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Diffstat (limited to 'Build/source/utils/asymptote/base/three_surface.asy')
-rw-r--r-- | Build/source/utils/asymptote/base/three_surface.asy | 1343 |
1 files changed, 1343 insertions, 0 deletions
diff --git a/Build/source/utils/asymptote/base/three_surface.asy b/Build/source/utils/asymptote/base/three_surface.asy new file mode 100644 index 00000000000..6e540c416ff --- /dev/null +++ b/Build/source/utils/asymptote/base/three_surface.asy @@ -0,0 +1,1343 @@ +import bezulate; + +int nslice=12; +real camerafactor=1.2; + +private real Fuzz=10.0*realEpsilon; +private real nineth=1/9; + +struct patch { + triple[][] P=new triple[4][4]; + triple[] normals; // Optionally specify 4 normal vectors at the corners. + pen[] colors; // Optionally specify 4 corner colors. + bool straight; // Patch is based on a piecewise straight external path. + bool3 planar; // Patch is planar. + + path3 external() { + return + P[0][0]..controls P[0][1] and P[0][2].. + P[0][3]..controls P[1][3] and P[2][3].. + P[3][3]..controls P[3][2] and P[3][1].. + P[3][0]..controls P[2][0] and P[1][0]..cycle; + } + + triple[] internal() { + return new triple[] {P[1][1],P[1][2],P[2][2],P[2][1]}; + } + + triple cornermean() { + return 0.25*(P[0][0]+P[0][3]+P[3][3]+P[3][0]); + } + + triple[] corners() {return new triple[] {P[0][0],P[0][3],P[3][3],P[3][0]};} + + real[] map(real f(triple)) { + return new real[] {f(P[0][0]),f(P[0][3]),f(P[3][3]),f(P[3][0])}; + } + + triple[] controlpoints() { + return new triple[] { + P[0][0],P[0][1],P[0][2],P[0][3], + P[1][0],P[1][1],P[1][2],P[1][3], + P[2][0],P[2][1],P[2][2],P[2][3], + P[3][0],P[3][1],P[3][2],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);} + triple BuPP(int j, real u) { + return bezierPP(P[0][j],P[1][j],P[2][j],P[3][j],u); + } + triple BuPPP(int j) {return bezierPPP(P[0][j],P[1][j],P[2][j],P[3][j]);} + + path3 uequals(real u) { + return straight ? Bu(0,u)--Bu(3,u) : + Bu(0,u)..controls Bu(1,u) and Bu(2,u)..Bu(3,u); + } + + 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);} + triple BvPP(int i, real v) { + return bezierPP(P[i][0],P[i][1],P[i][2],P[i][3],v); + } + triple BvPPP(int i) {return bezierPPP(P[i][0],P[i][1],P[i][2],P[i][3]);} + + path3 vequals(real v) { + return straight ? Bv(0,v)--Bv(3,v) : + Bv(0,v)..controls Bv(1,v) and Bv(2,v)..Bv(3,v); + } + + triple point(real u, real v) { + return bezier(Bu(0,u),Bu(1,u),Bu(2,u),Bu(3,u),v); + } + + // compute normal vectors for degenerate cases + private triple normal0(real u, real v, real epsilon) { + triple n=0.5*(cross(bezier(BvPP(0,v),BvPP(1,v),BvPP(2,v),BvPP(3,v),u), + bezier(BuP(0,u),BuP(1,u),BuP(2,u),BuP(3,u),v))+ + cross(bezier(BvP(0,v),BvP(1,v),BvP(2,v),BvP(3,v),u), + bezier(BuPP(0,u),BuPP(1,u),BuPP(2,u),BuPP(3,u),v))); + return abs(n) > epsilon ? n : + 0.25*cross(bezier(BvPP(0,v),BvPP(1,v),BvPP(2,v),BvPP(3,v),u), + bezier(BuPP(0,u),BuPP(1,u),BuPP(2,u),BuPP(3,u),v))+ + 1/6*(cross(bezier(BvPPP(0),BvPPP(1),BvPPP(2),BvPPP(3),u), + bezier(BuP(0,u),BuP(1,u),BuP(2,u),BuP(3,u),v))+ + cross(bezier(BvP(0,v),BvP(1,v),BvP(2,v),BvP(3,v),u), + bezier(BuPPP(0),BuPPP(1),BuPPP(2),BuPPP(3),v)))+ + 1/12*(cross(bezier(BvPPP(0),BvPPP(1),BvPPP(2),BvPPP(3),u), + bezier(BuPP(0,u),BuPP(1,u),BuPP(2,u),BuPP(3,u),v))+ + cross(bezier(BvPP(0,v),BvPP(1,v),BvPP(2,v),BvPP(3,v),u), + bezier(BuPPP(0),BuPPP(1),BuPPP(2),BuPPP(3),v)))+ + 1/36*cross(bezier(BvPPP(0),BvPPP(1),BvPPP(2),BvPPP(3),u), + bezier(BuPPP(0),BuPPP(1),BuPPP(2),BuPPP(3),v)); + } + + static real fuzz=1000*realEpsilon; + + triple normal(real u, real v) { + triple