13.6 Cylinder and Quadric Surfaces

CYLINDERS

Definition: A cylinder is a surface composed of all lines parallel to a given line that pass through a given curve. A cylinder does not need to be a "round tube" as the common use of the term cylinder suggests. The place z = 2, for example, fits the definition of a cylinder - all lines parallel to the y-axis through the line L: x = t, y = 0, z = 2.

 

                                        

Traces – the intersections with a plane parallel to one of the xy-, xz- or yz- planes. Helps identifies a surface. Here are some cylinders with conic sections as traces.

Problems:

1.  True or False: If one of x, y, z is missing from an equation, the surface is a cylinder.

2.  Identify and sketch a graph. Check sketch on Maple using implicitplot3d.

          a)  x + 2y = 6

          b)  yz = 4          

          c)  x²/9 + z²/25 = 1

          d)  z = cos(x)

QUADRIC SURFACES

The analogues of the three conics in two dimensions are quadric surfaces in three dimensions. Again, it will help to first determine traces.

Problems:

3.  By just looking at the equation how would you know what the surface was?

4.  Find the traces of the given surface in the planes x = k, y = k, z = k.  Then identify the surface and sketch graph.  Check your answer in Maple.

      a)  x²/25 – y²/4 – z²/16 = 1

      b)  y - 4x² = 16z²

      c)  y² = x² + z²

       d)  x² – 8x – y² – y + z² + 15 = 0

       e)  z = x² - y²

       f)  z² = x² + 4y²

       g)  z = y² + xy

3.  Identify and give an equation that would generate the approximate shape. Check you answer in MAPLE.

a)


b)

 

c)