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.
