Exploring Analyic Geometry with Mathematica®

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Polar Equation of a Circle

polarcir.html

Exploration

Show that the polar equation of a circle centered at "polarcir_1.gif" with radius R is given by

"polarcir_2.gif".

Approach

Represent the circle 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

Define a function to convert from polar coordinates to rectangular coordinates.

Point2D[PolarPoint2D[r_,theta_]] :=
   Point2D[{r*Cos[theta],r*Sin[theta]}];

Create the circle.

Clear[r1,t1,R];
P=Point2D[PolarPoint2D[r1,t1]];
C1=Circle2D[P,R]

"polarcir_3.gif"

Convert to a polynomial in polar coordinates.

Clear[x,y,r,t];
eq1=Polynomial2D[Quadratic2D[C1],{x,y}] /.
   {x->r*Cos[t],y->r*Sin[t]} //FullSimplify

"polarcir_4.gif"


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