Curves of Surface Intersection

Example #1 : The cylinder (x^2)+(y^2)=1 and the parabolic cylinder z = (x^2)

Graph the parametric representations of each surface on the same graph to see the curve of intersection

Pcylinder = ParametricPlot3D[{Sin[u],Cos[u],v},{u,0,2*Pi},{v,-1,2}, PlotStyle->Red];

Pparabolic = ParametricPlot3D[{u, v, u^2},{u,-2,2},{v,-2,2}, PlotStyle->Blue];


Show[Pcylinder,Pparabolic, PlotRange->{-1,2}, AxesLabel->{"x","y","z"}, PlotLabel->"z = x^2 and x^2 + y^2 = 1"]

Define parametric equations for the curve of intersection

x[t_]:=Sin[t]; y[t_]:=Cos[t];z[t_]:=Sin[t]^2;

Plot the curve of intersection.  First by itself and then superimposed on the two surfaces.

CParaCylinder = ParametricPlot3D[{x[t],y[t],z[t]},{t,-5,5},ColorFunction->Function[{x,y,z},Hue[.65*(1-z)]], AxesLabel->{"x","y","z"},PlotLabel->"Parametric Plot of Intersection Curve Between x^2 + y^2 = 1 and z = x^2", PlotStyle->{Thick}, BoxRatios->{1,1,1}]

Show[CParaCylinder,Pcylinder,Pparabolic]

Example #2 : The cone z = (x^2)+(y^2) and the plane z = 1 + y

Graph the parametric representations of each surface on the same graph to see the curve of intersection

Pcone = Plot3D[Sqrt[x^2+y^2],{x,-5,5},{y,-5,5}, PlotStyle->Blue, BoxRatios->{1,1,1}];

Pplane = Plot3D[1+y,{x,-5,5},{y,-5,5},PlotStyle->Red, BoxRatios->{1,1,1}];

Show[Pcone,Pplane, AxesLabel->{"x","y","z"}, PlotLabel->"Graph of cone z = Sqrt[x^2+y^2] and plane z = 1+y"]

Define parametric equations for the curve of intersection

x[t_]:=t; y[t_]:=(1/2)*(t^2-1);z[t_]:=(1/2)*(t^2+1);

Plot the curve of intersection.  First by itself and then superimposed on the two surfaces.

CParaCone = ParametricPlot3D[{x[t],y[t],z[t]},{t,-3.5,3.5},ColorFunction->Function[{x,y,z},Hue[.65*(1-z)]],
AxesLabel->{"x","y","z"},PlotLabel->"Parametric Plot of Intersection Curve Between z = Sqrt[x^2+y^2] and z = 1 + y", PlotStyle->{Thick, Yellow}, BoxRatios->{1,1,1}]

Show[CParaCone,Pcone,Pplane]