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

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]]

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]}

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

Solve[eq1,r]