Investigating Conics...
Eccentricity

This demo demonstrates the geometric effects of varying the eccentricity for conics.  We start with an ellipse with fixed vertices at x = -1 and x = 1 with eccentricity e = 0.  As the eccentricity varies, the foci and shape of the conic change, making a transition from ellipse to hyperbola.

For an ellipse centered at the origin,

.

The Mathematica code below generates an animation that illustrates the shape of the conic as the eccentricity varies. When e = 0, the equations for the ellipse show that c = 0 and a = b so the foci are coincident at the origin.  The "ellipse" is a circle.  As e increases, the foci separate and b decreases.  As e approaches 1, the ellipse becomes flatter and b approaches 0.  the equation for the ellipse does not apply at this point, since that would require division by zero.  In this case, the definition of ellipse requires the graph to be a line segment connecting the foci.  As e increases from1, the foci continue to separate and the resulting conic is a hyperbola.  Thus, we see that for 0 < e < 1, the conic is an ellipse; for e > 1, the conic is a hyperbola.

This can lead to a further discussion of whether there is a conic for which the eccentricity is exactly equal to 1.  The answer is yes; that conic is a parabola.  

The animation is shown below the Mathematica code.

[Graphics:Images/eccdemo_mma_gr_2.gif]
[Graphics:Images/eccdemo_mma_gr_3.gif]
[Graphics:Images/eccdemo_mma_gr_4.gif]
[Graphics:Images/eccdemo_mma_gr_5.gif]
[Graphics:Images/eccdemo_mma_gr_6.gif]
[Graphics:Images/eccdemo_mma_gr_7.gif]
[Graphics:Images/eccdemo_mma_gr_8.gif]


To download the Mathematica notebook, click here.

The Mathematica Notebook, adapted from a demo submitted by Dr. Colin Star, was created and submitted by

Elizabeth G. Carver and
Dr. Lila F. Roberts
Mathematics and Computer Science Department
Georgia Southern University
Statesboro, GA  30460
lroberts@gasou.edu

and is included in Demos with Positive Impact with their permission.


Converted by Mathematica      March 16, 2001