2450_Cylinders and Quadric Surfaces
Part 1: Cylinders
Most people are familiar with the idea of a cylinder, more specifically, the idea of a very specific type of cylinder which we call a right-circular cylinder which is shown in the following figure. In this case, the generating line is parallel to the -axis.
Right-Circular Cylinder
However, we now want to extend our idea of a cylinder. A cylinder is a surface that consists of all lines that are parallel to a give line and pass through a given plane curve.
This means that we will use the idea of a cylinder to encompass ANY set of parallel lines which pass through a given plane curve.
For example, consider the hyperbolic cylinder given by
which is shown in the figure below. In this case, the generating line is parallel to the -axis.
Hyperbolic Cylinder
Question 1:
Using the two examples above as references, with which axis is the generating line parallel in the graph of the hyperbolic cylinder ?
We can even have elliptic cylinders, such as
whose surface is given in the figure below. Here the generating line is parallel to the -axis.
Elliptic Cylinder
Part 2: Quadric Surfaces
Some more interesting surfaces, called quadric surfaces, consist of the graphs of second-degree equations in the three variables , , and .
There are six primary types of surfaces we want to discuss in this section, each being given by a second-degree equation as defined above.
The following are examples of the six standard forms of quadric surfaces.
Ellipsoid
Elliptic Paraboloid
Hyperbolic Paraboloid
Circular Cone
Hyperboloid of One Sheet
Hyperboloid of Two Sheets
The next couple of examples allow you to see how you can manipulate certain values in the equations to alter the shape of the surfaces.
The first example is the ellipsoid which now only allows you to change the value of the constants in the denominator, but also shows you the respective traces in the plane after these changes occur.
Ellipsoid
Question 3:
What surface results when ?
In the final figure of this activity, we look at a special case of how a changing a constant can change the entire surface from one of the primary six into another.
Question 4:
There are three distinct surfaces that occur as k varies between [-5,5]. What three surfaces do you see here and for what values of k do they occur?