Parametric Surfaces

Breather Surface

a=0.498888;vmax=47.1232;w=Sqrt[1-a^2];

Breather[u_,v_]:=Module[{d=a((w Cosh[a u])^2+(a Sin[w v])^2),x,y,z},x=-u+2w^2 Cosh[a u]Sinh[a u]/d;y=2w Cosh[a u](-w Cos[v]Cos[w v]-Sin[v]Sin[w v])/d;z=2w Cosh[a u](-w Sin[v]Cos[w v]+Cos[v]Sin[w v])/d;
{x,y,z}];

gr=ParametricPlot3D[Breather[u,v],{u,-10,10},{v,0,vmax},PlotPoints->320,ColorFunction->(Directive[ColorData["DeepSeaColors"][#5/vmax],Specularity[White,128]]&),Mesh->None,ColorFunctionScaling->False,ImageSize->{600,600},Boxed->False,Axes->False,SphericalRegion->True,ViewAngle->π/13,Background->Black,Lighting->"Neutral"]

Geometry Examples

The following examples all came from:
http://mathmuse.sci.ibaraki.ac.jp/geom/param1E.html

ParametricPlot3D[{((10 (Cos[u]+3) Cos[v])/(1+(1/u))),((10 (Cos[u]+3) Sin[v])/(1+(1/u))),u Sin[v]},{u,(π/10),70 π},{v,0,2 π},ViewPoint->{2,2,-2},PlotPoints->{75,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{Cos[u] Cos[v],Sin[u] v,Sin[u] Cos[v]},{u,-π,π},{v,-π,π},PlotPoints->80,Boxed->False,Axes->False,Mesh->None,ImageSize->{600,600},Background->Black,Lighting->"Neutral",PlotStyle->Directive[Orange,Specularity[White,50]]]

ParametricPlot3D[{((2 (Cos[u]+u Sin[u]) Sin[v])/(1+u u Sin[v] Sin[v])),((2 (Sin[u]-u Cos[u]) Sin[v])/(1+u u Sin[v] Sin[v])),Log[Tan[(v/2)]]+((2 Cos[v])/(1+u u Sin[v] Sin[v]))},{u,-4,4},{v,0.05,π},AspectRatio->Automatic,ViewPoint->{0.988,3.85,2.656},PlotPoints->120,Boxed->False,Axes->False,ImageSize->{600,600},MeshShading->{{Directive[RGBColor[.25,.43,.82],Specularity[White,128]],Directive[White,Specularity[White,128]]},{Directive[White,Specularity[White,128]],Directive[RGBColor[.25,.43,.82],Specularity[White,128]]}},Lighting->"Neutral",Background->Black,SphericalRegion->True,ViewAngle->π/15]

ParametricPlot3D[{Cos[t] (3+Cos[u]),Sin[t] (3+Cos[u]),Sec[2 t] Sin[2 u]},{t,0,2 π},{u,0,2 π},PlotPoints->{50,50},Boxed->False,Axes->False,ImageSize->{600,600},Mesh->80,MeshShading->{Flatten@Table[{Directive[ColorData["SouthwestColors"][i],Specularity[White,128]],FaceForm[None]},{i,0,1,.1}],Flatten@Table[{FaceForm[None],Directive[ColorData["SouthwestColors"][i],Specularity[White,128]]},{i,0,1,.1}]},Lighting->"Neutral",Background->Black,SphericalRegion->True,ViewAngle->π/10]

ParametricPlot3D[{(1/2) (r^2) Cos[2 phi]-Log[r],-phi-(1/2) (r^2) Sin[2 phi],2 r Cos[phi]},{r,0.005,2.},{phi,-2 π,3 π},ViewPoint->{-1.,-1.,1.},PlotPoints->{10,100},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{Sin[u] Cos[v],2 u Cos[v],2 v Cos[u]},{u,-π,π},{v,-π,π},ViewPoint->{-1.14,2.25,1.85},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{2 (3+Cos[v]) Cos[u],((2 Sin[2 u])/(1+Cos[(v^2)])),2 (E^(2 Sin[v]))},{u,0,2 π},{v,0,2 π},ViewPoint->{5.199,-3.074,2.9},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{t Sin[s] Cos[t],t Sin[s] Sin[t],1.5 t Cos[s]},{s,0,4 π},{t,0,2 π},ViewPoint->{2,1,0.7},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{(0.1+0.7 Sin[u]) 1.5 Sin[v],(0.1+0.7 Cos[u]) 1.5 Sin[v],0.5 Sin[u]+(v/5)},{u,0,2 π},{v,-3 π,3 π},ViewPoint->{0,-2,0},PlotPoints->120,Boxed->False,Axes->False,Mesh->None,ImageSize->{600,600},ColorFunction->(Directive[Specularity[White,128],ColorData["LakeColors"][#3]]&),Background->Black,SphericalRegion->True,ViewAngle->π/6,Lighting->Dynamic@{{"Directional",White,ImageScaled[{-.24,.01,.5}]}}]

