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Here is a page of
examples and demonstrations illustrating the
formation of a conic section as the intersection of a plane and a cone. |
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This
animation shows a conic section transforming from a circle to an ellipse to a
hyperbola as eccentricity changes from 0 to 4 (e = 1 which yields a parabola is
not seen and the ellipses disconnect).
Quicktime Version Polar
Version, eccentricity from 0 to 2, ellipses connect
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Here are
three videos demonstrating the effect of changing eccentricity.
Video 1 includes audio and some
introductory material followed by a Geometer's Sketchpad demonstration,
Video 2 includes some audio
along with a Geometer's Sketchpad demonstration, and
Video 3 includes no audio and
is just a short Geometer's Sketchpad demonstration.
Large Quicktime Version of Video 3
Medium Quicktime Version of Video
3
Small Quicktime Version of Video 3 |
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Here is
a Winplot Demo illustrating the shapes
of various ellipses and hyperbolas corresponding to various eccentricity
values. You may need to download the file to your desktop and then
use the freeware
Winplot to open the file (by opening Winplot, clicking on Window,
clicking on 2-dim, clicking on File, clicking on Open, and then opening
Eccentricity5W3). You can use the slider to vary the
value of E (eccentricity) from 0 to 5. When the eccentricity value
is between 0 and 1 you will see an ellipse along with its foci and the
endpoints of its major and minor axes. When the eccentricity value
is more than 1 you will see a hyperbola along with its foci, vertices,
endpoints of the conjugate axis, and its asymptotes. A default
parabola will appear when the eccentricity is 1 and we have a circle
when the eccentricity is 0. Winplot
Demo 2 shows the same conics and their eccentricities demonstrated
using the focus-directrix definition for ellipses and hyperbolas (one
focus and the directrix are pictured rather than two foci for the
ellipses and hyperbolas). The file to save and open is
EccentricityB. |
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Winplot Demo 3 demonstrates polar
equations of conics illustrating the focus/directrix definition.
The focus in each case is at the pole (origin), the directrix has
rectangular equation y = d, and e represents the eccentricity. Use
the sliders to vary the values of d and e. The file to save and
open is EccentricityPolar. The polar form of the equation of each
conic is
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Click on the ellipse and hyperbola below to see
animations demonstrating the foci definitions of each. Click on the
parabola to see an animation demonstrating its focus/directrix definition.
Click here
or here (with scales) to see a cute way of tracing out an
ellipse. Quicktime Version
Here is a nice java
sketchpad demo of this cute way of tracing out an ellipse.
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