=P.bboxonly); return L.align.is3D ? align(g,z,project(L.align.dir3,P)-project(O,P),L.p) : shift(z)*g; } transform3 alignshift(path3[] g, transform3 t=identity4, triple position, triple align) { if(determinant(t) == 0) return identity4; triple m=min(g); triple dir=rectify(inverse(t)*-align); triple a=m+realmult(dir,max(g)-m); return shift(-a); } transform3 alignshift(surface s, transform3 t=identity4, triple position, triple align) { if(determinant(t) == 0) return identity4; triple m=min(s); triple dir=rectify(inverse(t)*-align); triple a=m+realmult(dir,max(s)-m); return shift(-a); } transform3 aligntransform(path3[] g, transform3 t=identity4, triple position, triple align, pen p=currentpen) { if(determinant(t) == 0) return identity4; triple m=min(g); triple dir=rectify(inverse(t)*-align); triple a=m+realmult(dir,max(g)-m); return shift(position+align*labelmargin(p))*t*shift(-a); } transform3 aligntransform(surface s, transform3 t=identity4, triple position, triple align, pen p=currentpen) { if(determinant(t) == 0) return identity4; triple m=min(s); triple dir=rectify(inverse(t)*-align); triple a=m+realmult(dir,max(s)-m); return shift(position+align*labelmargin(p))*t*shift(-a); } void label(frame f, Label L, triple position, align align=NoAlign, pen p=currentpen, light light=nolight, string name="", render render=defaultrender, interaction interaction=LabelInteraction(), projection P=currentprojection) { bool prc=prc(); Label L=L.copy(); L.align(align); L.p(p); if(interaction.targetsize && settings.render != 0) L.T=L.T*scale(abs(P.camera-position)/abs(P.vector())); transform3 T=transform3(P); if(L.defaulttransform3) L.T3=T; if(is3D()) { bool lighton=light.on(); if(name == "") name=L.s; if(prc() && interaction.type == Billboard.type) { surface s=surface(texpath(L)); transform3 centering=L.align.is3D ? alignshift(s,L.T3,position,L.align.dir3) : identity4; transform3 positioning= shift(L.align.is3D ? position+L.align.dir3*labelmargin(L.p) : position); frame f1,f2,f3; begingroup3(f1,name,render); if(L.defaulttransform3) begingroup3(f3,render,position,interaction.type); else { begingroup3(f2,render,position,interaction.type); begingroup3(f3,render,position); } for(patch S : s.s) { S=centering*S; draw3D(f3,S,position,L.p,light,interaction); // Fill subdivision cracks if(prc && render.labelfill && opacity(L.p) == 1 && !lighton) _draw(f3,S.external(),position,L.p,light,interaction); } endgroup3(f3); if(L.defaulttransform3) add(f1,T*f3); else { add(f2,inverse(T)*L.T3*f3); endgroup3(f2); add(f1,T*f2); } endgroup3(f1); add(f,positioning*f1); } else { begingroup3(f,name,render); for(patch S : surface(L,position).s) { triple V=L.align.is3D ? position+L.align.dir3*labelmargin(L.p) : position; draw3D(f,S,V,L.p,light,interaction); // Fill subdivision cracks if(prc && render.labelfill && opacity(L.p) == 1 && !lighton) _draw(f,S.external(),V,L.p,light,interaction); } endgroup3(f); } } else { pen p=color(L.T3*Z,L.p,light,shiftless(P.T.modelview)); if(L.defaulttransform3) { if(L.filltype == NoFill) fill(f,path(L,project(position,P.t),P),p); else { frame d; fill(d,path(L,project(position,P.t),P),p); add(f,d,L.filltype); } } else for(patch S : surface(L,position).s) fill(f,project(S.external(),P,1),p); } } void label(picture pic=currentpicture, Label L, triple position, align align=NoAlign, pen p=currentpen, light light=nolight, string name="", render render=defaultrender, interaction interaction=LabelInteraction()) { Label L=L.copy(); L.align(align); L.p(p); L.position(0); pic.add(new