Non-uniqueness of Polar Coordinates

polarunq.html

Exploration

Show that the polar coordinates of a point (r,θ) are not unique as all points of the form

(r,θ+2 k π) and (-r,θ+(2k+1)π)

represent the same position in the plane for integer values of k.

Approach

Convert the given polar coordinates of the points to rectangular coordinates and demonstrate that the rectangular coordinates are coincident.

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

This is a function for converting a polar point to rectangular coordinates.

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

Convert the two points to rectangular coordinates.

Clear[r,theta,k];
pts={Point2D[PolarPoint2D[r,theta+2k*Pi]],
     Point2D[PolarPoint2D[-r,theta+(2k+1)Pi]]};

Simplifying shows that the points are identical for all values of k.

pts //Simplify

Discussion

The principal polar coordinates of a point (r,θ) are given when r>0 and 0≤θ<2π. These functions convert a PolarPoint2D to principal coordinates.

PolarPoint2D[r_?IsNegative2D,theta_]:=
   PolarPoint2D[-r,theta+Pi];
PolarPoint2D[r_,theta_]:=
   PolarPoint2D[r, PrimaryAngle2D[theta]] /;
theta=!=PrimaryAngle2D[theta]

Convert some polar points to principal form.

{PolarPoint2D[-1,Pi/2],PolarPoint2D[2,-Pi/3]}