Illuminations: Conic Section Explorer

# Conic Section Explorer

Explore the different conic sections and their graphs. Use the Cone View to manipulate the cone and the plane creating the cross section, and then observe how the Graph View changes.

### Instructions

 Use the tools along the bottom to manipulate the parameters of the double-napped cone and the plane creating the cross section: Increase or decrease Height to make the cone taller or shorter. Increase or decrease Slant to make the cone wider or narrower. Increase or decrease m to change the angle of the plane. Increase or decrease b to change the distance between the cone's center and the plane. Click Reset Parameters to reset all values to 1.   In the Cone View: Click and drag in the window to change the perspective. Click the magnifying glasses to zoom in and out, or Reset to see the default view.   In the Graph View: Click and drag in the window to move the center of the graph. Click the magnifying glasses to zoom in and out, or Reset to see the default view.

### Exploration

Change the parameters and manipulate the Cone View and Graph View to find the different conic sections. As you explore, try to answer these questions:
• How many different conic sections are there?
• How do the different conic sections look different in the Cone View? in the Graph View?
• What range or values do the parameters have for each conic section?
• What are the key features of each conic section?
• Can you create two identical Cone Views with different sets of parameters? Is there a relationship in the parameters for these two views? Do the Graph Views also look the same?
• When the cross section is an X, a line, or a point, it is called a "degenerate" case because the graphs are not unique to conic sections. Can you create these graphs? What are the parameters to create them?

Here are some suggested parameters to guide your exploration:
 Height Slant m b 1 0.5 1 1 1.5 0.5 1 1 1.5 0.5 0.5 1 1 1 0.5 1 1.5 1 0.5 1 2 1 0.5 1 2 1 0.5 0.5 1 1 0.5 0.5 1 1 0.1 0.5 1 1 0 0.5

### Lessons

 More and Better Mathematics for All Students
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