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Topic:  Shodor lessons 
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Subject:  RE: Shodor lessons 
Author:  Alan Cooper 
Date:  Jan 27 2006 
> Here's a slider I'd like to see, just for fun (and I have almost
> succeeded in creating one in Geometer's Sketchpad):
> an "eccentricity" slider. Slide from 0 through 3, say, and see a
> conic section change from a circle, through ellipses, to a parabola,
> then through hyperbolas. I can get the effect by using overhead
> transparencies with parallel lines and/or concentric circles...
I like the idea of doing it geometrically, either by projecting transparencies
or using GSP, but if you want an analytic version,
you could do it with the applet at
http://www.langara.bc.ca/~acooper/GeX/v2/Grapher.htm
Just enter the appropriate family of equations
(eg (1e^2)x^2+y^2=1 if you don't mind the intermediate case being
parallel lines rather than a parabola) and adjust the slider for e.
Or if you want to be more general you could enter
Ax^2+By^2+Cx+Dy+Fxy=1 and vary any of the parameters to your heart's
content.
These implicit forms get a bit ragged as you move the slider because recomputing
an implicit plot fast enough to animate is hard to do without making some
tradeoffs, but if you just click on the equation and press 'enter' again when
you have got to a case that you want to see more neatly then that should clean
it up.
Alternatively, you could use the polar form
r=1/(1+ecos(theta))
(which does give you the parabola as the intermediate case if that is what you
want)and in this case the graphs should stay reasonably nice looking throughout
the animation.
 
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