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.

Right-Circular Cylinder

However, we now want to extend our idea of what a cylinder can be. 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 parabolic cylinder given by

which is shown in the figure below.

Parabolic Cylinder

The cylinder consists of all the lines running parallel to the axis and through the parabola . To plot such a surface in Geogebra, you sould simply enter z = x^2 in the input line and press enter. Try plotting the parabolic cylinder in the graph below.

Question 1:

Using the two examples above as references, with which axis are the straight lines parallel in the graph of the parabolic cylinder ?

We can even have elliptic cylinders, such as

whose surface is given in the figure below.

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. These all have interesting connections to the idea of conic sections which will be discussed later in the semester. The following are examples of the six standard forms of quadric surfaces.

Ellipsoid

Elliptic Paraboloid

Hyperbolic Paraboloid

Cone

Hyperboloid of One Sheet

Hyperboloid of Two Sheets

To plot any of these in Geogebra, one simply enters the equation defining the quadric surface into the input line. For example, to plot the hyperboloid of two sheets given by in Geogebra, you would enter 1 = z^2 - x^2 - y^2 and hit enter. Geogebra understands that this is an implicitly defined surface and will draw it for you. The actual equations for these surfaces (given in your textbook) are more complicated than the examples I have given you here.

Question 2:

In the graph below, plot the surface given by the equation

What is the resulting surface?

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 one.

Question 4:

There are three distinct surfaces that occur as k varies between [-5,5]. What three surfaces do you see heree and for what values of k do they occur?

Question 5: