Lesson 7:

3-Dimensional Figures & Conics

By Carly Coffman

Elliptical Paraboloids

Many of the two-dimensional figures we look at, come from cross sections of three-dimensional figures.  We explored this in the first lesson when discovering conics.  Parabolas, Ellipse, Circles, and Hyperbolas are all cross sections of a double cone.  Now we will explore some other three-dimensional figures and cross sections.

The first one we will look at is called an elliptical paraboloid.  It is formed by the equation, .

Notice that in three-dimensional figures there are three unknowns, x, y, and z.  Take a look (you can adjust the values of x, y, and z on the right side):   Exploration 1

Open a Word Document, title it “Three-Dimensional Shapes”, and type your name and date under the title.  Answer all questions in complete sentences.

1)                What are all of the possible two-dimensional cross sections that can be formed from the elliptical paraboliod?  (Make sure to include the equation)

Hyperbolic Paraboloid

Now, let’s look at the hyperbolic Paraboloid.  The equation, , forms a hyperbolic paraboloid.   Exploration 2

2)               What are all of the possible two-dimensional cross sections that can be formed from the hyperbolic

paraboloid?  (Include the equation)

Look at the following cross-section:  Exploration 3

Did you include this cross-section shape?  If not, go back and include it in #2.

Other Three-Dimensional Figures

3)               List the four three-dimensional figures and determine whether they have cross-sections of parabolas, ellipse, circles, hyperbolas, or lines.

4)               Can you come up with any of your own three-dimensional figures or find any on the internet?  List equations for any that you invent yourself or discover on the world wide web.  You can try them out by typing them into the NuCalc link.    NuCalc

Print your Word Document and file in your portfolio or notebook.  You have now completed the three-dimensional exploration!