Thread: describe the matrix transformation

Hi folks,

SO, here is the problem:
A reflection in the line y = x - 1 is followed by an anticlockwise rotation of about the point (-1,1). Show that the resultant transformation has an invariant line, and give the equation of this line. Describe the resultant transformation in relation to this line.

For the first transformation I get :



For the second the rotation about the origin O is:



and about (-1,1) is:



The resultant of a process P that transforms x to x' (i.e. x' = Px) and is followed by a process Q that transforms x' to x'' (i.e. x'' = Qx') is the resultant process x'' = QPx. So:





The result of this transformation on the unit square OABC is as follows:
O(0,0) goes to O(0,2)
A(1,0) A'(0,3)
B(0,2) B'(1,4)
C(1,1) C'(1,5)

I have attached a rough sketch. It is not what I was hoping for. The answer is that the transform is a glide reflection in the line x = 1/2 with ((1/2,0) going to (1/2, 3). My transformation doesn't even preserve a closed shape. If you ignore the sequence of the letters OABC it looks like a shear along x = 1/2. Can anyone see what I am doing wrong?

Hello s_ingram

Originally Posted by s_ingram

Hi folks,

SO, here is the problem:
A reflection in the line y = x - 1 is followed by an anticlockwise rotation of about the point (-1,1). Show that the resultant transformation has an invariant line, and give the equation of this line. Describe the resultant transformation in relation to this line.

For the first transformation I get :



For the second the rotation about the origin O is:



and about (-1,1) is:



The resultant of a process P that transforms x to x' (i.e. x' = Px) and is followed by a process Q that transforms x' to x'' (i.e. x'' = Qx') is the resultant process x'' = QPx. So:





The result of this transformation on the unit square OABC is as follows:
O(0,0) goes to O(0,2)
A(1,0) A'(0,3)
B(0,2) B'(1,4)
C(1,1) C'(1,5)

I have attached a rough sketch. It is not what I was hoping for. The answer is that the transform is a glide reflection in the line x = 1/2 with ((1/2,0) going to (1/2, 3). My transformation doesn't even preserve a closed shape. If you ignore the sequence of the letters OABC it looks like a shear along x = 1/2. Can anyone see what I am doing wrong?

Thanks for showing us your full working. I think it's pretty well correct, except for the first transformation matrix. This doesn't represent the reflection, does it? Don't forget that a transformation represented by a single 2 x 2 matrix will always leave the origin fixed; and you want .

You have to reflect first in a parallel line through the origin (that's ) and then translate so that the origin moves to the required image. So that's:

Do you want to try again?

Grandad

Hi Grandad,

Thanks very much Grandad! Yes, I was worried about the very point you have made - how to represent a reflection about a line that doesn't go through the origin. I tried it again, using your suggestion and it comes out perfectly! Great!