Math Is Fun Forum

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Matrix transformations

I've been asked to find the matrix that reflects points across the line y = 2x.

I can get the answer by considering it as a rotation of

, but this is ugly.

How should it be done?

Thanks.

Last edited by Daniel123 (2008-09-21 05:08:27)

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Matrix transformations

Also, what exactly does "a shear of 30° in the direction of Oy" mean?

EDIT: Nevermind about this question - lines parallel to Ox are tilted 30° and stretched to maintain height. So the shear would be written:

Last edited by Daniel123 (2008-09-21 05:43:22)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Matrix transformations

Reflections are not rotations.

A shear of factor k in the y-axis increases the vertical height of a point by an amount depending on k and its distance from the y-axis; horizontal distances are unaffected. Precisely, this is multiplication by the matrix

Last edited by JaneFairfax (2008-09-21 05:48:37)

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Matrix transformations

Reflections are not rotations.

I know they're not, but isn't a relfection across y = 2x equivalent to the rotation I mentioned above?

How should the question be done?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Matrix transformations

I know they're not, but isn't a relfection across y = 2x equivalent to the rotation I mentioned above?

NO!! The matrix representing any rotation always has determinant +1 while that representing any reflection always has determinant −1. They can’t possibly be equal.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Matrix transformations

I know they're not, but isn't a relfection across y = 2x equivalent to the rotation I mentioned above?

NO!! The matrix representing any rotation always has determinant +1 while that representing any reflection always has determinant −1. They can’t possibly be equal.

Haven't come across determinants yet

I'll try and find the proper way of doing it.

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Matrix transformations

Isn't the following treating a reflection across a line to be a rotation through twice the angle between the line and the x-axis? Is there anything wrong with it?

The X coordinate of P' is given by:

, and the Y coordinate of P' is given by

. The same method can be applied to the j vector, giving the transformation matrix as:

In the case where y=2x,

, which can be used along with the double angle trig identities to give the transformation matrix:

, which I know is correct.

When I first did this in the case y=2x, I didn't do it generally but instead calculated the angle, which is why I said it was ugly. Now that I've done it generally, I can't see anything wrong with treating a reflection as a rotation.

Last edited by Daniel123 (2008-09-21 06:43:42)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Matrix transformations

All right, I just worked it out geometrically. A reflection about the line

would take the point

to the point

.

Hence the matrix representing this reflection is

           

EDIT: Well, I see that you’ve managed to work it out as well.

Last edited by JaneFairfax (2008-09-21 06:44:39)

Daniel123

Member

Registered: 2007-05-23

Posts: 663

Re: Matrix transformations

All right, I just worked it out geometrically. A reflection about the line

would take the point

to the point

.

Hence the matrix representing this reflection is

           

Beat you ever so slightly