void(frame f, transform3 t, picture pic2, projection P) { // Handle relative projected 3D alignments. bool prc=prc(); Label L=L.copy(); triple v=t*position; if(!align.is3D && L.align.relative && L.align.dir3 != O && determinant(P.t) != 0) L.align(L.align.dir*unit(project(v+L.align.dir3,P.t)-project(v,P.t))); if(interaction.targetsize && settings.render != 0) L.T=L.T*scale(abs(P.camera-v)/abs(P.vector())); transform3 T=transform3(P); if(L.defaulttransform3) L.T3=T; if(is3D()) { bool lighton=light.on(); if(name == "") name=L.s; if(prc && interaction.type == Billboard.type) { surface s=surface(texpath(L,bbox=P.bboxonly)); if(s.s.length > 0) { transform3 centering=L.align.is3D ? alignshift(s,L.T3,v,L.align.dir3) : identity4; transform3 positioning= shift(L.align.is3D ? v+L.align.dir3*labelmargin(L.p) : v); frame f1,f2,f3; begingroup3(f1,name,render); if(L.defaulttransform3) begingroup3(f3,render,v,interaction.type); else { begingroup3(f2,render,v,interaction.type); begingroup3(f3,render,v); } for(patch S : s.s) { S=centering*S; draw3D(f3,S,v,L.p,light,interaction); // Fill subdivision cracks if(prc && render.labelfill && opacity(L.p) == 1 && !lighton) _draw(f3,S.external(),v,L.p,light,interaction); } endgroup3(f3); if(L.defaulttransform3) add(f1,T*f3); else { add(f2,inverse(T)*L.T3*f3); endgroup3(f2); add(f1,T*f2); } endgroup3(f1); add(f,positioning*f1); } } else { begingroup3(f,name,render); for(patch S : surface(L,v,bbox=P.bboxonly).s) { triple V=L.align.is3D ? v+L.align.dir3*labelmargin(L.p) : v; draw3D(f,S,V,L.p,light,interaction); // Fill subdivision cracks if(prc && render.labelfill && opacity(L.p) == 1 && !lighton) _draw(f,S.external(),V,L.p,light,interaction); } endgroup3(f); } } if(pic2 != null) { pen p=color(L.T3*Z,L.p,light,shiftless(P.T.modelview)); if(L.defaulttransform3) { if(L.filltype == NoFill) fill(project(v,P.t),pic2,path(L,P),p); else { picture d; fill(project(v,P.t),d,path(L,P),p); add(pic2,d,L.filltype); } } else pic2.add(new void(frame f, transform T) { for(patch S : surface(L,v).s) fill(f,T*project(S.external(),P,1),p); }); } },!L.defaulttransform3); Label L=L.copy(); if(interaction.targetsize && settings.render != 0) L.T=L.T*scale(abs(currentprojection.camera-position)/ abs(currentprojection.vector())); path[] g=texpath(L,bbox=true); if(g.length == 0 || (g.length == 1 && size(g[0]) == 0)) return; if(L.defaulttransform3) L.T3=transform3(currentprojection); path3[] G=path3(g); G=L.align.is3D ? align(G,L.T3,O,L.align.dir3,L.p) : L.T3*G; pic.addBox(position,position,min(G),max(G)); } void label(picture pic=currentpicture, Label L, path3 g, align align=NoAlign, pen p=currentpen, light light=nolight, string name="", interaction interaction=LabelInteraction()) { Label L=L.copy(); L.align(align); L.p(p); bool relative=L.position.relative; real position=L.position.position.x; if(L.defaultposition) {relative=true; position=0.5;} if(relative) position=reltime(g,position); if(L.align.default) { align a; a.init(-I*(position <= sqrtEpsilon ? S : position >= length(g)-sqrtEpsilon ? N : E),relative=true); a.dir3=dir(g,position); // Pass 3D direction via unused field. L.align(a); } label(pic,L,point(g,position),light,name,interaction); } surface extrude(Label L, triple axis=Z) { Label L=L.copy(); path[] g=texpath(L); surface S=extrude(g,axis); surface s=surface(g); S.append(s); S.append(shift(axis)*s); return S; } restricted surface nullsurface; // Embed a Label onto a surface. surface surface(Label L, surface