r f( ) = sin
x
y
x
y
r
P
r
2
select
pinch
flip
MATH 164, Spring 2001 Due Date: Name(s):
Honors Project 11: Rectangular vs. Polar r-Coordinates
Objective
In trigonometry, you studied rectangular r-coordinates. These
are the coordinates with respect to which you drew graphs of the
trig functions, such as the one to the right of f() = sin .
In calculus, we study polar r-coordinates. These are defined by
the scheme illustrated by the figure to the right.
These are different coordinate systems. In this project we investigate the relationship between them.
Narrative
Rectangular and polar r-coordinates are related by a transformation (a topic to be discussed in detail in
later calculus courses). A transformation T : R2
 R2
is a mapping from one copy of R2
to another copy of
R2
which may be written -- at least in our case -- in any of the following 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.
You can think of a transformation of the plane as a function that
distorts the plane. The specific transformation T described above
can be described by first selecting a semi-infinite strip 2 units
wide,
and then pinching the interval on the -axis to a point.
Next, flip the resulting "fan"-shaped region over.
r
2
select
pinch
flip
unfold
x
y
Finally, unfold the resulting "fan".
Note how the vertical lines in the rectangular r-coordinate plane -- lines whose equations are of the form
 = 0 -- are transformed by T into rays that emanate from the origin, and how the horizontal lines in the
rectangular r-coordinate plane -- lines whose equations are of the form r = r0 -- are transformed by T
into circles whose centers are the origin.
Task
To test your understanding of this transformation, see if you can understand why the graphs of r = a sin n
and r = a cos n in rectangular r-coordinates (sine and cosine waves) are transformed into roses that have
2n-leaves, as  goes from 0 to 2 (only n of which are visible if n is odd). (Hint: How is the graph of r = sin 3
transformed by T? How about the graph of r = sin 3? To get the complete picture, you must use the fact
that every semi-infinite strip 2 units wide in the rectangular r-coordinate plane can be transformed by T.)
Present your work in the form of an essay supported by computer- and/or hand-drawn graphics.