Investigating Conics...



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.


(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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