ParametricPlot3D[{(2+0.5 Cos[u] Cos[v]) Cos[v] Sin[v],(2+0.3 Cos[u]) Sin[v],0.5 Sin[u] Sin[v]+(v/5)},{u,0,2 π},{v,-3 π,3 π},PlotPoints->{25,100},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{Cos[u],Sin[v],Cos[u-v]},{u,0,2 π},{v,0,2 π},ViewPoint->{-1.997,2.3,2.218},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500,PlotStyle->{{Orange,Specularity[White,20]}}]

ParametricPlot3D[{Cos[Sin[x]]+Sin[y] Cos[(x^2)],Sin[Cos[y]]-Cos[y] Sin[(x^2)],x+y},{x,-π,π},{y,-π,π},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

g1=ParametricPlot3D[{2 Sin[x],((Cos[3 x])/x)+(y/2),2 Cos[3 y]},{x,-π,-0.01},{y,1,2 π},PlotPoints->{25,50},Mesh->None];
g2=ParametricPlot3D[{2 Sin[x],((Cos[3 x])/x)+(y/2),2 Cos[3 y]},{x,0.01,π},{y,1,2 π},PlotPoints->{25,50},Mesh->None];
Show[{g1,g2},ViewPoint->{1.299,0.445,-8.885},Boxed->False,Axes->False,ImageSize->500]

ParametricPlot3D[{Cos[x+(y^2)]+Cos[x] Sin[(y^2)],Sin[x]+Cos[x],2 Cos[y]+Sin[(y^2)]},{x,0,2 π},{y,0,2 π},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{u Sin[t],u Cos[t],(t/3)},{t,0,20},{u,-1,1},PlotPoints->{50,25},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{(1+0.3 Cos[5 s]) Sin[s] Cos[t],(1+0.3 Cos[5 s]) Sin[s] Sin[t],(1+0.3 Cos[5 s]) Cos[s]},{s,0,2 π},{t,0,2 π},ViewPoint->{2,1,0.7},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{Cos[t] Cos[s+12 t],Cos[t] Sin[s+12 t],Sin[t]},{t,0,2 π},{s,-0.2,0.2},PlotPoints->{200,2},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{3 Sin[u]-(u^2) Sin[u],2 Cos[v]-3 Sin[v],-3 Cos[u] Cos[v]},{u,-π,π},{v,-π,π},ViewPoint->{4.234,-1.992,2.003},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{6 Cos[(u/3)] (1+Sin[u])+4 (1-(1/2) u Cos[u]) Cos[v+π],8 Cos[3 u] Sinh[Cos[2 v]],4 (1-((Cos[u])/2)) Sin[2 v]},{u,0,2 π},{v,0,2 π},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{(2+0.8 Cos[u]) Cos[v],(2+0.8 Cos[u]) Sin[v],0.8 Sin[u]+(v/2)},{u,0,2 π},{v,-3 π,3 π},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{Sin[t] Cos[p],Sin[t] Sin[p],Sin[2 t]},{t,0,π},{p,0,((3 π)/2)},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{Cos[t] (2+Cos[(t/2)] Sin[u]-Sin[(t/2)] Sin[4 u]),Sin[t] (2+Cos[(t/2)] Sin[u]-Sin[(t/2)] Sin[4 u]),Cos[(t/2)] Sin[2 u]+Sin[(t/2)] Cos[2 u]},{t,-(π/4),((3 π)/2)},{u,0,2 π},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

ParametricPlot3D[{Cos[t] (2+Cos[2 t] Sin[(u/2)]-Sin[2 t] Sin[2 u]),Sin[t] (2+Cos[2 t] Sin[(u/2)]-Sin[2 t] Sin[2 u]),Cos[2 t] Sin[u]+Sin[t] Cos[2 u]},{t,-π,((2 π)/2)},{u,0,2 π},PlotPoints->{100,100},Boxed->False,Axes->False,Mesh->None,ImageSize->500,PlotStyle->{{Orange,Specularity[White,128]}},Lighting->{{White,ImageScaled[{1,1,1}]}}]

ParametricPlot3D[{Cos[t] (Cos[((2 t)/3)] Sin[u]-Sin[((2 t)/3)] Sin[4 u]),Sin[t] (Cos[((2 t)/3)] Sin[4 u]-Sin[((2 t)/3)] Sin[u]),Cos[(t/3)] Sin[u]+Sin[(t/3)] Cos[u]},{t,-(π/4),((3 π)/2)},{u,0,2 π},PlotPoints->{50,50},Boxed->False,Axes->False,Mesh->None,ImageSize->500]

g1=ParametricPlot3D[{Sin[v] Cos[u],Sin[v] Sin[u],Cos[v]},{u,-π,π},{v,0,π},ViewPoint->{0.659,3.311,-0.226},PlotPoints->{50,25},Boxed->False,Axes->False,Mesh->None,ImageSize->500];
g2=ParametricPlot3D[{Sin[v] Cos[u],Sin[v] Sin[u],Cos[v]-3.5},{u,-π,π},{v,0,π},ViewPoint->{0.659,3.311,-0.226},PlotPoints->{50,25},Boxed->False,Axes->False,Mesh->None,ImageSize->500];
g3=ParametricPlot3D[{(1/2) Tan[v] Sin[u],(1/2) Tan[v] Cos[u],2 v-3},{u,-π,π},{v,0,(π/2)-0.01},ViewPoint->{0.659,3.311,-0.226},PlotPoints->{50,25},Boxed->False,Axes->False,Mesh->None,ImageSize->500];
g4=ParametricPlot3D[{(1/2) Tan[v] Sin[u],(1/2) Tan[v] Cos[u],2 v-3},{u,-π,π},{v,(π/2)+0.01,π},ViewPoint->{0.659,3.311,-0.226},PlotPoints->{50,25},Boxed->False,Axes->False,Mesh->None,ImageSize->500];
Show[{g1,g2,g3,g4},PlotRange->5]

