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MATH 261, Spring 2001 Due Date: Name(s):
Extra Project 16.9b: Transformation from Rectangular to Polar Coordinates
Objective
In this project we illustrate the geometry of the transformation from rectangular r-coordinates to polar
r-coordinates.
Narrative
In trigonometry we studied rectangular r-coordinates,
coordinates with respect to which we drew graphs of the
trig functions such as f() = sin .
Rectangular r-coordinates.
In second-semester calculus we studied polar r-coordinates,
coordinates with respect to which the graph of r = sin 
looked quite different.
Polar r-coordinates.
Rectangular r-coordinates are related to polar r-coordinates by a transformation T : R2
 R2
which may
be written in any of the following three equivalent ways:
T : (r, )  (x, y) = (r cos , r sin )
T(r, ) = (x, y) = (r cos , r sin )
x = r cos , y = r sin 
In this project, we discuss the geometry of this transformation.
Task
a) Type the command lines below into Maple in the order in which they are listed; they produce two test
patterns in rectangular r-coordinates, and the images of these test patterns in polar r-coordinates.
> # Project 16.9b: Transformation from Rectangular to Polar Coordinates
> restart;
> with(plottools):with(plots):
> f := transform((r,theta) -> [r*cos(theta),r*sin(theta)]):
> TP := plot(0,1,2,3,[1,t,t=0..3],r=0..1,color=blue):
> display(TP); display(f(TP));
> TP := plot(0,1,[1,t,t=0..1],[2,t,t=0..1],[3,t,t=0..1],r=0..3,color=blue):
> display(TP); display(f(TP));
b) By hand (without using Maple):
(i) sketch the test pattern in rectangular r-coordinates determined by  = 1,  = 2, r = 0, r = 1, r = 2,
r = 3, and its image in rectangular xy-coordinates under the transformation T.
(ii) sketch the test pattern in rectangular r-coordinates determined by r = 1, r = 2,  = 0,  = 1,  = 2,
 = 3, and its image in rectangular xy-coordinates under the transformation T.