s, real uoffset, real voffset, real height=0, bool bottom=true, bool top=true) { int nu=s.index.length; int nv; if(nu == 0) nu=nv=1; else { nv=s.index[0].length; if(nv == 0) nv=1; } path[] g=texpath(L); pair m=min(g); pair M=max(g); pair lambda=inverse(L.T*scale(nu-epsilon,nv-epsilon))*(M-m); lambda=(abs(lambda.x),abs(lambda.y)); path[] G=bezulate(g); path3 transpath(path p, real height) { return path3(unstraighten(p),new triple(pair z) { real u=uoffset+(z.x-m.x)/lambda.x; real v=voffset+(z.y-m.y)/lambda.y; if(((u < 0 || u >= nu) && !s.ucyclic()) || ((v < 0 || v >= nv) && !s.vcyclic())) { warning("cannotfit","cannot fit string to surface"); u=v=0; } return s.point(u,v)+height*unit(s.normal(u,v)); }); } surface s; for(path p : G) { for(path g : regularize(p)) { path3 b; bool extrude=height > 0; if(bottom || extrude) b=transpath(g,0); if(bottom) s.s.push(patch(b)); if(top || extrude) { path3 h=transpath(g,height); if(top) s.s.push(patch(h)); if(extrude) s.append(extrude(b,h)); } } } return s; } private real a=4/3*(sqrt(2)-1); private transform3 t1=rotate(90,O,Z); private transform3 t2=t1*t1; private transform3 t3=t2*t1; private transform3 i=xscale3(-1)*zscale3(-1); // Degenerate first octant restricted patch octant1x=patch(X{Y}..{-X}Y{Z}..{-Y}Z..Z{X}..{-Z}cycle, new triple[] {(1,a,a),(a,1,a),(a^2,a,1), (a,a^2,1)}); surface octant1(real transition) { private triple[][][] P=hsplit(octant1x.P,transition); private patch P0=patch(P[0]); private patch P1=patch(P[1][0][0]..controls P[1][1][0] and P[1][2][0].. P[1][3][0]..controls P[1][3][1] and P[1][3][2].. P[1][3][3]..controls P[1][0][2] and P[1][0][1].. cycle,O); // Set internal control point of P1 to match normals at P0.point(1/2,1). triple n=P0.normal(1/2,1); triple[][] P=P1.P; triple u=-P[0][0]-P[1][0]+P[2][0]+P[3][0]; triple v=-P[0][0]-2*P[1][0]+P[1][1]-P[2][0]+P[3][1]; triple w=cross(u,v+(0,0,2)); real i=0.5*(n.z*w.x/n.x-w.z)/(u.x-u.y); P1.P[2][1]=(i,i,1); return surface(P0,P1); } // Nondegenerate first octant restricted surface octant1=octant1(0.95); restricted surface unithemisphere=surface(octant1,t1*octant1,t2*octant1, t3*octant1); restricted surface unitsphere=surface(octant1,t1*octant1,t2*octant1,t3*octant1, i*octant1,i*t1*octant1,i*t2*octant1, i*t3*octant1); unitsphere.draw= new void(frame f, transform3 t=identity4, material[] m, light light=currentlight, render render=defaultrender) { material m=material(m[0],light); drawSphere(f,t,half=false,m.p,m.opacity,m.shininess,m.metallic,m.fresnel0, render.sphere); }; unithemisphere.draw= new void(frame f, transform3 t=identity4, material[] m, light light=currentlight, render render=defaultrender) { material m=material(m[0],light); drawSphere(f,t,half=true,m.p,m.opacity,m.shininess,m.metallic,m.fresnel0, render.sphere); }; restricted patch unitfrustum1(real ta, real tb) { real s1=interp(ta,tb,1/3); real s2=interp(ta,tb,2/3); return patch(interp(Z,X,tb){Y}..{-X}interp(Z,Y,tb)--interp(Z,Y,ta){X}..{-Y} interp(Z,X,ta)--cycle, new triple[] {(s2,s2*a,1-s2),(s2*a,s2,1-s2),(s1*a,s1,1-s1), (s1,s1*a,1-s1)}); } restricted surface unitfrustum(real ta, real tb) { patch p=unitfrustum1(ta,tb); return surface(p,t1*p,t2*p,t3*p); } restricted surface unitcone=surface(unitfrustum(0,1)); restricted surface unitsolidcone=surface(patch(unitcircle3)...unitcone.s); // Construct an approximate cone over an arbitrary base. surface cone(path3 base, triple vertex) {return extrude(base,vertex--cycle);} private patch unitcylinder1=patch(X{Y}..{-X}Y--Y+Z{X}..