Surfaces

Cylinders

Let C be a curve, then we define a cylinder to be the set of all lines through C and perpendicular to the plane that C lies in.

         


We can tell that an equation is a cylinder is it is missing one of the variables.

 

Quadric Surfaces

Recall that the quadrics or conics are lines , hyperbolas, parabolas, circles, and ellipses.  In three dimensions, we can combine any two of these and make a quadric surface.  For example

       

is a paraboloid since for constant z we get a circle and for constant x or y we get a parabola.  We use the suffix -oid to mean ellipse or circle.  We have:


   

  •        x2          y2          z2
                 +           +            =  1 is an ellipsoid     
          a2          b2          c2        

  •           x2         y2         z2
        -          -           +            =  1 is a hyperboloid of 2 sheets while     
            a2         b2          c2       

  •        x2          y2          z2
                 +           -            =  1 is a hyperboloid of 1 sheet     
          a2          b2          c2        

        

Surface of Revolution

Let y = f(x) be a curve, then the equation of the surface of revolution abut the x-axis is

        y2 + z2 = f(x)2



Example

Find the equation of the surface that is formed when the curve

        y   =   sin x          0  <  x  < p/2

is revolved around the y-axis.

Solution

This uses a different formula since this time the curve is revolved around the y-axis.  The circular cross section has radius sin-1 y and the circle is perpendicular to the y-axis.  Hence the equation is

        x2 + z2 = (sin-1 y)2

 

 

Cylindrical Coordinates

We can extend polar coordinates to three dimensions by

     x = rcosq
     y = rsinq
     z = z


Example  

We can write (1,1,3) in cylindrical coordinates.  We find 

       

and

         

so that the cylindrical coordinates are

        (, p/4, 3) 


Spherical Coordinates

An alternate coordinate system works on a distance and two angle method called spherical coordinates.  We let r denote the distance from the point to the origin, q represent the same q as in cylindrical coordinates, and f denote the angle from the positive z-axis to the point.  The picture tells us that

        r  =  r sin f  

and that 

        z  =  r cos f  

From this we can find

     x = rcosq = r sin f cosq
     y = r sinq = r sin f sinq
     z = r cos f


Immediately we see that 

        x2 + y2 + z2 = r2

We use spherical coordinates whenever the problem involves a distance from a source.

 

Example

convert the surface 

        z  =  x2 + y2

to an equation in spherical coordinates.  

Solution

We add z2 to both sides

        z + z2  =  x2 + y2 + z2 

Now it is easier to convert

        r cos f + r2 cos2 f  =  r2 

Divide by r to get

        cos f + r cos2 f  =  r

Now solve for r.

                      cos f                    cos f  
         r =                            =                     =  csc f  cot f   
                   1 - cos2 f                sin2 f  

 

 

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