Investigating Conics...

Eccentricity

 
 

Objective: To demonstrate the effect of varying eccentricity.

Level:  This demo can be presented in any study of analytic geometry that includes a discussion of the conic sections.  It is appropriate for Precalculus or Calculus. 

Prerequisites: Locus definitions of conic sections.

Platforms:

(MATLAB) The MATLAB M-file, eccdemo.m, generates a sequence of frames that illustrate changes in an ellipse as the eccentricity changes.  The M-file generates a movie that plays back once at one frame per second.  (MATLAB v 5.3 Release 11)

(MAPLE V)  The text file, eccdemo.txt, contains the code to generate a Maple V animation of the ellipse demo.  This text may be copied and pasted into a Maple worksheet. (Maple V Release 5)

(Mathematica 4)  A Mathematica notebook can be viewed by clicking here.  You can download the Mathematica notebook from that page.

(Mathcad v8)  A Mathcad worksheet can be viewed by clicking here.  You can download the Mathcad worksheet from that page.

 

Instructor's Notes:  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 and eccentricity e = 0.  As the eccentricity varies the foci and shape of the conic change, making a transition from ellipse to hyperbola.

Note that for an ellipse centered at the origin,

.

The computer codes generate 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.   The definition of ellipse requires the graph to be a line segment connecting the foci.   As e increases from 1,  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.

In the MATLAB generated animation, as each new conic is rendered, both the conic and the foci are shown in the figure.  The Mathematica version creates a similar animation.  The Maple V version animates the changing ellipse.



Credits:  This demo and the Maple V worksheet was submitted by 

Dr. Colin Starr
Department of Mathematics and Statistics
Stephen F. Austin State University

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


 

LFR 4/27/01   Last updated 5/19/2006 (DRH)

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