Transformations

Date: 12/14/95 at 20:32:29
From: Anonymous
Subject: A transformation of a rectangle

During my grade 10 enriched math class, we were instructed to make up
our own transformation for the following quadrilateral, using the x 
and y axis.  Point A= (1,1), Point B= (3,1), Point C= (3,4), and Point
D= (1,4).  I decided to use (2x,x+y^2).  I drew the transformation and
found that it had turned into a vertical paralellogram, with all sides
straight except for the 1 at the top (line CD).  I showed it to my 
teacher, and he said that more sides should have been curved.  When he
tried it he got the same transformation as me.  Another kid in my 
class got the same thing too.  What do you think about this?  Do you
think it is abnormal too?  Or is this just a strange thing that 
happens every once in a while?  Is this a different kind of 
transformation that I don't know about, besides the usual translations, 
reflections, rotations, and dilations?  If not what kind is it?

Date: 12/16/95 at 18:28:14
From: Doctor Ken
Subject: Re: A transformation of a rectangle

Hello!

First, let me say something about transformations in general.  All that
"transformation" means is a function that takes every point of the 
original set (sitting in some space) and maps it to some other point 
sitting in the same space.  So yes, there are lots of kinds of 
transformations other than the ones you mention.  Any function from 
R2 to R2 will do.

A lot of times people talk about "rigid" transformations.  These are the
ones that preserve distance between two points, and they are the 
translations, reflections, rotations, and glide reflections (a 
reflection and a translation together).  

Another kind of transformation is a "linear" transformation.  These are
transformations of the form f(x,y) = (ax + by, cx + dy).  Linear 
transformations include dilations and shear transformations.  They also 
include all the rigid transformations that map (0,0) to (0,0).  

So I guess that the answer to your question is that your function is 
indeed a transformation, but I don't think I'd call it any special kind 
of transformation.  The reason it's not a linear or rigid transformation 
is that you squared your y.

Let me check something with you, though.  Are you sure that the top edge 
is curved?  I'm not convinced, because for all the points between D and 
C, the output of the transformation will be of the form (2x, x + 16) 
where x starts at 1 and ends at 3.  That's a straight line: in the (u,v) 
plane (the coordinates I'm using for the output of your function), let 
u = 2x, and v = x+16.  Then x = v-16, and so we have u = 2(v-16), so 
v = u/2 - 16, and that's a line.  Did you mistype any of the vertices?

To help you see why you won't get any curving on this figure, I made a 
sketch using The Geometer's Sketchpad, and you can get it off the Web 
using the URL 

http://mathforum.org/dr.math/sketches/transform.gsp   

If you don't have a copy of Sketchpad, you can download a demo copy of 
it at the URL

http://mathforum.org/dynamic.html   

and it will let you look at the sketch.

Enjoy!

-Doctor Ken,  The Geometry Forum