A Variety Of Surfaces 

with(plots): 

Ellipsoid 

implicitplot3d(x^2/9+y^2/4+z^2/16=1,x=-5..5,y=-5..5,z=-5..5,axes=normal,numpoints=2500,labels=[x,y,z]); 

Hyperboloid of One Sheet 

implicitplot3d(x^2/9+y^2/4-z^2/16=1,x=-7..7,y=-7..7,z=-7..7,axes=boxed,numpoints=2500,labels=[x,y,z]); 

Hyperboloid of One Sheet Shifted in the x- and y-Directions 

implicitplot3d((x-1)^2/9+(y-3)^2/4-z^2/16=1,x=-7..7,y=-7..7,z=-7..7,axes=boxed,numpoints=2500,labels=[x,y,z]); 

Hyperboloid of Two Sheets 

implicitplot3d(-x^2/9-y^2/4+z^2/9=1,x=-10..10,y=-10..10,z=-10..10,axes=boxed,numpoints=2500,labels=[x,y,z]); 

Elliptic Cone (missing the tips) 

implicitplot3d(x^2/9+y^2/4-z^2/16=0,x=-7..7,y=-7..7,z=-7..7,axes=boxed,numpoints=10000,labels=[x,y,z]); 

Elliptic Paraboloid 

implicitplot3d(z=x^2/9+y^2/4,x=-8..8,y=-7..7,z=-2..7,axes=boxed,numpoints=2500,labels=[x,y,z]); 

Hyperbolic Paraboloid 

implicitplot3d(z=x^2/9-y^2/4,x=-8..8,y=-7..7,z=-6..7,axes=boxed,numpoints=2500,labels=[x,y,z]); 

Cylinder 

implicitplot3d(36=9*x^2+4*y^2,x=-8..8,y=-7..7,z=-2..7,axes=boxed,numpoints=2500,labels=[x,y,z]); 

Cylinder 

implicitplot3d(z=4*sin(x),x=-2*Pi..2*Pi,y=-7..7,z=-5..5,axes=boxed,numpoints=10000,labels=[x,y,z]); 

Surface of Revolution 

SurfOfRev:=implicitplot3d(y^2+z^2=(x^3+2*x^2-3*x+10)^2,x=-2..3,y=-46..46,z=-46..46,axes=boxed,numpoints=2500,labels=[x,y,z]): 

>	GeneratingCurve:=spacecurve([t,0,t^3+2*t^2-3*t+10],t=-2..3,color=blue,thickness=4):

 

>	display(SurfOfRev,GeneratingCurve);

 

Curve of Intersection--Level Curve 

HypPar:=implicitplot3d(z=x^2/9-y^2/4,x=-8..8,y=-7..7,z=-6..7,axes=boxed,numpoints=2500,labels=[x,y,z]): 

Plane:=implicitplot3d(z=1,x=-8..8,y=-7..7,z=-6..7,axes=boxed,numpoints=2500,labels=[x,y,z],color=green): 

>	LevelCurve1:=spacecurve([3*sec(t),2*tan(t),1],t=-1.17..1.17,color=blue,thickness=4):

 

>	LevelCurve2:=spacecurve([3*sec(t),2*tan(t),1],t=1.97..4.31,color=blue,thickness=4):

 

display(HypPar,Plane,LevelCurve1,LevelCurve2); 

Trace in the xz-coordinate plane 

Paraboloid:=implicitplot3d(z=x^2/9+y^2/4,x=-8..8,y=-7..7,z=-2..7,axes=boxed,numpoints=2500,labels=[x,y,z]): 

Plane2:=implicitplot3d(y=0,x=-8..8,y=-7..7,z=-2..7,axes=boxed,numpoints=2500,labels=[x,y,z],color=green): 

>	LevelCurve3:=spacecurve([t,0,t^2/9],t=-8..8,color=blue,thickness=4):

 

display(Paraboloid,Plane2,LevelCurve3); 

If You Can't Parameterize The Trace You Can Do This. 

Intersection:=implicitplot3d(z=x^2/9+y^2/4,x=-8..8,y=-0.2..0.2,z=-2..7,axes=boxed,numpoints=1600,labels=[x,y,z],color=green,scaling=constrained): 

display(Intersection); 

display(Paraboloid,Intersection);