1. Computer Graphics Matrices
CO2409 Computer Graphics
Weeks 6-7

2. Lecture Contents Meshes/Models & World/Model Space
Matrices for Positioning
Matrix Definitions
Matrix Arithmetic

3. Working with 3D Geometry
In 2D we looked at:
Conversion from geometry axes to viewport axes
2D geometry: positioning & drawing
We didn’t look at rotation or scaling of geometry / sprites
3D needs similar processes:
Converting between different 3D axes (e.g. world to camera view)
And finally into 2D viewport axes
Positioning of 3D geometry, drawing of 2D geometry
We will also look at rotation and scaling in 3D

4. Meshes and Models A mesh defines some 3D geometry
We define a model as a single instance of a mesh in a 3D scene
Like a sprite is an instance of a bitmap in a 2D scene
Several models may use the same mesh in the scene
Like several sprites using the same bitmap
Each model can be positioned differently
Position, rotation and scaling
Example of two models of a square mesh:
1 mesh, 2 models

5. World / Model Space
We specify our 3D scene using fixed axes XW YW & ZW
A global view of the world and everything in it
This is called world space or the world coordinate system
The axes are the world axes (in red)
An arbitrarily positioned model has:
A position P and local axes, X, Y & Z
The local axes are its own rightward, upward & forward directions
This is the model’s local space or local coordinate system
Also called model space
Compare to the two overlaid sets of axes we saw in 2D

6. Model Positioning
So the position and rotation of a 3D model can be specified by defining its local space:
Position: a 3D vertex P
Rotation: three 3D vectors – Right, Up and Forward (X,Y & Z)
What about scaling?
The length of the vectors X, Y & Z can define the scaling in that axis
1.0 = unscaled, 2.0 = double size etc.
Effectively scaling local space

7. Matrices for Positioning
We can put these 4 elements into a 4x3 matrix:
(RX, RY, RZ) is the model’s rightward direction, which is its local X axis
(UX, UY, UZ) is the up direction (local Y axis)
(FX, FY, FZ) the forward direction (local Z axis)
These are 3D vectors
(PX, PY, PZ) is the model’s position
All defined in world space units
Example on later slide
Or we can use a 3x4 matrix
Put everything in columns rather than rows
DirectX docs use rows, math books and OpenGL docs use columns. We will use rows. or

8. Alternative Written Form
Here the local X, Y & Z axes are written as R, U and F for right, up and forward:
Easier to understand
Can also use letters X, Y and Z to give this equivalent form:
I frequently use this form
Maths texts may present this matrix with different letters again
This is just presentation, the matrix always has the same basic form

9. Example Position Matrix
Consider this model (30° rotation):
Looking down Y axis - so it is not shown. Allows me to draw 2D diagram
Position, right, up and forward are given on diagram
All axes length 1 – no scaling
This gives us a model matrix:
Uniquely defines model’s position
This matrix can be used to manipulate the model
So we need to look at the mathematics of matrices…

10. Matrix Definition
A matrix (plural matrices) is a rectangular table of numbers:
Matrices can have any number of rows and columns.
A matrix with m rows and n columns is called an m by n matrix
The above matrices are 3x3 and 2x4
The value at the i-th row and j-th column is denoted Aij
In the example above: M23 = 5 and N14 = 2

11. Simple Matrices A 1xn or nx1 matrix can represent a set of coordinates
E.g. a vector or vertex/point:
A square matrix is one with a equal number of rows and columns, e.g. M from last slide
A 1x1 matrix is simply a number

12. Simple Matrix Arithmetic
Addition - equal size matrices can be added:
Subtraction – similar to addition:
Multiplication by a scalar:

13. Matrix Multiplication
We can multiply two matrices R = AB if:
Number of columns in A = number of rows in B
Result has same number of rows as A and columns as B
So if multiplying 2x3 & 3x4 matrices, we get 2x4 matrix result
Values in the result matrix are given by this formula:
Rij = Ai1B1j + Ai2B2j AinBnj
Stated differently, the value in row i and column j of R is equal to the dot product of i-th row of A & j-th column of B
Need to practice this to understand it…

14. Matrix Multiplication Properties
Matrix multiplication is associative:
(AB)C = A(BC)
And distributive:
(A + B)C = AC + BC, C(A + B) = CA + CB
This is the same as scalars
But matrix multiplication is not commutative:
AB ≠ BA (in general)
Very important for transformations later
The order in which you multiply matrices has an effect on the result matrix

