Linear Transformations

Introduction

As mentioned earlier, matrices are used for transformation of objects (shifting, scaling, shearing, and rotation) in 2D and 3D coordinate spaces. In this lesson there is a bit of “jumping around” with the material. The rationale for this is that students need to understand how translation and scaling work in terms of 2D and 3D coordinate spaces, with the 3D coordinate space using a 4D homogeneous space, before covering rotations.

Homogeneous Matrices

Key Concepts

The key concepts for this part of the lesson are:

The differences between 2D and 3D coordinate spaces for transformations

Lesson

In a 2D coordinate system the z-axis is not shown (the z-axis is typically for a 3D coordinate system) but this axis plays a key role in transformations, and rotations (which will be covered in the next lesson). In the 2D system consider the z-axis as the focal point of all transformations. For example if an object needs to be moved from its current position to another position in the 2D plane there needs to be a standard reference point for the required transformation. This point will use the z-axis. Here it is important that the instructor choose whether to use row or column vectors as this choice needs to be consistent; the other option is to use both so that the students see both.

In a 2D system that is then extended to a 3D system the z-axis is used but there is no one focal axis that remains constant for any transformations, or rotations; in 3D there are now three axis that can be used for rotations, versus the implied z-axis in a 2D system. To get around this we extend the 3D system, $(x,y,z)$ to a 4D system that is $(x,y,z,w)$ ; similarly, the 2D system can be extended from $(x,y)$ to $(x,y,w)$ to keep everything consistent as the z-axis is not really part of the 2D system. Typically the value of the w coordinate is 1. For example the point (2,3,-4) is a 3D system is represented as:

$\left[\begin{array}{cccc}2&3&-4&1\end{array}\right]$ OR $\left[\begin{array}{c}2\\3\\-4\\1\end{array}\right]$

Similarly the homogeneous equivalent for the point (4, -1) in a 2D system is:

$\left[\begin{array}{ccc}4&-1&1\end{array}\right]$ OR $\left[\begin{array}{c}4\\-1\\1\end{array}\right]$

Transformations

Key Concepts

The key concepts for this part of the lesson are:

Construct a transformation matrix in 2D and 3D
Shift a point (representing a single point of an object) in 2D and 3D space

Lesson

As noted at the end of the last lesson two matrices were shown, with each representing a different graphics API system, which are shown below:

$T_{DX}^T={\left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\T_{x}&T_{y}&T_{z}&1\end{array}\right]}^T=\left[\begin{array}{cccc}1&0&0&T_{x}\\0&1&0&T_{y}\\0&0&1&T_{z}\\0&0&0&1\end{array}\right]=T_{GL}$

Looking at T_DX or T_GL you should see that each 4x4 matrix uses the w coordinate axis. These matrices were introduced as transformations, but the term transformation was not formally explained. In this part of the lesson the transformation represented by these matrices is used to shift an object from one relative position to another position at a given displacement from the original position. Note that the T_X, T_Y, and T_Z values represent components of the transformation.

By example in a 2D system there is a requirement to move an object from its current position to a position that is $(\Delta{x},\Delta{y})=(3,4)$ from its original position. The transformation matrix for this action would be:

$\left[\begin{array}{ccc}1&0&T_{x}\\0&1&T_{y}\\0&0&1\end{array}\right]=\left[\begin{array}{ccc}1&0&3\\0&1&4\\0&0&1\end{array}\right]$

Given one point of the object at (-2, 4) the translated point would be:

$\left[\begin{array}{ccc}1&0&3\\0&1&4\\0&0&1\end{array}\right]\times{\left[\begin{array}{c}-2\\4\\1\end{array}\right]=\left[\begin{array}{c}(1)(-2)+(0)(4)+(3)(1)\\(0)(-2)+(1)(4)+(4)(1)\\(0)(-2)+(0)(4)+(1)(1)\end{array}\right]=\left[\begin{array}{c}1\\8\\1\end{array}\right]$

As with the majority of calculations that are done in this course the question that needs to be asked is whether the answer makes sense. Given the point (-2,4) if we simply add the transformation (3,4) we would get (1,8) using simple vector addition:

$\left[\begin{array}{c}-2\\4\end{array}\right]+\left[\begin{array}{c}3\\4\end{array}\right]=\left[\begin{array}{c}1\\8\end{array}\right]$

The use of the w-axis in this 2D example is a bit of an overkill but the power of the w-axis lies when doing 3D rotations, but before this is covered what about 3D transformations? Given a point (-2,5,3) of an object in 3D space. If the object is required to move by $(\Delta{x},\Delta{y},\Delta{z})=(-2, -3,5)$ then the transformation matrix would be:

