MATH 164, Spring 2001 Due Date: Name(s):
Project 11.4a: Polar Curves in Cartesian Coordinates
Objective
To illustrate how Maple can be used to plot polar curves in Cartesian coordinates.
Narrative
If you have not already done so, read Section 11.4 of the text.
The curve whose polar equation is r = r(),   [, ], can be plotted using the Cartesian coordinate
parametrization
x = r cos  = r() cos , y = r sin  = r() sin ,   [, ].
Tasks
1. a) Type the command lines below into Maple in the order in which they are listed. These commands
plot the polar curve r = 3 sin 2,   [0, 2].
> # Project 11.4a: Polar Curves in Cartesian Coordinates
> restart;
> # Part a
> r := t -> 3*sin(2*t);
> plot([r(t)*cos(t),r(t)*sin(t),t=0..2*Pi],scaling=constrained);
b) Continue by typing the command lines below into Maple in the order in which they are listed. These
commands plot the polar curve r = 2 cos 3,   [0, 2].
> # Part b
> r := t -> 2*cos(3*t);
> plot([r(t)*cos(t),r(t)*sin(t),t=0..2*Pi],scaling=constrained);
At this time, make a hard-copy of your typed input and Maple's responses. Then, ...
2. a) On the graphic you created in part (a) of Task 1, label by hand:
i) the points at which t = 0, /4, /2, 3/4, , 5/4, 3/2, 7/4, 2, and
ii) the direction in which t increases from the point at which t = 0.
3. a) Referring to the graphic you created in part (b) of Task 1, what minimum range of t values produces
a 3-leaf rose?
b) On the graphic you created in part (b) of Task 1, label by hand:
i) the points at which t = 0, /6, /3, /2, 2/3, 5/6, , and
ii) the direction in which t increases from the point at which t = 0.
Comments
Some other interesting curves you might like to investigate include curves defined parametrically by x(t) =
r(t) cos kt, y(t) = r(t) sin kt where k is a real constant.