Conjugating Transformations
Conjugating a Rotation
To determine what happens to a point, P, when we rotate it about a point, C, that is not at the origin of the Cartesian plane, O; we can first translate C to O, then rotate about O, then undo the translation. See how this works in the example below, for a counterclockwise rotation of 90 degrees about C.
Note that C=(h,k) and P=(X,Y).
Conjugating a Reflection
Now, let's try using conjugation to determine what happens when we reflect over a line that does not pass through the origin. The given line of reflection, f, is y=sqrt(3)x+3.
Step 1: Translate the line to pass through the origin
In this new reference frame, the line, f, is given by the equation y=sqrt(3) x.
Step 2: Rotate to align with the x-axis
To rotate, we use a rotation around the origin by where .