$\left[\begin{array}{cccc}1&0&0&-2\\0&1&0&-3\\0&0&1&5\\0&0&0&1\end{array}\right]\times{\left[\begin{array}{c}-4\\2\\8\\1\end{array}\right]}=\left[\begin{array}{c}-6\\-1\\13\\1\end{array}\right]$

Scaling

Key Concepts

The key concepts for this part of the lesson are:

Construct a scaling matrix in 2D and 3D
Scale a point (representing a single point of an object) in 2D and 3D space

Lesson

Similar to transforming (shifting) an object in 2D and 3D space it is possible to scale an object, either uniformly or otherwise, in 2D and 3D space. This type of matrix takes the form of:

2D: $S=\left[\begin{array}{ccc}S_{x}&0&0\\0&S_{y}&0\\0&0&1\end{array}\right]$ 3D: $S=\left[\begin{array}{cccc}S_{x}&0&0&0\\0&S_{y}&0&0\\0&0&S_{z}&0\\0&0&0&1\end{array}\right]$

Note that this is one instance where the Row and Column major matrices are the same. To demonstrate this use the following example. Given an object in 3D space with one of its points at (4,-1,2) and this object is scaled by $(S_{x},S_{y},S_{z})=(3,2,4)$ .

$\left[\begin{array}{cccc}3&0&0&0\\0&2&0&0\\0&0&4&0\\0&0&0&1\end{array}\right]\times{\left[\begin{array}{c}4\\-1\\2\\1\end{array}\right]}=\left[\begin{array}{c}12\\-2\\8\\1\end{array}\right]$

Combination Transformations

Key Concepts

The key concepts for this part of the lesson are:

Create a Scaling and Transformation matrix in 2D and 3D
Calculate the results of a combination matrix in 2D and 3D

Lesson

As was learned in the previous lesson it is possible to multiply two matrices to create a new matrix. It was a general rule that $A\times{B}\neq{B\times{A}}$ , unless one of the matrices was a special matrix, such as the Identity matrix. In this part of the lesson the students will combine the S and the T matrices, which will be shown that can be done in any order, to create a combination matrix. This is possible due to the elements of each matrix that have values of 0.

2D

In this part create a scaling matrix S, and a transformation matrix T and multiply them $S\times{T}$ and $T\times{S}$ (at this point, you should be able to do most of the multiplication without having to write out each step):

$S\times{T}=\left[\begin{array}{ccc}S_{x}&0&0\\0&S_{y}&0\\0&0&1\end{array}\right]\times{\left[\begin{array}{ccc}1&0&T_{x}\\0&1&T_{y}\\0&0&1\end{array}\right]}=\left[\begin{array}{ccc}S_{x}&0&T_{x}\\0&S_{y}&T_{y}\\0&0&1\end{array}\right]$

$T\times{S}=\left[\begin{array}{ccc}1&0&T_{x}\\0&1&T_{y}\\0&0&1\end{array}\right]\times{\left[\begin{array}{ccc}S_{x}&0&0\\0&S_{y}&0\\0&0&1\end{array}\right]}=\left[\begin{array}{ccc}S_{x}&0&T_{x}\\0&S_{y}&T_{y}\\0&0&1\end{array}\right]$

3D

Using the knowledge gained from the 2D examples above the combination matrices in 3D are:

$S\times{T}=T\times{S}=\left[\begin{array}{cccc}S_{x}&0&0&T_{x}\\0&S_{y}&0&T_{y}\\0&0&S_{z}&T_{z}\\0&0&0&1\end{array}\right]$

For example a point on an object is 2D space, $(-2,4)$ is being transformed by scaling it by $(2,3)$ and shifting it by $(3,4)$ :

$\left[\begin{array}{ccc}2&0&3\\0&3&4\\0&0&1\end{array}\right]\times{\left[\begin{array}{c}-2\\4\\1\end{array}\right]}=\left[\begin{array}{c}-1\\16\\1\end{array}\right]$

Exercises & Assignments

Complete the Matrix Transforms worksheet. Once complete proceed to Moodle to complete Knowledge Check 06 - Matrix Transforms (strongly recommended to be completed prior to attempting Lab 2).

Linear Transformations

Introduction

Homogeneous Matrices

Key Concepts

Lesson

Transformations

Key Concepts

Lesson

Scaling

Key Concepts

Lesson

Combination Transformations

Key Concepts

Lesson

2D

3D

Exercises & Assignments

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