CO1301 Games Concepts Week 23 Matrices

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CO1301 - Games ConceptsWeek 23Matrices

• Gareth Bellaby

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Reading

• Rabin, Introduction to Game Development
• 4.1 "Mathematical Concepts"
• Van Verthe, Essential Mathematics for Games
• Chapter 2 "Linear Transformations and Matrices"
• Chapter 3 "Affine Transformations".
• Frank Luna, Introduction to 3D Game Programming
  with DirectX 9.0c A Shader Approach
• Chapter 2 "Matrix Algebra"
• Wendy Stahler, Fundamentals of Math and Physcis
  for Games Programmers, Pearson, Prentice Hall,
  2006.

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Model Positioning Reminder

• Each model has three axes definied x, y and z.
• The axes of the model are local to the model.
• The axes are relative to the origin of the model,
  i.e. ( 0, 0, 0 ).
• Each model also has a position. The position is
  in world coordinates. The position is the
  location of its local origin.

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3D Models in graphics

• Recorded as a 4x4 matrix.

x-axis
y-axis
z-axis
position
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Matrices

• Singular matrix
• Plural matrices
• A matrix is a rectangular table of numbers.
• A matrix is composed of rows and columns.

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Why are matrices important?

• A console typically runs at 30 frames per second.
  A PC game may run at 60 frames per second.
• You therefore only have something in the region
  of between 0.017 to 0.033 of a second to perform
  all of the operations you need to carry out for
  the game. This includes AI, game logic, physics
  and, of course, updating and rendering the scene.
• Computer games need to be efficient.
• Matrices are efficient.

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Why are matrices important?

• Matrices are an essential part of graphics.
• Transformations are carried out using matrices.
• Matrices are a key element of linear algebra and
  hence of computer graphics.
• A single matrix can express a complex
  transformation.
• For example, a single matrix can rotate a model
  in all 3 axes simultaneously, whilst at the same
  time moving and scaling.
• Matrices can be combined together. The resulting
  matrix will perform all of the operations of the
  individual matrices but in one single
  mathematical operation.

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Rendering a model

• How is a model rendered on the screen?
• A model is made up of polygons. In practice these
  are triangles.
• A triangle is stored using the coordinates of
  each of its 3 points - the vertices. (e.g. look
  at an .x file for vertex lists).
• The coordinates of the vertices are in local
  space- relative to the origin of the model.
  Another way of saying this is that the
  coordinates of the vertices of the triangles are
  in model space.
• The model is also rotated - along 3 different
  axis.

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Rendering a model

• So
• Every vertex needs to be transformed into world
  space (3 rotations, a transformation, possibly
  scaling).
• The screen is 2D....
• The model needs projecting from 3D into 2D.
• The camera has an imaginary screen somewhere in
  front of it. The model (now in world coordinates)
  is converted from 3D space into the 2D space of
  the screen. The model is projected onto this
  screen.

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Transformation

• When you have a go with DirectX over summer you
  will see that matrices are used to perform this
  transformation the world, view and projection
  matrices. You will also see that the matrices can
  be combined into a single matrix that carries out
  all of the maths in one operation.
• There are calls within DirectX to do all of the
  operations.
• You need to understand the background.
• You also need to understand the maths because
  some of the more interesting graphics requires
  other types of matrix operations.

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Matrices

• Most important thing to remember with matrices
  is
• Row - Column
• First manifestation of row-column is the size the
  matrix. You write the numbers of rows first and
  the number of columns second.
• So a 2x3 matrix is composed of 2 rows and 3
  columns.

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Matrix Sizes
1x3

• Rows by columns.

2x3
3x1
4x4
3x2
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Matrix Addition

• Matrices can be summed together.
• Addition is done component by component (the same
  way as vectors).
• This only works if the matrices are the same size.

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Matrix Addition
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Matrix Addition

• Subtraction is identical to addition.

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Scalar multiplication

• Scalar multiplication is simply when each
  component of a matrix is multiplied by a single
  value, in effect scaling it.

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Matrix Multiplication

• Matrices can be multiplied together.
• Remember with matrices
• Row - Column
• The second manifestation of row-column is matrix
  multiplication when two matrices are multiplied
  together the rows of the first matrix are
  combined with the columns of the second matrix.
• The number of columns in the first matrix has to
  be equal to the number of rows in the second.

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Step guide

• Take a row of the first matrix.
• Take a column from the second matrix.
• The row is rotated clockwise so that it sits
  above the column.
• The components of each will pair up, row
  component with column component (if this does not
  happen then the matrices cannot be multiplied).
• Multiply the pairs together.
• Sum all of the results.
• The final result is one component of the new
  matrix.

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Matrix Multiplication example
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Matrix Multiplication

• The location of the result in the new matrix is
  the position where the row and the column
  intersect.

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Matrix Multiplication example
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Matrix Multiplication

• Anyone notice what operation is being performed
  here?

• It is the dot product of row i from A with column
  j from B.

• Multiplying a matrix by a compatible vector will
  perform a linear transformation on the vector.
• Multiplying matrices together will produce a
  single matrix that performs their combined linear
  transformations.

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Matrix Multiplication

• The size of the row must be equal to the size of
  the column in order to do matrix multiplication.
• Or to put this the other way around. The number
  of columns in the first matrix has to be equal to
  the number of rows in the second.
• Examine the dimensions of the matrices. The two
  inside numbers must be equal
• 2x3 . 3x1
• The size of the resulting matrix will be the two
  outside numbers. So the result of 2x3 . 3x1 will
  be a 2x1 matrix.

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Properties of Matrix Multiplication

• (AB)C A(BC)
• (A B)C AC BC

• AB ? BA
• This is important in graphics because matrices
  are multiplied and it is important to get the
  order right .

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Transpose

• The transpose of a matrix simply interchanges the
  rows and the columns of the matrix.
• It is important because vectors in graphics can
  be ordered by rows (DirectX) or columns (OpenGL
  and HLSL).
• Represented by a superscript T. The transpose of
  matrix A is AT.

• So a nxm matrix becomes an nxm matrix.

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The Identity Matrix

• The identity matrix is a special matrix. The
  identity matrix is a square matrix that has zeros
  for all elements expect along the main diagonal.
• Multiplying a matrix by the identity matrix
  leaves the matrix unchanged.
• For example used as a starting point before
  additional operations are added to a matrix.

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Vectors

• A matrix can have just one column or just one
  row.
• A row or column matrix can be used to represent a
  vector.

vector is
row matrix is

• DirectX uses row vectors (OpenGL uses column
  vectors).

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Transforming a Vector

• If you multiply a row matrix by a 3x3 matrix, the
  result is another row matrix.
• The 3x3 matrix therefore acts to transform the
  row matrix.
• However, a 3x3 matrix can only produce some of
  the transformations required in 3D graphics.
• So another component is added at the end to
  produce a row matrix with a length of 4. The
  transformation matrix now be a 4x4 matrix.

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Rotation using a matrix

• Matrix multiplication is used to perform
  rotations.
• Well start with Euler rotation. In year 3 you'll
  look at a faster alternative.
• Euler is pronounced "oiler".

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Rotation using a matrix