Computer Graphics Matrix Hierarchies Animation

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Computer GraphicsMatrix Hierarchies / Animation

• CO2409 Computer Graphics
• Week 21

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Lecture Contents

• Model Animation
• Model/Matrix Hierarchies
• Limitations of Hierarchies
• Rendering a Model Hierarchy
• Matrix Stacks
• Process

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Model Animation

• So far we have looked at individual models
• Each a fixed piece of geometry
• No moving parts
• We have animated these models
• By moving and rotating them each frame
• However, now we will focus on manipulating
  (animating) geometry that is
• Made of several rigid parts or
• Flexible with an underlying skeleton

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Rigid Body Animation

• We start with models made up of several rigid
  parts that can move with respect to each other
• Mainly mechanical models, such as vehicles,
  doors, guns, robots etc.
• A common assumption for such models is that the
  parts form a hierarchy
• A tree structure defining how the parts are
  connected

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Matrix Hierarchies

• In such a hierarchy
• Each part has a parent (or is the root of the
  tree)
• A part can have any number of children (including
  0)
• Each part in the hierarchy has a world matrix
• Defining its position and orientation - just like
  a model
• But the world matrix for each part is stored
  relative to its parent

• So each part is defined in the local space of its
  parent
• Root is stored in world space
• Implies that child parts inherit their parents
  movement

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Matrix Hierarchy Diagram

• Such hierarchies are sometimes called Matrix
  Hierarchies or Transform Hierarchies

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Building Hierarchies

• The position of a childs origin determines where
  it will pivot relative to its parent
• The orientation of its axes will determines how
  it will rotate (in X, Y and Z)
• So the parts matrix defines the joint with its
  parent

• Must ensure that we (or the artists) build the
  hierarchies and part matrices correctly
• To allow required animation

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Limitations of Hierarchies

• Most multi-part objects fit naturally into a
  hierarchical form, but not all
• A bicycle chain which link is the root?
• Also assumes each part has only one parent that
  controls its movement
• Not true when multiple forces involved
• Train carriage with two engines
• Two people carrying a stretcher
• Need more complex solution for these cases
• Use soft body methods or physics engine

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Rendering Hierarchies

• Need to render a hierarchy of model parts
• Each with its own parent-relative world matrix
• Can simply make the existing rendering code
  recursive the code for each part is
• Get world matrix by combining this parts matrix
  with the parents world matrix
• Render part with combined matrix
• Repeat process for each child part
• This process dictates a depth-first traversal of
  the hierarchy tree structure
• To pass matrices from parent to child easily (see
  lab)

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Rendering Hierarchies Matrix Stack

• Recursion is inefficient for a real-world app
  that has many models with many parts
• We can convert this recursive process into an
  iterative process
• To help us we use a Matrix Stack
• To store the matrices for ancestors of the
  current part
• DirectX provides such a feature
• Not difficult to write our own if necessary
• This is an efficient LIFO structure for pushing
  and popping matrices

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Rendering Hierarchies Efficiently

• Put parts into a list in depth-first order
• Done in advance
• Also store depth in the hierarchy of each part
• Each part has its parent-relative world matrix
• Call it the local matrix
• In the example will use L0, L1 etc.

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Rendering Hierarchies Efficiently

• Iterating through the list is now the same as
  traversing the tree depth-first
• At each step, will keep track of
• Absolute world matrix call it W
• Current depth in hierarchy call it CurrentDepth
• Initialise the process
• Set W the identity / unit matrix
• Set CurrentDepth 0
• Start with empty matrix stack

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Rendering Hierarchies cont

• For each part x in the hierarchy list
• Get depth of part x, call it NewDepth
• If NewDepth gt CurrentDepth, push W onto stack
• If NewDepth CurrentDepth, do nothing to stack
• If NewDepth lt CurrentDepth, pop matrices from
  stack
• Pop (CurrentDepth NewDepth) times
• E.g. CurrentDepth 3, NewDepth 1, pop stack
  twice
• Set W Matrix at top of stack Lx
• Render part x using world matrix W
• CurrentDepth NewDepth

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Rendering Hierarchies Example

• Starting with W identity(I) CurrentDepth 0
• 0. NewDepth 1 gt CurrentDepth push W
  (StackI)
• W Stack Top L0 L0 render base with
  world matrix L0
• 1. NewDepth 2 gt current push W (StackI,L0)
• W Stack Top L1 L0.L1 render arm
  with matrix L0.L1
• 2. NewDepth 3 gt current push W
  (StackI,L0,L0.L1)
• W Stack Top L2 L0.L1.L2 render grip1
  with matrix L0.L1.L2
• 3. NewDepth 3 current no change
  (StackI,L0,L0.L1)
• W Stack Top L3 L0.L1.L3 render grip2
  with matrix L0.L1.L3
• 4. NewDepth 2 lt current pop stack 1 time
  (StackI,L0)
• W Stack Top L4 L0.L4 render switch
  with matrix L0.L4

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Rendering Hierarchies Example

• Looking just at the render steps
• Render base with world matrix L0
• Render arm with matrix L0.L1
• Render grip1 with matrix L0.L1.L2
• Render grip2 with matrix L0.L1.L3
• Render switch with matrix L0.L4
• These are the correct combined matrices