Exploring Analyic Geometry with Mathematica®

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Polar Equation of an Ellipse

polarell.html

Exploration

Show that the polar equation of an ellipse with a horizontal major axis and centered at (0,0) is given by

"polarell_1.gif"

where a and b are the lengths of the semi-major and semi-minor axes, respectively.

Approach

Create the ellipse in rectangular coordinates. Convert the equation to polar coordinates.

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

Create a quadratic representing the ellipse.

Clear[a,b];
Q1=Quadratic2D[Ellipse2D[{0,0},a,b,0]]

"polarell_2.gif"

Convert the rectangular equation to a polar equation.

Clear[x,y,r,theta]
eq1=Equation2D[Q1,{x,y}] /.
    {x->r*Cos[theta],y->r*Sin[theta]}

"polarell_3.gif"

Put into the desired form by solving for r (taking the positive result).

Solve[eq1,r]

"polarell_4.gif"


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
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