3D Plots of Quadric Surfaces 

Given a first-degree equation in three (or fewer) variables, we get a plane: 

(1)

 

Given a second-degree equation in two variables, we get a cylinder. 

For example: (parabolic cylinder), (circular cylinder): 

(2)

 

(3)

 

(4)

 

A quadric surface is the graph of a second-degree equation in (exactly) three variables. Thus its equation fits into the general form: 

(5)

 

Using a translation and a rotation, this equation can be put into a standard form which looks like one of the following: 

or  

(6)

 

(7)

 

Quadric surfaces only come in six flavors. Here they are... 

Ellipsoids:  

(8)

 

Of course, if then we have  

(9)

 

which is a Sphere (a very special kind of ellipsoid). 

Elliptic Paraboloids:  

(10)

 

Cones:  

(11)

 

Hyperboloids of One Sheet:  

(12)

 

Hyperboloids of Two Sheets:  

(13)

 

Hyperbolic Paraboloids:  

(14)