Thread: Transofrmation

You are working on points in a 2D space. You have a triangle with the points (0,0),(2,1),(1,2). Now the question is: Give the following transformations as matrices and use these on the triangle:

1. Translation along (1,3) followed by a rotation along the z-axis by 45 degrees.
2. Scaling by 50% followed by a transformation of (2,1).

Here is my attempt but got stuck at the beginning.

The first one; a translation of (1,3) will give the points of the triangle as (1,3);(3,4);(2,5) now rotation along the z-axis, is that not a 3D space?

Hello Keep

Originally Posted by Keep

You are working on points in a 2D space. You have a triangle with the points (0,0),(2,1),(1,2). Now the question is: Give the following transformations as matrices and use these on the triangle:

1. Translation along (1,3) followed by a rotation along the z-axis by 45 degrees.
2. Scaling by 50% followed by a transformation of (2,1).

Here is my attempt but got stuck at the beginning.

The first one; a translation of (1,3) will give the points of the triangle as (1,3);(3,4);(2,5) now rotation along the z-axis, is that not a 3D space?

1 The $\displaystyle z$-axis is perpendicular to the $\displaystyle x-y$ plane, and passes through the origin. So a rotation along the $\displaystyle z$-axis is simply a rotation in the $\displaystyle x-y$ plane about the origin. Every point in the plane will be rotated on to another point in the plane, and so it is really a 2D transformation.

The matrix that will rotate about the origin through $\displaystyle 45^o$ is: $\displaystyle \begin{pmatrix} \dfrac{1}{\sqrt2} & -\dfrac{1}{\sqrt2}\\\dfrac{1}{\sqrt2} & \dfrac{1}{\sqrt2}\end{pmatrix}$

The translation is described by adding the column matrix $\displaystyle \binom{1}{3}$ to the column vectors representing the original points; i.e.

$\displaystyle \binom00 + \binom13 = \binom13,\quad \binom21 + \binom13 = \binom34, \quad\binom12 + \binom13 = \binom25$

2 I presume the scaling has centre the origin, in which case the matrix you need is: $\displaystyle \begin{pmatrix} \frac{1}{2} & 0\\0 & \frac{1}{2}\end{pmatrix}$

Can you complete this now?

Grandad

Originally Posted by Grandad

Hello Keep1 The $\displaystyle z$-axis is perpendicular to the $\displaystyle x-y$ plane, and passes through the origin. So a rotation along the $\displaystyle z$-axis is simply a rotation in the $\displaystyle x-y$ plane about the origin. Every point in the plane will be rotated on to another point in the plane, and so it is really a 2D transformation.

The matrix that will rotate about the origin through $\displaystyle 45^o$ is: $\displaystyle \begin{pmatrix} \dfrac{1}{\sqrt2} & -\dfrac{1}{\sqrt2}\\\dfrac{1}{\sqrt2} & \dfrac{1}{\sqrt2}\end{pmatrix}$

The translation is described by adding the column matrix $\displaystyle \binom{1}{3}$ to the column vectors representing the original points; i.e.

$\displaystyle \binom00 + \binom13 = \binom13,\quad \binom21 + \binom13 = \binom34, \quad\binom12 + \binom13 = \binom25$

2 I presume the scaling has centre the origin, in which case the matrix you need is: $\displaystyle \begin{pmatrix} \frac{1}{2} & 0\\0 & \frac{1}{2}\end{pmatrix}$

Can you complete this now?

Grandad

I first translated it by (1,3) so that wil give the points $\displaystyle \binom13,\quad\binom34,\quad\binom25,\quad$

Now rotating this along the z-axis by 45 degrees will give $\displaystyle \begin{pmatrix} {-1.4} & -0.7 & -2.1\\2.8 & {4.9}&{4.9}\end{pmatrix}$

I don't know whether the scaling is by the origin; we weren't given so I presume that it is by the origin. So scaling by 50% basically means dividing by two, right? Which will give $\displaystyle \begin{pmatrix} {-0.7} & -0.35 & -1.05\\1.4 & {2.45}&{2.45}\end{pmatrix}$?

Thank you!