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Polar Coordinates of Points in the Euclidean Plane tex2html_wrap_inline893 .

We assign coordinates to points in tex2html_wrap_inline895 so that we can locate them. Let P be a point in tex2html_wrap_inline895 , and tex2html_wrap_inline551 be the line segment between P and the origin. Let r be the distance between P and the origin. (So r is the length of tex2html_wrap_inline551 .) Let tex2html_wrap_inline715 be the angle, measured in the counterclockwise direction, that tex2html_wrap_inline551 makes with the x-axis. (See the figure below.)

The point P lies on the circle of radius r, centered at the origin, given by the equation tex2html_wrap_inline923 . We can verify that tex2html_wrap_inline925 and tex2html_wrap_inline927 satisfy this equation. tex2html_wrap_inline929

tex2html_wrap_inline931 are called the polar coordinates for the point P.

polar.epsTwo sets of polar coordinates tex2html_wrap_inline931 and tex2html_wrap_inline937 describe precisely the same point when one of the following cases occurs.

  1. r=r'=0, and tex2html_wrap_inline715 and tex2html_wrap_inline943 are arbitrary.
  2. tex2html_wrap_inline945 , r=r', and tex2html_wrap_inline949 , where k is an integer.
  3. tex2html_wrap_inline945 , r'=-r, and tex2html_wrap_inline957 , where k is an integer.

Carl Lee
Wed Apr 21 08:17:28 EDT 1999