Parametric equations with two parameters

Example 1: The Great Wall of China

The following notebook must be evaluated in the order given.This is because it reinitializes the variables x, y, and z.

(*Evaluate First*)
Needs["VectorFieldPlots`"]

(*The Line Integral*)
(*The Parametric Equations for the Line Integral Problem*)
x[t_]=Cos[t];
y[t_]=4Sin[t];
F[X_,Y_]:=Sin[X]+Cos[Y]+3;
Q1:=ParametricPlot3D[{x[t],y[t],0},{t,0,((3π)/2)}]
Q2:=ParametricPlot3D[{x[t],y[t],F[x[t],y[t]]+0.05},{t,0,((3π)/2)}]
Q3:=Plot3D[F[X,Y],{X,-2π,2π},{Y,-2π,2π},Mesh->False]
Q4:=ParametricPlot3D[{x[t],y[t],u F[x[t],y[t]]},{t,0,((3π)/2)},{u,0,1},Boxed->False,Axes->False,AspectRatio->1]
Q5:=Plot3D[0,{X,-2π,2π},{Y,-2π,2π},Mesh->False]

(*The Curve and Projection of the Curve onto the surface*)
Show[{Q1,Q2,Q3,Q5},Boxed->False,Axes->False,AspectRatio->1,PlotRange->All]

(*The Surface calculated by the Line Integral*)
GraphicsRow[{Show[{Q1,Q2,Q4},Boxed->False,Axes->False,AspectRatio->1,PlotRange->All],Show[{Q1,Q2,Q4,Q5},Boxed->False,Axes->False,AspectRatio->1,PlotRange->All]}]

Example 2: The Sphere

r[u_,v_]:={2Sin[v]Cos[u],2Sin[v]Sin[u],2Cos[v]}

Manipulate[Show[Table[ParametricPlot3D[r[u,v],{u,0,2π}],{v,0,vMax,(π/5)}],PlotRange->{{-3,3},{-3,3},{-3,3}}],{{vMax,(π/5),"max value of v"},(π/5),π,(π/5)}]

Manipulate[Show[Table[ParametricPlot3D[r[u,v],{v,0,π}],{u,0,uMax,(π/5)}],PlotRange->{{-3,3},{-3,3},{-3,3}}],{{uMax,(π/5),"max value of u"},(π/5),2π,(π/5)}]

Manipulate[
(*Mesh*)
Show[{Table[ParametricPlot3D[r[u,v],{u,0,2π}],{v,0,vMax,(π/5)}],Table[ParametricPlot3D[r[u,v],{v,0,π}],{u,0,uMax,(π/5)}]},PlotRange->{{-3,3},{-3,3},{-3,3}},AxesLabel->{x,y,z},Ticks->None]
,{{uMax,(π/5),"max value of u"},(π/5),2π,(π/5)},{{vMax,(π/5),"max value of v"},(π/5),π,(π/5)}]

Manipulate[
GraphicsRow[{

(*Region*)
ParametricPlot[{u,v},{u,0,uMax},{v,0,vMax},PlotRange->{{0,2π},{0,π}},Ticks->{{π,2π},{π}},AxesLabel->{u,v},Frame->False,Mesh->False],

(*Mesh*)
Show[{Table[ParametricPlot3D[r[u,v],{u,0,2π}],{v,0,vMax,(π/5)}],Table[ParametricPlot3D[r[u,v],{v,0,π}],{u,0,uMax,(π/5)}]},PlotRange->{{-3,3},{-3,3},{-3,3}},AxesLabel->{x,y,z},Ticks->None],

(*Final Graph*)
ParametricPlot3D[r[u,v],{u,0,2π},{v,0,π},PlotRange->{{-3,3},{-3,3},{-3,3}},AxesLabel->{x,y,z},Ticks->None]
}] (*end of Graphics Row*)

,{{uMax,(π/5),"max value of u"},(π/5),2π,(π/5)},{{vMax,(π/5),"max value of v"},(π/5),π,(π/5)}]

Manipulate[
GraphicsRow[{

(*Region*)
ParametricPlot[{u,v},{u,uMax-(π/5),uMax},{v,vMax-(π/5),vMax},PlotRange->{{0,2π},{0,π}},Ticks->{{π,2π},{π}},AxesLabel->{u,v},Frame->False,Mesh->False],

(*Mesh*)
Show[{Table[ParametricPlot3D[r[u,v],{u,0,2π}],{v,vMax-(π/5),vMax,(π/5)}],Table[ParametricPlot3D[r[u,v],{v,0,π}],{u,uMax-(π/5),uMax,(π/5)}]},PlotRange->{{-3,3},{-3,3},{-3,3}},AxesLabel->{x,y,z},Ticks->None],

(*Final Graph*)
ParametricPlot3D[r[u,v],{u,uMax-(π/5),uMax},{v,vMax-(π/5),vMax},PlotRange->{{-3,3},{-3,3},{-3,3}},AxesLabel->{x,y,z},Ticks->None]
}] (*end of Graphics Row*)

,{{uMax,(π/5),"interval in u"},(π/5),2π,(π/5)},{{vMax,(π/5),"interval in v"},(π/5),π,(π/5)}]

Example 3: The Lattice

r[u_,v_]:={(2+Sin[v])Cos[u],(2+Sin[v])Sin[u],u+Cos[v]}

Manipulate[Show[Table[ParametricPlot3D[r[u,v],{u,0,4π}],{v,0,vMax,(π/5)}],PlotRange->{{-3,3},{-3,3},{0-1,4π+1}}],{{vMax,(π/5),"max value of v"},(π/5),2π,(π/5)}]

Manipulate[Show[Table[ParametricPlot3D[r[u,v],{v,0,2π}],{u,0,uMax,(π/5)}],PlotRange->{{-3,3},{-3,3},{0-1,4π+1}}],{{uMax,(π/5),"max value of u"},(π/5),4π,(π/5)}]

Manipulate[
Show[{Table[ParametricPlot3D[r[u,v],{u,0,4π}],{v,0,vMax,(π/5)}],Table[ParametricPlot3D[r[u,v],{v,0,2π}],{u,0,uMax,(π/5)}]},PlotRange->{{-3,3},{-3,3},{0-1,4π+1}},AxesLabel->{x,y,z},Ticks->None],
{{uMax,(π/5),"max value of u"},(π/5),4π,(π/5)},{{vMax,(π/5),"max value of v"},(π/5),2π,(π/5)}]

Manipulate[
GraphicsRow[{

(*Region*)
ParametricPlot[{u,v},{u,0,uMax},{v,0,vMax},PlotRange->{{0,4π},{0,2π}},Ticks->{{π,2π},{π,2π,3π,4π}},AxesLabel->{u,v},Frame->False,Mesh->False],

(*Mesh*)
Show[{Table[ParametricPlot3D[r[u,v],{u,0,4π}],{v,0,vMax,(π/5)}],Table[ParametricPlot3D[r[u,v],{v,0,2π}],{u,0,uMax,(π/5)}]},PlotRange->{{-3,3},{-3,3},{0-1,4π+1}},AxesLabel->{x,y,z},Ticks->None],

(*Final Graph*)
ParametricPlot3D[r[u,v],{u,0,4π},{v,0,2π},PlotRange->{{-3,3},{-3,3},{0-1,4π+1}},AxesLabel->{x,y,z},Ticks->None]
}] (*end of Graphics Row*)

,{{uMax,(π/5),"max value of u"},(π/5),4π,(π/5)},{{vMax,(π/5),"max value of v"},(π/5),2π,(π/5)}]