6.5 - Transformation Matrices¶

The previous three lessons described the basic transformations that can be applied to models: translation, scaling, and rotation. These transformations can be combined to produce complex motion. But we need an easy and efficient way to combine these transformations. The solution is matrices!

This lesson will review the basics of matrix math and show you how to combine transformations using matrices. Matrices are used for almost all computer graphics calculations, including camera manipulation and the projection of your 3D scene onto a 2D viewing window. Therefore, this is a critical section of material that you need to master.

It is not the purpose of this tutorial to provide mathematical proofs for these concepts, but rather to give you a solid foundation in how to use matrix math to create computer graphics. If you desire a formal education in matrix math, I suggest you study the linear algebra course at Kahn academy, but that level of expertise is not required to master computer graphics.

Matrix Basics¶

A system of equations can be written in matrix format by separating out the coefficients of the equations from the variables. Let’s take the general equation for rotation and put it into matrix form. Please note that the x, y, and z values are our “variables” and the fn values are the “coefficients” of the equation terms. To manipulate a graphics model for a computer graphics scene, we will create the transformation by choosing appropriate fn values and then apply the transformation to every vertex in a model. The fn values are constant for the rendering of a single frame of animation, but they typically change for the next animation frame.

The equations:

f1*x + f2*y + f3*z = x'
f4*x + f5*y + f6*z = y'
f7*x + f8*y + f9*z = z'


looks like this in matrix format:

f1
f4
f7
f2
f5
f8
f3
f6
f9
*x
y
z
=x'
y'
z'
Eq1

The matrix equations presented in these tutorials can be executed by clicking on the operator symbols in the equation. Try it now by clicking on the multiplication symbol in the equation above. A new version of the equation will be displayed below the original equation, where the matrices before and after the operation symbol have been replaced by a single matrix that is the result of the operation. Matrices are multiplied as follows: each element in the result is calculated by taking the corresponding row of the left matrix and the corresponding column of the right matrix and multiplying the individual corresponding elements and then adding up the terms. If you place your cursor over any element in the newly displayed matrix, the corresponding rows and columns that were used to calculate that term will be highlighted in the original matrices. Perhaps it would be easier to visualize these issues if the matrix contained numbers instead of symbols. Perform the matrix multiplication below by clicking on the multiplication symbol and then hover your mouse over each term of the result.

1
5
6
0
2
1
3
0
0
*x
y
z
=x'
y'
z'
Eq2

The equation used to calculate each term is intentionally shown in the result so that you can see where each value comes from. Selecting the “-” button to the right of the equation will reduce each term to its simplest calculated value. Please use this functionality throughout these tutorials to closely examine how matrices work. If you click on the equal sign in the equation, each side of the equation will be reduced to its simplest form. The “X” button to the right will remove the generated equation from the web page.

Basic Transformations In Matrix Format¶

From the previous lesson you learned that a scaling transformation is performed by multiplying the vertex components like this, where (x,y,z) is a vertex, and (x’,y’z’) is a transformed vertex:

x * sx = x'
y * sy = y'
z * sz = z'


These scaling equations can be written in matrix format like this:

sx
0
0
0
sy
0
0
0
sz
*x
y
z
=x'
y'
z'
Eq3

Multiply the matrices and reduce the equations to see the equivalence. It may seem crazy to take such simple equations and make them more complex by using matrices. But you will see the power of matrices shortly.

Rotation transformations can easily be written in matrix format. Let’s put the equations for rotation about the Z axis in matrix format:

x * cos(angle) + y * -sin(angle) = x'
x * sin(angle) + y *  cos(angle) = y'
z = z'

cos(angle)
sin(angle)
0
-sin(angle)
cos(angle)
0
0
0
1
*x
y
z
=x'
y'
z'
Eq4

Now we come to the hard one – translation. Notice that in each of the previous examples, the transformation of each vertex component was some combination of the original x, y, and z values. But for translation, we simply want to add a value to each component. The matrices we have used so far have no way to do this. So we need a larger matrix. Instead of a 3 by 3 matrix, we use a 4 by 4 like this:

The translation equations are:

x + tx = x'
y + ty = y'
z + tz = z'


The equivalent equations in matrix format are:

1
0
0
0
0
1
0
0
0
0
1
0
tx
ty
tz
1
*x
y
z
1
=x'
y'
z'
1
Eq5

• The additional component of 1 at the end of the (x,y,z) column vector guarantees that the offsets (tx, ty, tz) will be unchanged.
• The one’s down the diagonal guarantee that the original (x,y,z) values are included in the results unchanged.
• The last row of the transformation matrix (0,0,0,1) guarantees that the 1 component at the end of the (x,y,z) value remains a 1 in the result.

