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)
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
(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
(Mathematica 4) A Mathematica
notebook can be viewed by clicking here. You
can download the Mathematica notebook from that page.
v8) A Mathcad worksheet can be viewed by clicking here.
You can download the Mathcad worksheet from that page.
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
increases, the foci separate and b decreases. As
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.