MATH 261, Spring 2001 Due Date: Name(s):
Project 16.9a*: Transformations of the Plane
Objective
The objective of this project is to give you some experience with the geometry of some simple transformations
of the plane.
Narrative
In this project we illustrate the geometry of some simple transformations of the plane by applying them
to a simple test pattern.
Task
1. Type the following commands into Maple; they define the test pattern TestPat.
> # Project 16.9a*: Transformations of the Plane
> restart;
> with(plottools):with(plots):
> L1 := [t,0,t=0..1]: L2 := [1,t,t=0..1]:
> L3 := [1-t,1,t=0..1]: L4 := [0,1-t,t=0..1]:
> L5 := [t,1+t,t=0..0.5]: L6:=[t,2-t,t=0.5..1]:
> TestPat := plot({L1,L2,L3,L4,L5,L6},color=blue,scaling=constrained):
> display(TestPat);
2. Type the following commands into Maple; they define the transformation T(x, y)  (2x, 1.5y), and
display the test pattern and its image under T.
> T := transform((x,y) -> [2*x,1.5*y]): display({TestPat,T(TestPat)});
3. Repeat Task 2 using the transformations:
a) T(x, y)  (-x, y),
b) T(x, y)  (x + 2, y + 1),
c) T(x, y)  (

2
2 (x - y),

2
2 (x + y)),
d) T(x, y)  (x - y + 3, 2x + y),
e) T(x, y)  (x + 2, x2
+ y).
At this time make a hard-copy of your typed input and Maple's responses. Then, ...
4. on each graphic you produced (on both the test pattern and the transformed test pattern), draw by
hand at least a door -- with a doorknob -- and a window. (Note: The point of including the doorknob
is to make sure there is some assymetry in your test pattern.)
Comments
We were careful to describe the segments of our test pattern by parametric curves. If, instead of doing
this, we had plotted points and "connected the dots" then -- as long as T takes lines to lines (as was the
case with the first five transformations above) -- we would have noticed no difference; however, there would
have been a difference with a transformation such as the sixth transformation above since it does not take
lines to lines.