{-Y}X+Z--cycle); restricted surface unitcylinder=surface(unitcylinder1,t1*unitcylinder1, t2*unitcylinder1,t3*unitcylinder1); drawfcn unitcylinderDraw(bool core) { return new void(frame f, transform3 t=identity4, material[] m, light light=currentlight, render render=defaultrender) { material m=material(m[0],light); drawCylinder(f,t,m.p,m.opacity,m.shininess,m.metallic,m.fresnel0, m.opacity == 1 ? core : false); }; } unitcylinder.draw=unitcylinderDraw(false); private patch unitplane=patch(new triple[] {O,X,X+Y,Y}); restricted surface unitcube=surface(reverse(unitplane), rotate(90,O,X)*unitplane, rotate(-90,O,Y)*unitplane, shift(Z)*unitplane, rotate(90,X,X+Y)*unitplane, rotate(-90,Y,X+Y)*unitplane); restricted surface unitplane=surface(unitplane); restricted surface unitdisk=surface(unitcircle3); unitdisk.draw= new void(frame f, transform3 t=identity4, material[] m, light light=currentlight, render render=defaultrender) { material m=material(m[0],light); drawDisk(f,t,m.p,m.opacity,m.shininess,m.metallic,m.fresnel0); }; void dot(frame f, triple v, material p=currentpen, light light=nolight, string name="", render render=defaultrender, projection P=currentprojection) { if(name == "" && render.defaultnames) name="dot"; pen q=(pen) p; real size=0.5*linewidth(dotsize(q)+q); transform3 T=shift(v)*scale3(size); draw(f,T*unitsphere,p,light,name,render,P); } void dot(frame f, triple[] v, material p=currentpen, light light=nolight, string name="", render render=defaultrender, projection P=currentprojection) { if(v.length > 0) { // Remove duplicate points. v=sort(v,lexorder); triple last=v[0]; dot(f,last,p,light,name,render,P); for(int i=1; i < v.length; ++i) { triple V=v[i]; if(V != last) { dot(f,V,p,light,name,render,P); last=V; } } } } void dot(frame f, path3 g, material p=currentpen, light light=nolight, string name="", render render=defaultrender, projection P=currentprojection) { dot(f,sequence(new triple(int i) {return point(g,i);},size(g)), p,light,name,render,P); } void dot(frame f, path3[] g, material p=currentpen, light light=nolight, string name="", render render=defaultrender, projection P=currentprojection) { int sum; for(path3 G : g) sum += size(G); int i,j; dot(f,sequence(new triple(int) { while(j >= size(g[i])) { ++i; j=0; } triple v=point(g[i],j); ++j; return v; },sum),p,light,name,render,P); } void dot(picture pic=currentpicture, triple v, material p=currentpen, light light=nolight, string name="", render render=defaultrender) { pen q=(pen) p; real size=0.5*linewidth(dotsize(q)+q); pic.add(new void(frame f, transform3 t, picture pic, projection P) { triple V=t*v; dot(f,V,p,light,name,render,P); if(pic != null) dot(pic,project(V,P.t),q); },true); triple R=size*(1,1,1); pic.addBox(v,v,-R,R); } void dot(picture pic=currentpicture, triple[] v, material p=currentpen, light light=nolight, string name="", render render=defaultrender) { if(v.length > 0) { // Remove duplicate points. v=sort(v,lexorder); triple last=v[0]; bool group=name != "" || render.defaultnames; if(group) begingroup3(pic,name == "" ? "dots" : name,render); dot(pic,last,p,light,partname(0,render),render); int k=0; for(int i=1; i < v.length; ++i) { triple V=v[i]; if(V != last) { dot(pic,V,p,light,partname(++k,render),render); last=V; } } if(group) endgroup3(pic); } } void dot(picture pic=currentpicture, explicit path3 g, material p=currentpen, light light=nolight, string name="", render render=defaultrender) { dot(pic,sequence(new triple(int i) {return point(g,i);},size(g)), p,light,name,render); } void