15. Square Matrices Square matrices can always be multiplied
The result is a square matrix of the same size
A square matrix with only 1’s down the diagonal has special properties:
Such a matrix does not change other matrices under multiplication
This is called the identity matrix
Or the unit matrix (similarity with the number 1)

16. Transformations (3x3)
A vector or vertex can be represented as a 1x3 matrix:
Multiply by a 3x3 matrix we get another 1x3 matrix:
The vertex/vector changes (transforms) into a new one
This square matrix is a transformation matrix or a transform
Almost all the vertex processing needed for 3D graphics can be performed with transformation matrices
However, we will see that 3x3 matrices cannot perform some of the transformations we need (movement)

17. Transformations (4x4)
So we add another element (p) to the vector / vertex to get a 1x4 matrix:
Then multiply by a 4x4 matrix to perform a transformation
Take the right column from the unit matrix (see earlier slide)
Note the transformation leaves the extra element p unchanged
This kind of 4x4 transformation matrix can perform all of the transformations we need for 3D (including movement)
This form is called an Affine Transformation

18. The 4th Element / Mesh Transforms
The 4th element p specifies if the x, y and z values represent a vertex (point) or a vector
1 = vertex, 0 = vector
Previously we couldn’t distinguish between them (both just x,y,z)
For example:
If we transform all the vertices & normals in a mesh then the entire mesh will be transformed
Recall that normals are just vectors
So we can use these transformations to manipulate entire pieces of geometry

19. Advanced Note: Homogenous Coordinates
A coordinate in 3D is just an x,y,z point. E.g. (10,15,-5)
The 4 element coordinates introduced on the last slide are called homogenous 3D coordinates
They have x,y,z values plus an extra 4th value
To convert a homogenous coordinate to a point, you divide the x, y and z by the 4th element.
The 4th element is only used like this in creating perspective (see later), and at all other times it will only be 1 or 0
If the 4th element is 1, dividing by 1 leaves the x,y,z the same
If the 4th element is 0, you can’t divide by 0 indicating that this is not a point, but a vector
This is only used for advanced work, but is mentioned here since some texts refer to homogenous coordinates

20. Translation Matrices
Translation describes movement of a vertex or mesh
The diagram shows movement along the vector V(p,q,r)
This transformation can be performed with the matrix on the right
The unit matrix with V written into the bottom row
We multiply each vertex in the mesh by this matrix
So the whole mesh is translated

21. Using a Translation Matrix
This is the full matrix multiplication on a point:
Note that a translation matrix has no effect on a vector
Vectors have no position – moving them makes no sense
E.g. the 3 blue vectors in the diagram are all the same even thought drawn in different places

22. Rotation Matrices
A rotation matrix rotates a vertex or mesh around an axis
Most often around the X, Y or Z axis
Could be the world axes or the local axes of a mesh (will see this later)
The diagram shows a rotation around the Z-axis
This Z rotation can be performed with the matrix on the right:
cos and sin appear since we are working with a circle

23. Using a Rotation Matrix
The rotate-Z matrix is used like this:
(not showing full formula for result x & y for simplicity)
Several important notes:
Z values don’t change when rotating around the Z axis
The rotation is around the origin
Not around the centre of the triangle
Vectors are affected by rotation
For example, the edges of the triangle are vectors - notice they are rotated too

24. Other Rotation Matrices
Can rotate around the X & Y axes using similar matrices
The three basic rotation matrices are:
Note how the row (& column) of the axis of rotation is unchanged in each version

25. Scaling Matrices
A simple scaling matrix scales the geometry around the origin
The matrix is like the unit matrix with the scaling value on the diagonal:
Note: Scaling is centred on the origin
Note: Vectors are affected by scaling
We can form a non-regular scaling matrix (different scaling in X, Y & Z):

26. Combining Transformations
We can combine transformations by multiplying their 4x4 matrices together
E.g. A scale followed by a translation:
The resulting matrix can then be used as a combined transformation for vertices / vectors:

27. Order of Transformations
The order of transformations is important
Use same example reversed, translation followed by a scale:
The result matrix is different
The translation has been scaled, as it occurred first
Recall that matrices are not commutative
I.e. AB ≠ BA in general
So it is very important to consider the order of your transformations before combining
[And to do any multiplications in the exam carefully!]

28. Creating Combined Transformations
Say we want a scaling matrix that will scale around a given point rather than the origin
We can do this with three basic transforms:
Translate the point to the origin
Scale (around the origin)
Translate the origin back to the point
The result matrix is potentially useful
Would have been difficult to guess