n=cross(bezier(BvP(0,v),BvP(1,v),BvP(2,v),BvP(3,v),u), + bezier(BuP(0,u),BuP(1,u),BuP(2,u),BuP(3,u),v)); + real epsilon=fuzz*change2(P); + return (abs(n) > epsilon) ? n : normal0(u,v,epsilon); + } + + triple normal00() { + triple n=9*cross(P[0][1]-P[0][0],P[1][0]-P[0][0]); + real epsilon=fuzz*change2(P); + return abs(n) > epsilon ? n : normal0(0,0,epsilon); + } + + triple normal01() { + triple n=9*cross(P[0][3]-P[0][2],P[1][3]-P[0][3]); + real epsilon=fuzz*change2(P); + return abs(n) > epsilon ? n : normal0(0,1,epsilon); + } + + triple normal11() { + triple n=9*cross(P[3][3]-P[3][2],P[3][3]-P[2][3]); + real epsilon=fuzz*change2(P); + return abs(n) > epsilon ? n : normal0(1,1,epsilon); + } + + triple normal10() { + triple n=9*cross(P[3][1]-P[3][0],P[3][0]-P[2][0]); + real epsilon=fuzz*change2(P); + return abs(n) > epsilon ? n : normal0(1,0,epsilon); + } + + pen[] colors(material m, light light=currentlight) { + bool nocolors=colors.length == 0; + if(normals.length > 0) + return new pen[] {light.color(normals[0],nocolors ? m : colors[0]), + light.color(normals[1],nocolors ? m : colors[1]), + light.color(normals[2],nocolors ? m : colors[2]), + light.color(normals[3],nocolors ? m : colors[3])}; + if(planar) { + triple normal=normal(0.5,0.5); + return new pen[] {light.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])}; + } + return new pen[] {light.color(normal00(),nocolors ? m : colors[0]), + light.color(normal01(),nocolors ? m : colors[1]), + light.color(normal11(),nocolors ? m : colors[2]), + light.color(normal10(),nocolors ? m : colors[3])}; + } + + triple bound(real m(real[], real), triple b) { + real x=m(new real[] {P[0][0].x,P[0][1].x,P[0][2].x,P[0][3].x, + P[1][0].x,P[1][1].x,P[1][2].x,P[1][3].x, + P[2][0].x,P[2][1].x,P[2][2].x,P[2][3].x, + P[3][0].x,P[3][1].x,P[3][2].x,P[3][3].x},b.x); + real y=m(new real[] {P[0][0].y,P[0][1].y,P[0][2].y,P[0][3].y, + P[1][0].y,P[1][1].y,P[1][2].y,P[1][3].y, + P[2][0].y,P[2][1].y,P[2][2].y,P[2][3].y, + P[3][0].y,P[3][1].y,P[3][2].y,P[3][3].y},b.y); + real z=m(new real[] {P[0][0].z,P[0][1].z,P[0][2].z,P[0][3].z, + P[1][0].z,P[1][1].z,P[1][2].z,P[1][3].z, + P[2][0].z,P[2][1].z,P[2][2].z,P[2][3].z, + P[3][0].z,P[3][1].z,P[3][2].z,P[3][3].z},b.z); + return (x,y,z); + } + + triple min3,max3; + bool havemin3,havemax3; + + void init() { + havemin3=false; + havemax3=false; + } + + triple min(triple bound=P[0][0]) { + if(havemin3) return minbound(min3,bound); + havemin3=true; + return min3=bound(minbound,bound); + } + + triple max(triple bound=P[0][0]) { + if(havemax3) return maxbound(max3,bound); + havemax3=true; + return max3=bound(maxbound,bound); + } + + triple center() { + return 0.5*(this.min()+this.max()); + } + + pair min(projection P, pair bound=project(this.P[0][0],P.t)) { + return minbound(controlpoints(),P.t,bound); + } + + pair max(projection P, pair bound=project(this.P[0][0],P.t)) { + return maxbound(controlpoints(),P.t,bound); + } + + void operator init(triple[][] P, triple[] normals=new triple[], + pen[] colors=new pen[], bool straight=false, + bool3 planar=default) { + init(); + this.P=copy(P); + if(normals.length != 0) + this.normals=copy(normals); + if(colors.length != 0) + this.colors=copy(colors); + this.planar=planar; + this.straight=straight; + } + + void operator init(pair[][] P, triple plane(pair)=XYplane, + bool straight=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; + } + + void operator init(patch s) { + operator init(s.P,s.normals,s.colors,s.straight); + } + + // A constructor for a convex cyclic path3 of length <= 4 with optional + // arrays of 4 internal points, corner normals, and pens. + void