dot(picture pic=currentpicture, path3[] g, material p=currentpen, light light=nolight, string name="", render render=defaultrender) { int sum; for(path3 G : g) sum += size(G); int i,j; dot(pic,sequence(new triple(int) { while(j >= size(g[i])) { ++i; j=0; } triple v=point(g[i],j); ++j; return v; },sum),p,light,name,render); } void dot(picture pic=currentpicture, Label L, triple v, align align=NoAlign, string format=defaultformat, material p=currentpen, light light=nolight, string name="", render render=defaultrender) { Label L=L.copy(); if(L.s == "") { if(format == "") format=defaultformat; L.s="("+format(format,v.x)+","+format(format,v.y)+","+ format(format,v.z)+")"; } L.align(align,E); L.p((pen) p); dot(pic,v,p,light,name,render); label(pic,L,v,render); } void pixel(picture pic=currentpicture, triple v, pen p=currentpen, real width=1) { real h=0.5*width; pic.add(new void(frame f, transform3 t, picture pic, projection P) { triple V=t*v; if(is3D()) drawpixel(f,V,p,width); if(pic != null) { triple R=h*unit(cross(unit(P.vector()),P.up)); pair z=project(V,P.t); real h=0.5*abs(project(V+R,P.t)-project(V-R,P.t)); pair r=h*(1,1)/mm; fill(pic,box(z-r,z+r),p,false); } },true); triple R=h*(1,1,1); pic.addBox(v,v,-R,R); } pair minbound(triple[] A, projection P) { pair b=project(A[0],P); for(triple v : A) b=minbound(b,project(v,P.t)); return b; } pair maxbound(triple[] A, projection P) { pair b=project(A[0],P); for(triple v : A) b=maxbound(b,project(v,P.t)); return b; } pair minbound(triple[][] A, projection P) { pair b=project(A[0][0],P); for(triple[] a : A) { for(triple v : a) { b=minbound(b,project(v,P.t)); } } return b; } pair maxbound(triple[][] A, projection P) { pair b=project(A[0][0],P); for(triple[] a : A) { for(triple v : a) { b=maxbound(b,project(v,P.t)); } } return b; } triple[][] operator / (triple[][] a, real[][] b) { triple[][] A=new triple[a.length][]; for(int i=0; i < a.length; ++i) { triple[] ai=a[i]; real[] bi=b[i]; A[i]=sequence(new triple(int j) {return ai[j]/bi[j];},ai.length); } return A; } // Draw a NURBS curve. void draw(picture pic=currentpicture, triple[] P, real[] knot, real[] weights=new real[], pen p=currentpen, string name="", render render=defaultrender) { P=copy(P); knot=copy(knot); weights=copy(weights); pic.add(new void(frame f, transform3 t, picture pic, projection Q) { if(is3D()) { triple[] P=t*P; bool group=name != "" || render.defaultnames; if(group) begingroup3(f,name == "" ? "curve" : name,render); draw(f,P,knot,weights,p); if(group) endgroup3(f); if(pic != null) pic.addBox(minbound(P,Q),maxbound(P,Q)); } },true); pic.addBox(minbound(P),maxbound(P)); } // Draw a NURBS surface. void draw(picture pic=currentpicture, triple[][] P, real[] uknot, real[] vknot, real[][] weights=new real[][], material m=currentpen, pen[] colors=new pen[], light light=currentlight, string name="", render render=defaultrender) { if(colors.length > 0) m=mean(colors); m=material(m,light); bool lighton=light.on(); P=copy(P); uknot=copy(uknot); vknot=copy(vknot); weights=copy(weights); colors=copy(colors); pic.add(new void(frame f, transform3 t, picture pic, projection Q) { if(is3D()) { bool group=name != "" || render.defaultnames; if(group) begingroup3(f,name == "" ? "surface" : name,render); triple[][] P=t*P; draw(f,P,uknot,vknot,weights,m.p,m.opacity,m.shininess,m.metallic, m.fresnel0,colors); if(group) endgroup3(f); if(pic != null) pic.addBox(minbound(P,Q),maxbound(P,Q)); } },true); pic.addBox(minbound(P),maxbound(P)); }