12.6 Cylinders and Quadric Surfaces

Cylinders

Definition: A cylinder is a surface that consists of all lines that are parallel to a given line and pass through a given curve.

Example: a right-circular cylinder (y^2)+(z^2)=4.

ContourPlot3D[(y^2)+(z^2)==4,{x,-1,5},{y,-3,3},{z,-3,3}, AxesLabel->Automatic]

Example: a parabolic cylinder z=(x^2).

ContourPlot3D[z==(x^2),{x,-3,3},{y,-3,3},{z,-1,5}, AxesLabel->Automatic]

Example: a sinusoidal cylinder z=sin(x).

ContourPlot3D[z==Sin[x],{x,-3Pi,3Pi},{y,-10,10},{z,-5,5}, AxesLabel->Automatic]

Quadric Surfaces

Definition: a Quadric Surface is the graph of a second-degree polynomial equation in the variables x, y, and z.

The most general form for this type of equation is:

A (x^2)+B (y^2)+C (z^2)+D x y+E y z+F x z +G x+H y+I z+J=0.

However, through the use of translations, rotations and change of variables it can be reduced to one of the two forms:

a (x^2)+b (y^2)+c (z^2)+d=0, or z=a (x^2)+b (y^2).

See page 854 for a table of the different types of quadric surfaces.  We will look at a few examples here.

Example: elliptic paraboloid

p1=ContourPlot3D[z==4 (x^2)+(y^2),{x,-4,4},{y,-4,4},{z,-1,10}, AxesLabel->Automatic];
p2=ContourPlot[4(x^2)+(y^2),{x,-3,3},{y,-3,3},FrameLabel->Automatic];
GraphicsGrid[{{p1,p2}}]

Example: a hyperbolic paraboloid

ContourPlot3D[x==(z^2)-(y^2),{x,-3,3},{y,-3,3},{z,-3,3},AxesLabel->Automatic]

Note that the “level curves” are parabolas when slicing for constant z and constant y and are hyperbolas for constant x.