Gumdrop

gr=ContourPlot3D[4(x^4+(y^2+z^2)^2)+17 x^2 (y^2+z^2)-20 (x^2+y^2+z^2)+17==0,{x,-2,2},{y,-2,2},{z,-2,2},ContourStyle->Directive[White,Specularity[White,128]],PlotPoints->80,Mesh->None,Boxed->False,Axes->False,Background->Black,Lighting->{{"Directional",RGBColor[.25,.43,.82], ImageScaled[{0, 1, 0}]}, 
     {"Directional",RGBColor[.49, 0, 0], ImageScaled[{1, -1, 0}]}, {"Directional",RGBColor[.6, .73, 0.36], ImageScaled[{-1, -1, 0}]}},SphericalRegion->True,ViewAngle->π/10,ImageSize->{600,600}]

Richmond Surface

Richmond[n_,z_]:={-1/(2z)-z^(2n+1)/(4n+2),-I/(2z)+I z^(2n+1)/(4n+2),z^n/n};

ParametricPlot3D[Evaluate@Re[Richmond[5,r Exp[I theta]]],{r,0.53,1.187},{theta,0,2Pi},PlotPoints->{160,320},SphericalRegion->True,ViewAngle->π/10,Boxed->False,Background->Black,Axes->False,ImageSize->{600,600},Mesh->20,MeshFunctions->{Sin[#5]^4+ Cos[#5]^4+Sin[#4]^4+Cos[#4]^4&},MeshStyle->GrayLevel[.2],MeshShading->{FaceForm[None],Directive[Gray,Specularity[White,30]],Directive[Blue,Specularity[White,30]]},Lighting->{{"Directional",RGBColor[.25,.43,.82], ImageScaled[{0, 1, 0}]}, 
     {"Directional",RGBColor[.49, 0, 0], ImageScaled[{1, -1, 0}]}, {"Directional",RGBColor[.6, .73, 0.36], ImageScaled[{-1, -1, 0}]},{"Directional",GrayLevel[.5], ImageScaled[{1, 1, 1}]}}]

Stacked Torii

Inspired from http://www.microscopy-uk.org.uk/mag/indexmag.html?http://www.microscopy-uk.org.uk/mag/artjan06/bjmaths.html

gr=ParametricPlot3D[Evaluate@{{Cos[t] (3+(Sin[1.5t]^2+.3)Cos[u]),Sin[t] (3+(Sin[1.5t]^2+.3)Cos[u]),(Sin[1.5t]^2+.3)Sin[u]},{Cos[t] (3+(Sin[3t]^2+.3)Cos[u]),Sin[t] (3+(Sin[3t]^2+.3)Cos[u]),3+(Sin[3t]^2+.3)Sin[u]},{Cos[t] (3+(Sin[4 t]^2+.3)Cos[u]),Sin[t] (3+(Sin[4 t]^2+.3)Cos[u]),6+(Sin[4 t]^2+.3)Sin[u]},{Cos[t] (3+(Sin[5 t]^2+.3)Cos[u]),Sin[t] (3+(Sin[5 t]^2+.3)Cos[u]),9+(Sin[5 t]^2+.3)Sin[u]},{Cos[t] (3+(Sin[6 t]^2+.3)Cos[u]),Sin[t] (3+(Sin[6 t]^2+.3)Cos[u]),12+(Sin[6 t]^2+.3)Sin[u]}},{t,0,2 Pi},{u,0,2 Pi},Mesh->{120,60},BoundaryStyle->Directive[GrayLevel[.2],Thickness[0]],PlotStyle->{Directive[RGBColor[.25,.43,.82],Specularity[White,50]],Directive[RGBColor[.6,.73,.36],Specularity[White,50]],Directive[RGBColor[.27,.2,.75],Specularity[White,50]],Directive[RGBColor[.48,.11,.56],Specularity[White,50]],Directive[RGBColor[.88,.63,.23],Specularity[White,50]]},MeshStyle->Directive[GrayLevel[.2],Thickness[0]],PlotPoints->30,ImageSize->{600,600},Lighting->{{"Ambient",GrayLevel[.1]},{"Directional",GrayLevel[.7],ImageScaled[{1,1,1}]},{"Directional",RGBColor[.7,.45,0],ImageScaled[{-1,1,1}]}},Background->Black,Boxed->False,Axes->False,SphericalRegion->True,ViewAngle->π/10]