Exploring Analyic Geometry with Mathematica®

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Polar Equations

polareqn.html

Exploration

A curve in polar coordinates may have more than one equation. A given point may have either of two general coordinate representations

(r,θ+2k π)

(-r,θ+(2k+1)π)

for any integer k. Hence a given curve r=f(θ) may have either of the two equation forms

r=f(θ+2k π)

-r=f(θ+(2k+1)π).

The first equation reduces to r=f(θ) when k=0, but may lead to an entirely different equation of the same curve for another value of k. Similarly, the second equation may yield other equations of the curve.  Show that in spite of the potential for multiple equations in polar coordinates, a linear equation A x + B y + C = 0 has only one representation in polar coordinates given by

r(A cos θ+B sin θ)+C=0.

Approach

Derive an equation for a linear equation in polar coordinates using the primary form (r,θ). Investigate and compare the primary form to the equation derived from the forms (r,θ+2k π) and (-r,θ+(2k+1)π).

Initialize

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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 the primary form of a linear equation in polar coordinates.

Clear[A1,B1,C1,x,y];
A1*x+B1*y+C1 /.
   {x->r*Cos[t],
    y->r*Sin[t]}

"polareqn_1.gif"

Compare to the form (r,θ+2k π), using two trigonometric identities.

Clear[r,t,k];
A1*x+B1*y+C1 //.
   {x->r*Cos[t+2k*Pi],
    y->r*Sin[t+2k*Pi],
    Cos[t+2k*Pi]->Cos[t],
    Sin[t+2k*Pi]->Sin[t]}

"polareqn_2.gif"

Compare to the form (-r,θ+(2k+1)π), using two trigonometric identities.

A1*x+B1*y+C1 //.
   {x->-r*Cos[t+(2k+1)*Pi],
    y->-r*Sin[t+(2k+1)*Pi],
    Cos[t+(2k+1)*Pi]->-Cos[t],
    Sin[t+(2k+1)*Pi]->-Sin[t]}

"polareqn_3.gif"


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