operator init(path3 external, triple[] internal=new triple[], + triple[] normals=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; + + int L=length(external); + 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])); + if(normals.length > 0) normals.append(array(3,normals[0])); + } else if(L == 2) { + external=external--cycle--cycle; + if(colors.length > 0) colors.append(array(2,colors[0])); + if(normals.length > 0) normals.append(array(2,normals[0])); + } else if(L == 3) { + external=external--cycle; + if(colors.length > 0) colors.push(colors[0]); + if(normals.length > 0) normals.push(normals[0]); + } + if(normals.length != 0) + this.normals=copy(normals); + 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)); + } else straight=false; + + P=new triple[][] { + {point(external,0),postcontrol(external,0),precontrol(external,1), + point(external,1)}, + {precontrol(external,0),internal[0],internal[1],postcontrol(external,1)}, + {postcontrol(external,3),internal[3],internal[2],precontrol(external,2)}, + {point(external,3),precontrol(external,3),postcontrol(external,2), + point(external,2)} + }; + } + + // A constructor for a convex quadrilateral. + void operator init(triple[] external, triple[] internal=new triple[], + triple[] normals=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(normals.length != 0) + this.normals=copy(normals); + 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[0],external[1]-delta[0],external[1]}, + {external[0]-delta[3],internal[0],internal[1],external[1]+delta[1]}, + {external[3]+delta[3],internal[3],internal[2],external[2]-delta[1]}, + {external[3],external[3]-delta[2],external[2]+delta[2],external[2]} + }; + } +} + +patch operator * (transform3 t, patch s) +{ + patch S; + for(int i=0; i < 4; ++i) { + triple[] si=s.P[i]; + triple[] Si=S.P[i]; + for(int j=0; j < 4; ++j) + Si[j]=t*si[j]; + } + + transform3 t0=shiftless(t); + for(int i=0; i < s.normals.length; ++i) + S.normals[i]=t0*s.normals[i]; + + S.colors=copy(s.colors); + S.planar=s.planar; + S.straight=s.straight; + return S; +} + +patch reverse(patch s) +{ + patch S; + S.P=transpose(s.P); + if(s.normals.length > 0) + S.normals= + new triple[] {s.normals[0],s.normals[3],s.normals[2],s.normals[1]}; + if(s.colors.length > 0) + S.colors=new pen[] {s.colors[0],s.colors[3],s.colors[2],s.colors[1]}; + S.planar=s.planar; + S.straight=s.straight; + return S; +} + +struct surface { + patch[] s; + + 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]); + } + + void operator init(triple[][][] P, triple[][] normals=new triple[][], + pen[][] colors=new pen[][], bool3 planar=default) { + s=sequence(new patch(int i) { + return patch(P[i],normals.length == 0 ? new triple[] : normals[i], + colors.length == 0 ? new pen[] : colors[i],planar); + },P.length); + } + + void colors(pen[][] palette) { + for(int i=0; i < s.length; ++i) { + pen[] palettei=palette[i]; + s[i].colors=new pen[] {palettei[0],palettei[1],palettei[2],palettei[3]}; + } + } + + 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); + } + + // A constructor for a possibly nonconvex cyclic path in a given plane. + void operator init (path p, triple plane(pair)=XYplane, + bool checkboundary=true) { + if(!cyclic(p)) + abort("cyclic path expected"); + + int L=length(p); + + if(L > 4) { + for(path g : bezulate(p)) + s.append(surface(g,plane,checkboundary).s); + return; + } + + pair[][] P(path 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 new pair[][] { + {point(p,0),postcontrol(p,0),precontrol(p,1),point(p,1)}, + {precontrol(p,0),internal[0],internal[1],postcontrol(p,1)}, + {postcontrol(p,3),internal[3],internal[2],precontrol(p,2)}, + {point(p,3),precontrol(p,3),postcontrol(p,2),point(p,2)} + }; + } + + bool