Intersections of Surfaces

The intersection of a cone and a plane

plane = ParametricPlot3D[{x,t,1+t},{x,-5,5},{t,-5,5},ColorFunction->"RustTones"]

Here is the cone in the cylindrical parameterization:

r=z;
cone = ParametricPlot3D[{r Cos[theta],r Sin[theta],z},{theta,0,2Pi},{z,0,4},ColorFunction->"CherryTones"]

intersection = ParametricPlot3D[{t,(t^2-1)/2,(t^2+1)/2},{t,-5,5},PlotStyle->{Red,Thickness[0.02]}]

Show[plane,cone,intersection]

The intersection of a cylinder and a plane

plane = ParametricPlot3D[{x,t,2-t},{x,-2,2},{t,-3,3},ColorFunction->"CoffeeTones"]

Here is the cylinder in the cartesian cylindrical parameterization:

r = 1;
cylinder = ParametricPlot3D[{r Cos[theta],r Sin[theta],z},{theta,0,2Pi},{z,0,4},ColorFunction->"SolarColors"]

intersection = ParametricPlot3D[{Sqrt[1-(t-2)^2],2-t,t},{t,1,3},PlotStyle->{Red,Thickness[0.02]}]

intersection = ParametricPlot3D[{Cos[t],Sin[t],2-Sin[t]},{t,0,2Pi},PlotStyle->{Red,Thickness[0.02]}]

Show[plane,cylinder,intersection]

The intersection of a cylinder and a cone

r1 = 1;
cylinder1= ParametricPlot3D[{x,r1*Cos[theta],r1* Sin[theta]},{theta,0,2Pi},{x,-4,4},ColorFunction->"LakeColors"]

Here is the cylinder in the cartesian cylindrical parameterization:

r2 = 1;
cylinder2= ParametricPlot3D[{r2* Cos[theta],r2* Sin[theta],z},{theta,0,2Pi},{z,-4,4},ColorFunction->"SandyTerrain"]

Clear[r]
cone1=(y^2)+(z^2)==((Subscript[r, 1])^2)
cone2=(x^2)+(y^2)==((Subscript[r, 2])^2)

Solve[{cone1,cone2},{x,y,z}]

Which we can parameterize using y=t although we would need to do this four times, once for each solution.

(Subscript[r, 1])=(Subscript[r, 2])=1;
intersection1 = ParametricPlot3D[{-(Sqrt[-(t^2)+(Subscript[r, 2]^2)]),t,-(Sqrt[-(t^2)+(Subscript[r, 1]^2)])},{t,-1,1},PlotStyle->{Red,Thickness[0.02]}]
intersection2 = ParametricPlot3D[{-(Sqrt[-(t^2)+(Subscript[r, 2]^2)]),t,(Sqrt[-(t^2)+(Subscript[r, 1]^2)])},{t,-1,1},PlotStyle->{Red,Thickness[0.02]}];
intersection3 = ParametricPlot3D[{(Sqrt[-(t^2)+(Subscript[r, 2]^2)]),t,-(Sqrt[-(t^2)+(Subscript[r, 1]^2)])},{t,-1,1},PlotStyle->{Red,Thickness[0.02]}];
intersection4 = ParametricPlot3D[{(Sqrt[-(t^2)+(Subscript[r, 2]^2)]),t,(Sqrt[-(t^2)+(Subscript[r, 1]^2)])},{t,-1,1},PlotStyle->{Red,Thickness[0.02]}];

Show[cylinder1,cylinder2,intersection1,intersection2,intersection3,intersection4,PlotRange->{{-1,1},{-1,1},{-1,1}},AxesLabel->{"x","y","z"}]

YAI (Yet Another Intersection)

surface1= ParametricPlot3D[{(t^2+1),t*Sin[theta],Sin[theta]},{theta,0,2Pi},{t,-3,3},ColorFunction->"LakeColors",AxesLabel->{"x","y","z"}]

Here is the cylinder in the cartesian cylindrical parameterization:

surface2= ParametricPlot3D[{t*Cos[theta]+4, Sin[theta],t},{theta,0,2Pi},{t,-2,2},ColorFunction->"SandyTerrain",AxesLabel->{"x","y","z"}]

Show[surface1,surface2,PlotRange->{All,All,All}]

For the first surface, we have:

eq1=x==(t^2+1)
eq2=y==t*Sin[theta]
eq3=z==Sin[theta]
Eliminate[{eq1,eq2,eq3},{t,theta}]

For the second surface we have:

eq1=x==t*Cos[theta]+4
eq2=y== Sin[theta]
eq3=z==t
Eliminate[{eq1,eq2,eq3},{t,theta}]

Let's pick the last and see what we can learn:

Solve[{(y^2)+(z^2)==x (z^2),((((z-(y^2) z))^2)/(((4-x))^2))==1-(y^2)},{x,y,z}]

Which we can parameterize using z=t although we would need to do this six times, once for each solution.

(Subscript[r, 1])=(Subscript[r, 2])=1;
intersection1 = ParametricPlot3D[{((1+(z^2))/(z^2)),-1,z},{z,-1,1},PlotStyle->{Red,Thickness[0.02]}]
intersection2 = ParametricPlot3D[{((1+(z^2))/(z^2)),1,z},{z,-1,1},PlotStyle->{Red,Thickness[0.02]}]
intersection3 = ParametricPlot3D[{(1/2) (8-(z^4)-z (Sqrt[4-12 (z^2)+(z^6)])),-((Sqrt[6 (z^2)-(z^6)-(z^3) (Sqrt[4-12 (z^2)+(z^6)])])/(Sqrt[2])),z},{z,-1,1},PlotStyle->{Red,Thickness[0.02]}]
intersection4 = ParametricPlot3D[{(1/2) (8-(z^4)-z (Sqrt[4-12 (z^2)+(z^6)])),((Sqrt[6 (z^2)-(z^6)-(z^3) (Sqrt[4-12 (z^2)+(z^6)])])/(Sqrt[2])),z},{z,-1,1},PlotStyle->{Red,Thickness[0.02]}]
intersection5 = ParametricPlot3D[{(1/2) (8-(z^4)+z (Sqrt[4-12 (z^2)+(z^6)])),-(Sqrt[3 (z^2)-((z^6)/2)+(1/2) (z^3) (Sqrt[4-12 (z^2)+(z^6)])]),z},{z,-1,1},PlotStyle->{Red,Thickness[0.02]}]
intersection6 = ParametricPlot3D[{(1/2) (8-(z^4)+z (Sqrt[4-12 (z^2)+(z^6)])),(Sqrt[3 (z^2)-((z^6)/2)+(1/2) (z^3) (Sqrt[4-12 (z^2)+(z^6)])]),z},{z,-1,1},PlotStyle->{Red,Thickness[0.02]}]

Show[surface1,surface2,intersection1,intersection2,intersection3,intersection4,intersection5,intersection6,PlotRange->{{1,5},{-1,1},{-1,1}},AxesLabel->{"x","y","z"}]

Notice that some extraneous solut5ions crept in when we solved the two equations. See:

(y^2)+(z^2)==x (z^2)/.{y->-1,x->((1+(z^2))/(z^2))}
((((z-(y^2) z))^2)/(((4-x))^2))==1-y/.{y->-1,x->((1+(z^2))/(z^2))}

The first two are extraneous. The intersections are given by:

Show[surface1,surface2,intersection3,intersection4,intersection5,intersection6,PlotRange->{{1,5},{-2,2},{-1,1}},AxesLabel->{"x","y","z"}]