Exploring Analyic Geometry with Mathematica®

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Similar Ellipses

ellsim.html

Exploration

All ellipses of equal eccentricity are essentially similar in that by a proper choice of scales (and axes) they can be made to coincide. Show this property is true for two ellipses of equal eccentricity centered at the origin.

Approach

Construct two ellipses with equal eccentricity. Show that one can be scaled to coincide with the other.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Construct the two ellipses by vertex points.

Clear[e1,a,a1,a2];
{E1,E2}={Ellipse2D[{Point2D[-a1,0],Point2D[a1,0]},e1],
         Ellipse2D[{Point2D[-a2,0],Point2D[a2,0]},e1]} /.
        Sqrt[a_^2]->a

"ellsim_1.gif"

Scale the first to coincide with the second

Scale2D[E1,a2/a1]

"ellsim_2.gif"

Discussion

This is a plot of a pair of similar ellipses.

Sketch2D[{E1,E2} /. {a1->1,a2->2,e1->.75}]

"ellsim_3.gif"

Graphics saved as "ellsim01.eps".


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