straight=piecewisestraight(p); + if(L <= 3 && straight) { + s=new patch[] {patch(P(p),plane,straight)}; + return; + } + + // 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*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=surface(subpath(g,0,cut)--cycle,plane,checkboundary).s; + s.append(surface(subpath(g,cut,L)--cycle,plane,checkboundary).s); + 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; + } + } + + pair[][] P=P(p); + + if(straight) { + s=new patch[] {patch(P,plane,straight)}; + return; + } + + // 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)) + s=new patch[] {patch(P,plane)}; + return; + } + + 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[0][j]-P[1][j])*P[0][i]).y; // u=0 + c[1] += w*(conj(P[i][3])*(P[j][3]-P[j][2])).y; // v=1 + c[2] += w*(conj(P[3][j]-P[2][j])*P[3][i]).y; // u=1 + c[3] += w*(conj(P[i][0])*(P[j][1]-P[j][0])).y; // v=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(BvP(0,v),BvP(1,v),BvP(2,v),BvP(3,v),u))* + bezier(BuP(0,u),BuP(1,u),BuP(2,u),BuP(3,u),v)).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={0,t,1,t}; + real[] V={t,1,t,0}; + 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; + } + + // 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; + + while(true) + for(int i=0; i < L; ++i) + if(!straight(p,i) && split(p,i+unitrand())) return; + } + + 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[], + triple[] normals=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,normals,colors,planar=true)); + return; + } + + // Construct a surface from a possibly nonconvex planar cyclic path3. + if(planar != false && internal.length == 0 && normals.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,normals,colors,planar)); + return; + } + + // Path is not planar; split into patches. + real factor=1/L; + pen[] p; + triple[] n; + bool nocolors=colors.length == 0; + bool nonormals=normals.length == 0; + triple center; + for(int i=0; i < L; ++i) + center += point(external,i); + center *= factor; + if(!nocolors) + p=new pen[] {mean(colors)}; + if(!nonormals) + n=new triple[] {factor*sum(normals)}; + // 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, + nonormals ? n : concat(normals[i:end+1],n), + nocolors ? p : concat(colors[i:end+1],p),planar)); + i=end; + } + s.push(patch(subpath(external,i,L)--center--cycle, + nonormals ? n : concat(normals[i:],normals[0:1],n), + nocolors ? p : concat(colors[i:],colors[0:1],p),planar)); + } + + void operator init(path3 external, triple[] internal=new triple[], + triple[] normals=new triple[], pen[] colors=new pen[], + bool3 planar=default) { + s=new patch[]; + construct(external,internal,normals,colors,planar); + } + + void operator init(explicit path3[] external, + triple[][] internal=new triple[][], + triple[][] normals=new triple[][], + pen[][] colors=new pen[][], bool3 planar=default) { + s=new patch[]; + for(int i=0; i < external.length; ++i) + construct(external[i], + internal.length == 0 ? new triple[] : internal[i], + normals.length == 0 ? new triple[] : normals[i], + colors.length == 0 ? new pen[] : colors[i],planar); + } + + void push(path3 external, triple[] internal=new triple[], + triple[] normals=new triple[] ,pen[] colors=new pen[], + bool3 planar=default) { + s.push(patch(external,internal,normals,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]; + 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; + + for(int i=0; i < L; ++i) { + path3 h=subpath(g,i,i+1); + path3 r=reverse(h); + triple max=max(h); + triple min=min(h); + triple perp=perp(max-c,axis); + real fuzz=epsilon*max(abs(max),abs(min)); + if(abs(perp) < fuzz) + perp=perp(min-c,axis); + perp=unit(perp); + triple normal=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=Tk*h{dirj}..