Quadratic Surfaces  --   Rectangular, Cylindrical and Spherical Coordinates

 

 

In a Nut Shell:  Besides planes in three dimensions there exist a family of cylinders

and quadric surfaces that are of interest in mathematics.  A quadric surface is the

graph of a second-degree equation in three variables  x, y, and z.  You may be asked

to identify and draw figures of these cylindrical and quadric surfaces.

 

 

 

 

 

Definition of Cylinders    A cylinder is a surface that consists of all lines (called rulings)

that are parallel to a given line and pass through a given plane curve.

 

Three examples include:

 

x2  + y2  =  a                    a cylinder with its axis along the z-axis of radius a

 

y2  + z2  =  a                     a cylinder with its axis along the x-axis of radius a

 

z  =  x2                              a parabolic cylinder with rulings parallel to the y-axis

 

 

 

 

 

The family of quadric surfaces include the ellipsoid, the elliptic paraboloid, the

elliptic cone, the hyperboloid of one sheet, the hyperboloid of two sheets, and the

hyperbolic paraboloid.  The table below gives expressions for each quadric surface.

 

 

Table of Quadric Surfaces   (Current software lets you plot these surfaces.)

 

(x/a)2  +  (y/b)2  +  (z/c)2   =  1              ellipsoid

 

(x/a)2  +  (y/b)2  =  (z/c)                    elliptic paraboloid

 

(x/a)2  +  (y/b)2  =  (z/c)2                  elliptical cone

 

(x/a)2  +  (y/b)2  -  (z/c)2   = 1            hyperboloid of one sheet

 

(z/c)2  -  (x/a)2  -  (y/b)2  =   1           hyperboloid of two sheets

 

(y/b)2   -  (x/a)2  =   z/c     c > 0        hyperbolic paraboloid

 

 

 

Click here to continue with a discussion of coordinate systems.

 



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