Question? Do we really need that last row in the transformation matrix? Could we do this?

1
0
0
0
1
0
0
0
1
tx
ty
tz
*x
y
z
1
=x'
y'
z'
Eq6

From a pure mathematical perspective, yes you can. However, our goal is to create a single, consistent format for applying a series of transformations. In addition, we need the ability to undo (or reverse) transformations, which requires that our transformation matrices be square. Therefore, the 4th row in a transformation matrix is required.

The extra value added to a vertex at the end, the trailing 1, is called the homogeneous coordinate. The standard convention is to call this the w component. Therefore, a vertex in homogeneous coordinates looks like (x,y,z,w). The w component is useful for more than just translation, and we will discuss those uses in future lessons. But for now, notice that the w component implements (and controls) translation. For a vertex, we always want the w component to be 1. However, remember that a vector has a magnitude and a direction, but no location. A vector can’t be translated! When we represent a vector using homogeneous coordinates, the w value need to be zero! Since we rarely store the homogeneous component in memory to reduce memory usage, you will have to add the homogeneous component when it is needed. Just remember:

• For vertices use (x,y,z,1), which allows for scaling, rotation, and translation.
• For vectors use <dx,dy,dz,0>, which allows for scaling and rotation (but not translation).

Putting this all together gives us the following consistent way to perform our three basic transformations:

Scale:

sx
0
0
0
0
sy
0
0
0
0
sz
0
0
0
0
1
*x
y
z
1
=x'
y'
z'
1
Eq7

Translate:

1
0
0
0
0
1
0
0
0
0
1
0
tx
ty
tz
1
*x
y
z
1
=x'
y'
z'
1
Eq8

Rotate angle degrees about the Z axis:

cos(angle)
sin(angle)
0
0
-sin(angle)
cos(angle)
0
0
0
0
1
0
0
0
0
1
*x
y
z
1
=x'
y'
z'
1
Eq9

Rotate angle degrees about the Y axis:

cos(angle)
0
-sin(angle)
0
0
1
0
0
sin(angle)
0
cos(angle)
0
0
0
0
1
*x
y
z
1
=x'
y'
z'
1
Eq10

Rotate angle degrees about the X axis:

1
0
0
0
0
cos(angle)
sin(angle)
0
0
-sin(angle)
cos(angle)
0
0
0
0
1
*x
y
z
1
=x'
y'
z'
1
Eq11

Rotate angle degrees about any axis defined as <ux,uy,uz>:

Let’s derive a transformation for rotating about any axis by combining the transformations we have already created. This will give you an example of how basic transformations can be combined to form more complex transformations. If we want to rotate about an axis defined by <ux, uy, uz>, then we can accomplish this by performing the following sequence of transformations:

1. Rotate about the Z axis to place the vector <ux, uy, uz> in the Z-X plane. Let’s call this new vector <ux’, uy’, uz’>.
2. Then rotate about the Y axis to place <ux’, uy’, uz’> along the X axis.
3. Then rotate about the X axis the desired angle.
4. Then rotate about the Y axis to place <ux’, uy’, uz’> back to its original location.
5. Then rotate about the Z axis to place <ux, uy, uz> back in its original location.

This series of 5 rotations will provide the visual affect of rotating a model about the axis <ux,uy,uz>. But we don’t want to do all of the 5 transformations over and over again for each vertex. We want a single transformation that will produce the visual motion we desire. We can accomplish this by multiplying the 5 matrices together before we start rendering, and then use a single transformation matrix to perform the desired rotation. To make this idea clear, lets perform the 5 transformations above in the order they are specified. The order is critical, because if you change the order, you will get a very different result.