{dirp}Tp*r{-dirp}..{-dirj}cycle; + Tk=Tp; + dirj=dirp; + s[++m]=color == null ? patch(G) : + patch(G,new pen[] {color(i,j),color(i+1,j),color(i+1,j+w), + color(i,j+w)}); + } + } + } + + 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]; + 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 for a subpatch of P on [u,1] x [v,1]. +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); + + u=1.0-u; + + triple[] c7=split(c3[2],c2[2],c1[2],c0[2],u); + triple[] c8=split(c3[3],c2[3],c1[3],c0[3],u); + triple[] c9=split(c3[4],c2[4],c1[4],c0[4],u); + triple[] c10=split(P3[3],P2[3],P1[3],P0[3],u); + + return new triple[][] {{c7[2],c8[2],c9[2],c10[2]}, + {c7[1],c8[1],c9[1],c10[1]}, + {c7[0],c8[0],c9[0],c10[0]}, + {c3[2],c3[3],c3[4],P3[3]}}; +} + +// Return the control points for a subpatch of P on [0,u] x [0,v]. +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); + + u=1.0-u; + + triple[] c4=split(P3[0],P2[0],P1[0],P0[0],u); + triple[] c5=split(c3[0],c2[0],c1[0],c0[0],u); + triple[] c6=split(c3[1],c2[1],c1[1],c0[1],u); + triple[] c7=split(c3[2],c2[2],c1[2],c0[2],u); + + return new triple[][] { + {P0[0],c0[0],c0[1],c0[2]}, + {c4[4],c5[4],c6[4],c7[4]}, + {c4[3],c5[3],c6[3],c7[3]}, + {c4[2],c5[2],c6[2],c7[2]}}; +} + +patch subpatch(patch s, real ua, real va, real ub, real vb) +{ + assert(ua >= 0 && va >= 0 && ub <= 1 && vb <= 1 && ua < ub && va < vb); + return patch(subpatchbegin(subpatchend(s.P,ub,vb),ua/ub,va/vb), + s.straight,s.planar); +} + +triple point(patch s, real u, real v) +{ + return s.point(u,v); +} + +void draw3D(frame f, patch s, material m, light light=currentlight) +{ + if(s.colors.length > 0) + m=mean(s.colors); + bool lighton=light.on(); + if(!lighton && !invisible((pen) m)) + m=emissive(m); + real granularity=m.granularity >= 0 ? m.granularity : defaultgranularity; + draw(f,s.P,s.straight,m.p,m.opacity,m.shininess,granularity, + s.planar ? s.normal(0.5,0.5) : O,lighton,s.colors); +} + +void tensorshade(transform t=identity(), frame f, patch s, + material m, light light=currentlight, projection P) +{ + tensorshade(f,box(t*s.min(P),t*s.max(P)),m.diffuse(), + s.colors(m,light),t*project(s.external(),P,1), + t*project(s.internal(),P)); +} + +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=light, + projection P=currentprojection) +{ + if(is3D()) { + for(int i=0; i < s.s.length; ++i) + draw3D(f,s.s[i],surfacepen[i],light); + pen modifiers=thin()+squarecap; + for(int k=0; k < s.s.length; ++k) { + pen meshpen=meshpen[k]; + if(!invisible(meshpen)) { + meshpen=modifiers+meshpen; + real step=nu == 0 ? 0 : 1/nu; + for(int i=0; i <= nu; ++i) + draw(f,s.s[k].uequals(i*step),meshpen,meshlight); + step=nv == 0 ? 0 : 1/nv; + for(int j=0; j <= nv; ++j) + draw(f,s.s[k].vequals(j*step),meshpen,meshlight); + } + } + } else { + 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[] {abs(camera-s.s[i].cornermean()),i}; + + depth=sort(depth); + + light.T=shiftless(P.modelview()); + + // Draw from farthest to nearest + while(depth.length > 0) { + real[] a=depth.pop(); + int i=round(a[1]); + tensorshade(t,f,s.s[i],surfacepen[i],light,P); + pen meshpen=meshpen[i]; + if(!invisible(meshpen)) + draw(f,t*project(s.s[i].