Collinear Polar Coordinates

polarcol.html

Exploration

Show that the points , and in polar coordinates are collinear if and only if

.

Approach

Convert the given polar coordinates of the points to rectangular coordinates and then apply the condition for collinearity

.

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]}];

Define three arbitrary points in polar coordinates.

Clear[r1,r2,r3,theta1,theta2,theta3];
p1=Point2D[PolarPoint2D[r1,theta1]];
p2=Point2D[PolarPoint2D[r2,theta2]];
p3=Point2D[PolarPoint2D[r3,theta3]];

Apply the condition for collinearity.

Simplify[
   Det[{
      {XCoordinate2D[p1],YCoordinate2D[p1],1},
      {XCoordinate2D[p2],YCoordinate2D[p2],1},
      {XCoordinate2D[p3],YCoordinate2D[p3],1}
       }]
   ]

Discussion

Here's a function based on the equation above that returns True if three points in polar coordinates are collinear.

IsCollinear2D[
   p1:PolarPoint2D[r1_,theta1_],
   p2:PolarPoint2D[r2_,theta2_],
   p3:PolarPoint2D[r3_,theta3_]]:=
IsZero2D[-r1*r2*Sin[theta1-theta2]+
          r1*r3*Sin[theta1-theta3]-
          r2*r3*Sin[theta2-theta3]]

Show that the polar coordinate points (1,π/3), (3,π/3) and (5,4π/3) are collinear using the new function.

p1=PolarPoint2D[1,Pi/3];
p2=PolarPoint2D[3,Pi/3];
p3=PolarPoint2D[5,4*Pi/3];
IsCollinear2D[p1,p2,p3]