The Cartesian Plane
Representing Transformations with Coordinates
One way to represent how an isometry (or any other transformation of space) transforms the plane (or any space) is through analytic geometry:
(A) Introduce a coordinate system in the plane
(B) Identify points, P, in the plane by their coordinates: P(x,y)
(C) Specify the coordinates for the new point, P', to which P is mapped by the isometry: P'(x',y').
So the isometry is represented by the mapping P(x,y)-->P'(x',y').
Use the applet below to determine where the point, P(x,y), is mapped by the following isometries:
1) Reflection over any of the lines of reflection shown in the applet;
2) Rotation about the origin by 90 degrees or 180 degrees;
3) Translation in any direction.
Conjugating Transformations
Now, what if we tried to represent a more complicated transformation, like a reflection over the line y=x+4? Where would the point P(x,y) go then? One idea we might try is to reduce this transformation into the prior transformation by, first, translating the line of reflection down 4 so that it passes through the origin. But then we would need to undo this transformation after performing the reflection. This is called conjugation: a-1ba, where a and b are transformations and a-1 is the inverse transformation of a.
Does this work in the present case? How can you check?
Would this conjugation strategy always work? Can you justify it?