We need to calculate 2 angles of rotation that will get the axis of rotation aligned with the X axis. Let i be the angle in step one, j be the angle for step two, and k be the angle for step 3. And let’s use s() and c() to represent the sin and cosine functions. The transformation looks like this:

c(-i)
s(-i)
0
0
-s(-i)
c(-i)
0
0
0
0
1
0
0
0
0
1
*c(-j)
0
-s(-j)
0
0
1
0
0
s(-j)
0
c(-j)
0
0
0
0
1
*1
1
0
0
0
c(k)
s(k)
0
0
-s(k)
c(k)
0
0
0
0
1
*c(j)
0
-s(j)
0
0
1
0
0
s(j)
0
c(j)
0
0
0
0
1
*c(i)
s(i)
0
0
-s(i)
c(i)
0
0
0
0
1
0
0
0
0
1
*x
y
z
1
=x'
y'
z'
1
Eq12

Perform the matrix multiplications in the above equation to see what the single transformation is equal to. Note that the resulting answer is in terms of sin and cos functions because the equations do not contain specific numerical values, but in a specific instances, you would have a single 4-by-4 matrix with 16 numeric values that would perform your desired transformation. If the model you were transforming contained 10,000 vertices, reducing your complex transformations to a single 4-by-4 matrix saves a huge amount of computation.

It should be noted that a rotation about an axis <ux, uy, uz> can be calculated using simpler equations by combining like terms in the above equations. For operations that are common, such as rotating about a specific axis, the calculations are simplified to their simplest form before programming them into algorithms. But for the general case, complex motion will be created by forming a series of 4x4 matrix transformations and combining them into a single transformation matrix.

Let’s take a look at some basic properties of matrices.

Basic Properties of Matrices - Order Matters!¶

A matrix represents a system of equations. Therefore, only a small set of operations make sense. The fundamental operation is multiplication. We defined how matrices are multiplied in the above discussion. The important thing to understand is that the order of multiplication matters. In general, M1*M2 != M2*M1. Experiment with the following example.

2
5
4
-3
*-4
-4
3
8
!=-4
-4
3
8
*2
5
4
-3
Eq13

From a computer graphics perspective it is easy to understand that matrix order matters. For example, physically take some object, assume it is located at the origin, and perform these transformations on it:

1. Move it 2 units down the X axis.
2. Then rotate it about the Z axis by 90 degrees.

Now, perform the transformations in reverse order:

1. Rotate the object 90 degrees about the Z axis.
2. Then move it 2 units down the X axis.

The object ends up in a totally different place!

Let’s perform these transformations in matrix format. The equation below moves an object 2 units down the x axis and then rotates 90 degrees about the Z axis:

cos(90)
sin(90)
0
0
-sin(90)
cos(90)
0
0
0
0
1
0
0
0
0
1
*1
0
0
0
0
1
0
0
0
0
1
0
2
0
0
1
*x
y
z
1
=x'
y'
z'
1
Eq14

while this equation performs the rotation first and then the translation:

1
0
0
0
0
1
0
0
0
0
1
0
2
0
0
1
*cos(90)
sin(90)
0
0
-sin(90)
cos(90)
0
0
0
0
1
0
0
0
0
1
*x
y
z
1
=x'
y'
z'
1
Eq15

Multiply and simplify the two equations to see that the transformations indeed are totally different! The matrix that is closest to the (x,y,z,1) vector is the transformation that happens first. Then the transformation to the left of that, and so on. Therefore, you must order the transformations from right to left in your calculations to get your desired order of transformations.

The Identity Matrix and the Matrix Inverse¶

An identity matrix will not change the values of another matrix if the two matrices are multiplied together. This is identical to multiplying a single value by one. Experiment with the following two examples.

1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
*x
y
z
1
=x
y
z
1
Eq16
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
*2
8
0
1
-3
-4
5
2
5
3
-6
3
7
2
5
4
=2
8
0
1
-3
-4
5
2
5
3
-6
3
7
2
5
4
Eq17

An identity matrix is represented by a capital I.

