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
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