# Transformation help

• Aug 2nd 2010, 05:47 AM
Glitch
Transformation help
The question:
Show that the order in which transformations is important by first sketching the result when the triangle A(1,0) B(1,1) C(0,1) is rotated by 45 degrees about the origin and then translated by (1,0), and then sketch the result when it is first translated then rotated.

I'm not sure how to determine its position after rotation. Do I have to use a transformation matrix? If so, how would I apply it to this question? Thanks.
• Aug 2nd 2010, 05:56 AM
Stuck Man
I presume you are doing a positive rotation which is anticlockwise. You can measure 45 degrees with a pertractor and then draw a faint line. Then a compass can be used to find the new coordinates.
• Aug 2nd 2010, 05:58 AM
Glitch
Ok, thanks. So there's no need to do it accurately with algebra?
• Aug 2nd 2010, 06:19 AM
Stuck Man
You need to create a matrix to multiply. Use the matrix where the top row is cosx -sinx and the bottom row is sinx cosx.
• Aug 2nd 2010, 06:25 AM
HallsofIvy
Well, we can't say what your teacher wants!

A rotation about the origin of $\displaystyle \theta$ degrees corresponds to a matrix multiplication with matrix $\displaystyle \begin{bmatrix}cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{bmatrix}$. That is the same as saying $\displaystyle x'= x cos(\theta)- y sin(\theta)$, $\displaystyle y'= x sin(\theta)+ y'cos(\theta)$

Here, $\displaystyle \theta= 45$ so $\displaystyle x'= x\frac{\sqrt{2}}{2}- y\frac{\sqrt{2}}{2}$, $\displaystyle y'= x\frac{\sqrt{2}}{2}+ y\frac{\sqrt{2}}{2}$

That is, (1, 0) is mapped into $\displaystyle \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, (1, 1) is mapped into $\displaystyle \left( 0, \sqrt{2}\right)$ and (0, 1) is mapped into $\displaystyle \left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$.

You should be able to see how that matches your graph. Now, "translate by (1, 0)" by adding 1 to each x coordinate.

It should be easy to see why moving the object away from the origin before rotating is different from rotating and then moving the object.
• Aug 2nd 2010, 06:28 AM
Glitch
Ahh, thank you. So basically I apply the rotation matrix on each of the co-ordinates?
• Aug 2nd 2010, 06:56 AM
earboth
Quote:

Originally Posted by Glitch
The question:
Show that the order in which transformations is important by first sketching the result when the triangle A(1,0) B(1,1) C(0,1) is rotated by 45 degrees about the origin and then translated by (1,0), and then sketch the result when it is first translated then rotated.

I'm not sure how to determine its position after rotation. Do I have to use a transformation matrix? If so, how would I apply it to this question? Thanks.

I've attached a sketch:

1. The thick outlined triangle is the original.

2. In red: First rotation, then translation.

3. In blue: First translation, then rotation.
• Aug 2nd 2010, 06:58 AM
Glitch
Thanks earboth, that looks like what I got using the protractor/compass method. :)