CO1301 Games Concepts Week 22 Cross Product

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CO1301 - Games ConceptsWeek 22Cross Product
the Model Matrix

• Gareth Bellaby

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End of Semester

• Into the last three weeks of teaching (including
  this week).
• The last day of teaching is Friday, 25th April.
• Exam period begins 5th May. Look dates of exams
  up on the University web site.
• You do not have an exam in this module.
• I will teach for all of the remaining three weeks
  and you are expected to attend both lectures and
  practicals.

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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 1 "Vector Algebra"

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• Topic 1
• Reminder about vectors and dot product followed
  by cross product

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Vectors

• A vector is a directed edge. In 3 dimensions a
  vector has three components v(x, y, z)

For example the vector a (2, 2, 0)
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Length of a vector

• The length of a vector can be calculated from its
  components.

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The normalised vector

• A normalised vector is a vector whose length is
  1.
• Also known as the unit vector.
• A vector can be normalised by dividing each of
  its components by its length

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Dot product

• The dot product of two vectors v and w is

• The relationship between two vectors can be
  calculated using the dot product.
• The dot product of two vectors expresses the
  angle between the vectors

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Cross Product

• The cross product (like the dot product) takes
  its name from the symbol used a single cross
  between two vectors.

• Pronounced "The cross product of v and w", or
  just the shorthand of "the cross of v and w ".
• The cross product takes two vectors and produces
  a single vector as its result.

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Cross Product
Cross Product

• Vector v

Vector w
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Cross Product

• The cross product of two vectors is the vector
  which are orthogonal (perpendicular or at
  right-angles) to both of the vectors.
• There will always be two vectors which are
  orthogonal to our original two vectors one
  sticking upwards and the other downwards
  (relative to the original vectors). The cross
  product is not commutative. The order of the
  vectors produces the direction.

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Why is it important to you?

• What is interesting about the cross product?

• The cross product of two vectors v and w would
  produce the local axis of rotation between the
  two vectors.

• For example the cross product is used to
  construct a LookAt() function and to implement
  shader techniques such as normal mapping and
  parallax mapping.

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• Topic 2
• Introducing Transformations

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Model Positioning Reminder

• A 3D model has its own local space
• three 3D vectors X,Y Z
• These are the local axes of the model.

• So when you move a model according to its local z
  axis you are calling up the local axes of the
  model in order to do the movement.

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Model Positioning

• These define the local rotation of the model,
  e.g. rotating around its local y-axis, etc.
• The following terms are less commonly used
  nowadays but just in case you come across them
• rotation around the local X is pitch (up and
  down)
• rotation around the local Y is yaw (side to side)
  
• rotation around the local Z is roll

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3D Models in graphics

• 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 ).
• The position of the model are its coordinates in
  world space. The position is the location of its
  local origin.

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3D Models in graphics

• Every model has 4 items of data
• Vector representing the x-axis (relative to model
  origin)
• Vector representing the y-axis (relative to model
  origin)
• Vector representing the z-axis (relative to model
  origin)
• Vector containing the coordinates of the model
  (relative to the world origin)
• Why is it done in this way?
• Is it possible to do this any other way?

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3D Models in graphics

• Each model within the TL-Engine has a floating
  point array associated with it. The array is
  actually a matrix and we'll talk about matrices
  in a while.
• It can be considered to be a 4 by 4 array

float matrix44
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3D Models in graphics

• The first row of the array is the model's local
  x-axis.
• The second row of the array is the model's local
  y-axis.
• The third row of the array is the model's local
  z-axis.
• The final row of the array are the model's
  coordinates.

• Ignore the final component of the vectors for now.

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3D Models in graphics

• As a 4x4 array

• Initialisation in C

float matrix44 0.2, 0.4, 0.1, 0 ,
1, 1, 1, 0 , 0.3, 0.3, 0.3 0 , 10,
3, 7, 1
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Transforming the model

• The matrix can be obtained using the GetMatrix()
  method.

float matrix44 box-gtGetMatrix(
matrix00 )

• The fourth row of the array are the coordinates
  of the model. The fourth row is accessed by
  setting the first subscript of the array to 3.

matrix30 // Px matrix31 //
Py matrix32 // Pz
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Transforming the model

• So doing the following would set the coordinates
  of the model to ( 4, 6, 3 )

matrix30 4.0f matrix31
6.0f matrix32 3.0f

• The matrix can be set using the SetMatrix()
  method.

box-gtSetMatrix( matrix00 )

• And the box (in this case) would move to location
  ( 4, 6, 3 ).

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Scaling

• The axes also provide a scaling. The length of
  the vectors X, Y Z can define the scaling in
  that axis
• 1.0 normal, 2.0 double size etc.
• Effectively scaling local space
• So if you obtain the matrix of the model and
  multiply the components of the first row by 2,
  you will double the size of the model along the
  x-axis.

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Scaling
float matrix44 box-gtGetMatrix(
matrix00 ) matrix01
2.0f matrix02 2.0f matrix03
2.0f box-gtSetMatrix( matrix00 )
To scale the model in all of its axes, you would
need to multiply the second and third rows of the
array in the same way as the first, the effect of
this would be to reproduce the Scale() method.
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• Topic 3
• Matrices

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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 to you?

• Matrices are an essential part of graphics.
• Transformations are carried out using matrices.
• Matrices are key element of linear algebra and
  hence of computer graphics.

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Matrices

• 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.