A 4-by-4 matrix performs a transformation on a set of vertices. There is often a need to reverse the transformation to get the original values back. In algebra, the way you undo an addition is to subtract. For example, examine how the 5 is moved to the other side of the equation by subtraction it from both sides:

x + 5 = x'
x + 5 - 5 = x' - 5
x = x' - 5


In a similar manner, the way to undo multiplication is to divide, (or multiply times the reciprocal). For example, examine how the multiplication by 5 is moved to the other side of the equation by dividing by 5.

x * 5 = x'
(x * 5) / 5 = x' / 5
x = x' / 5
x = x' * (1/5)


Division for matrices is performed by multiplying by a matrix inverse. Given a matrix M, if you multiply it by its inverse, the result is the identity matrix. The notation M-1 represents the inverse of M. A matrix inverse will produce an identity matrix regardless of the order of the matrix multiplication. That is,

M
*M-1
=M-1
*M
=I
Eq18

An arbitrary 4-by-4 matrices may or may not have an inverse. However, if you create a transformation which is a combination of scaling, rotation, and/or translation, the resulting 4-by-4 matrix will always have an inverse. We will discuss how a matrix inverse is used in later lessons.

Matrix math follows the same simple rules as algebra. If you have an equation, you must always perform the same operation on both sides of the equation to maintain its equality. However, since the order of matrix multiplication matters, if you pre-multiply one side of an equation by a matrix, make sure you pre-multiply the other side of the equation by the same matrix. Consider the following equation:

S
*T
=U
Eq19

In the following manipulation of this equation, the first and second equations are valid, while the third is not valid.

B
*S
*T
=B
*U
Eq20 - is valid because B is pre-multiplied
S
*T
*B
=U
*B
Eq21 - is valid because B is post-multiplied
S
*T
*B
=B
*U
Eq22 - is invalid because the multiplications are inconsistent

Matrix Conventions¶

The fundamental issue with computer graphics transformations is their order. As we have already discussed, this correlates to the ordering of your matrix multiplications. By convention WebGL (and the OpenGL system it was derived from) orders the transformations from right to left. This is because of the way we created the initial equations. We started by positioning the transformation matrix to the left of the (x,y,z,w) vertex. There is another way you can perform the same multiplication. You can put the (x,y,z,w) vertex at the front of the equation, like this:

x
y
z
1
*1
0
0
tx
0
1
0
ty
0
0
1
tz
0
0
0
1
=x'
y'
z'
1
Eq19

Notice that the translation matrix in the above example had to move the translation values to the last row of the matrix. In fact, every transformation matrix we have discussed (except scaling) will have a different format if you post-multiply the transformations. And if you use this convention, the transformations are applied from left to right, not right to left. The important thing is that you select a convention and use it consistently. Never mix the conventions! For this entire tutorial, we will use the WebGL/OpenGL convention of pre-multiplying the transformations times the vertices.

A note of caution. It is easy to search the web and get conflicting information about graphic transformations because there are two ways to structure the transformations, pre-multiplying or post-multiplying. OpenGL uses the pre-multiplying convention while Microsoft’s Direct3D uses the post-multiplying convention. Therefore, make sure that you understand which convention a web page is assuming so you don’t get confused by conflicting information.

Glossary¶

vector
In mathematics, a vector is any ordered list of values. In computer graphics a vector represents a direction in 3D space.
matrix
The coefficients of a system of equations with the variables removed.
linear algebra
The mathematical concepts and theory concerning vectors and matrices.
matrix multiplication
An algorithm for multiplying two matrices to produce a single matrix.
square matrix
A matrix that has the same number of rows as columns.
transformation matrix
A 4x4 matrix with values in specific locations to perform a specific computer graphics operation.
pre-multiply matrix
The matrix goes on the left side of the multiplication operator.
post-multiply matrix
The matrix goes on the right side of the multiplication operator.
identity matrix
A square matrix with 1’s down the diagonal and zeros in all other positions. Multiplication of a matrix times an identity matrix does not change the original matrix using either pre or post multiplication.
matrix inverse
A matrix that is derived from another matrix such that the multiplication of the original matrix and its inverse results in an identity matrix. A matrix inverse can be either pre or post multiplied to get an identity matrix. Multiplying by a matrix inverse is equivalent to division in algebra.

Extra Resources¶

If your understanding of matrix math is still weak, students have recommended this math is fun web site.

Next Section - 6.6 - A Matrix Code Library - Learn_webgl_matrix