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=light, + 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,P); +} + +void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1, + material[] surfacepen, pen[] meshpen=nullpens, + light light=currentlight, light meshlight=light) +{ + if(s.empty()) return; + + 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); + } else if(pic != null) + pic.add(new void(frame f, transform T) { + draw(T,f,S,nu,nv,surfacepen,meshpen,light,meshlight,P); + },true); + if(pic != null) { + 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) { + pen meshpen=meshpen[k]; + if(!invisible(meshpen)) { + 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=light) +{ + material[] surfacepen={surfacepen}; + pen[] meshpen={meshpen}; + surfacepen.cyclic(true); + meshpen.cyclic(true); + draw(pic,s,nu,nv,surfacepen,meshpen,light,meshlight); +} + +void draw(picture pic=currentpicture, surface s, int nu=1, int nv=1, + material[] surfacepen, pen meshpen, + light light=currentlight, light meshlight=light) +{ + pen[] meshpen={meshpen}; + meshpen.cyclic(true); + draw(pic,s,nu,nv,surfacepen,meshpen,light,meshlight); +} + +surface extrude(path p, triple axis=Z) +{ + static patch[] allocate; + path3 G=path3(p); + path3 G2=shift(axis)*G; + return surface(...sequence(new patch(int i) { + return patch(subpath(G,i,i+1)--subpath(G2,i+1,i)--cycle); + },length(G))); +} + +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) 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) 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) +{ + surface s=surface(texpath(L)); + return L.align.is3D ? align(s,L.T3,position,L.align.dir3,L.p) : + shift(position)*L.T3*s; +} + +path[] path(Label L, pair z=0, projection P) +{ + path[] g=texpath(L); + if(L.defaulttransform3) { + return L.align.is3D ? align(g,z,project(L.align.dir3,P)-project(O,P),L.p) : + shift(z)*g; + } else { + path3[] G=path3(g); + return L.align.is3D ? shift(z)*project(align(G,L.T3,O,L.align.dir3,L.p),P) : + shift(z)*project(L.T3*G,P); + } +} + +void label(frame f, Label L, triple position, align align=NoAlign, + pen p=currentpen, light light=nolight, + projection P=currentprojection) +{ + Label L=L.copy(); + L.align(align); + L.p(p); + if(L.defaulttransform3) + L.T3=transform3(P); + if(is3D()) { + for(patch S : surface(L,position).s) + draw3D(f,S,L.p,light); + } else { + if(L.filltype == NoFill) + fill(f,path(L,project(position,P.t),P), + light.color(L.T3*Z,L.p,shiftless(P.modelview()))); + else { + frame d; + fill(d,path(L,project(position,P.t),P), + light.color(L.T3*Z,L.p,shiftless(P.modelview()))); + add(f,d,L.filltype); + } + } +} + +void label(picture pic=currentpicture, Label L, triple position, + align align=NoAlign, pen p=currentpen, light light=nolight) +{ + Label L=L.copy(); + L.align(align); + L.p(p); + L.position(0); + path[] g=texpath(L); + if(g.length == 0 || (g.length == 1 && size(g[0]) == 0)) return; + pic.add(new void(frame f, transform3 t, picture pic, projection P) { + triple v=t*position; + if(L.defaulttransform3) + L.T3=transform3(P); + if(is3D()) + for(patch S : surface(L,v).s) + draw3D(f,S,L.p,light); + if(pic != null) { + if(L.filltype == NoFill) + fill(project(v,P.t),pic,path(L,P), + light.color(L.T3*Z,L.p,shiftless(P.modelview()))); + else { + picture d; + fill(project(v,P.t),d,path(L,P), + light.color(L.T3*Z,L.p,shiftless(P.modelview()))); + add(pic,d,L.filltype); + } + } + },!L.defaulttransform3); + + 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) +{ + Label L=Label(L,align,p); + bool relative=L.position.relative; + real position=L.position.position.x; + pair Align=L.align.dir; + bool alignrelative=L.align.relative; + if(L.defaultposition) {relative=true; position=0.5;} + if(relative) position=reltime(g,position); + if(L.align.default) { + alignrelative=true; + Align=position <= 0 ? S : position >= length(g) ? N : E; + } + label(pic,L,point(g,position), + alignrelative && determinant(currentprojection.t) != 0 ? + -Align*project(dir(g,position),currentprojection.t)*I : L.align); +} + +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; + +private real a=4/3*(sqrt(2)-1); +private transform3 t=rotate(90,O,Z); +private transform3 t2=t*t; +private transform3 t3=t2*t; +private transform3 i=xscale3(-1)*zscale3(-1); + +restricted patch octant1=patch(X{Z}..{-X}Z..Z{Y}..{-Z}Y{X}..{-Y}cycle, + new triple[] {(1,a,a),(a,a^2,1),(a^2,a,1), + (a,1,a)}); + +restricted surface unithemisphere=surface(octant1,t*octant1,t2*octant1, + t3*octant1); +restricted surface unitsphere=surface(octant1,t*octant1,t2*octant1,t3*octant1, + i*octant1,i*t*octant1,i*t2*octant1, + i*t3*octant1); + +restricted patch unitfrustum(real t1, real t2) +{ + real s1=interp(t1,t2,1/3); + real s2=interp(t1,t2,2/3); + return patch(interp(Z,X,t2)--interp(Z,X,t1){Y}..{-X}interp(Z,Y,t1)-- + interp(Z,Y,t2){X}..{-Y}cycle, + new triple[] {(s2,s2*a,1-s2),(s1,s1*a,1-s1),(s1*a,s1,1-s1), + (s2*a,s2,1-s2)}); +} + +// Return a unitcone constructed from n frusta (the final one being degenerate) +surface unitcone(int n=6) +{ + surface unitcone; + unitcone.s=new patch[4*n]; + real r=1/3; + for(int i=0; i < n; ++i) { + patch s=unitfrustum(i < n-1 ? r^(i+1) : 0,r^i); + unitcone.s[i]=s; + unitcone.s[n+i]=t*s; + unitcone.s[2n+i]=t2*s; + unitcone.s[3n+i]=t3*s; + } + return unitcone; +} + +restricted surface unitcone=unitcone(); +restricted surface unitsolidcone=surface(patch(unitcircle3)...unitcone.s); + +private patch unitcylinder1=patch(X--X+Z{Y}..{-X}Y+Z--Y{X}..{-Y}cycle); + +restricted surface unitcylinder=surface(unitcylinder1,t*unitcylinder1, + t2*unitcylinder1,t3*unitcylinder1); + +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); + +void dot(frame f, triple v, material p=currentpen, + light light=nolight, projection P=currentprojection) +{ + pen q=(pen) p; + if(is3D()) { + material m=material(p,p.granularity >= 0 ? p.granularity : dotgranularity); + for(patch s : unitsphere.s) + draw3D(f,shift(v)*scale3(0.5*linewidth(dotsize(q)+q))*s,m,light); + } else dot(f,project(v,P.t),q); +} + +void dot(frame f, path3 g, material p=currentpen, + projection P=currentprojection) +{ + for(int i=0; i <= length(g); ++i) dot(f,point(g,i),p,P); +} + +void dot(frame f, path3[] g, material p=currentpen, + projection P=currentprojection) +{ + for(int i=0; i < g.length; ++i) dot(f,g[i],p,P); +} + +void dot(picture pic=currentpicture, triple v, material p=currentpen, + light light=nolight) +{ + pen q=(pen) p; + real size=0.5*linewidth(dotsize(q)+q); + pic.add(new void(frame f, transform3 t, picture pic, projection P) { + if(is3D()) { + material m=material(p,p.granularity >= 0 ? p.granularity : + dotgranularity); + for(patch s : unitsphere.s) + draw3D(f,shift(t*v)*scale3(size)*s,m,light); + } + if(pic != null) + dot(pic,project(t*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) +{ + for(int i=0; i < v.length; ++i) dot(pic,v[i],p); +} + +void dot(picture pic=currentpicture, explicit path3 g, material p=currentpen) +{ + for(int i=0; i <= length(g); ++i) dot(pic,point(g,i),p); +} + +void dot(picture pic=currentpicture, path3[] g, material p=currentpen) +{ + for(int i=0; i < g.length; ++i) dot(pic,g[i],p); +} + +void dot(picture pic=currentpicture, Label L, triple v, align align=NoAlign, + string format=defaultformat, material p=currentpen) +{ + 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); + label(pic,L,v); +} |