Exploring Analyic Geometry with Mathematica®

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Polar Equations



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)


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

r=f(θ+2k 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.


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).


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Create the primary form of a linear equation in polar coordinates.

A1*x+B1*y+C1 /.


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

A1*x+B1*y+C1 //.


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

A1*x+B1*y+C1 //.


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