1. Computer Graphics Matrix Hierarchies / Animation
CO2409 Computer Graphics
Week 21

2. Lecture Contents Model Animation Model/Matrix Hierarchies
Limitations of Hierarchies
Rendering a Model Hierarchy
Matrix Stacks
Process

3. Model Animation So far we have looked at individual models:
Each a fixed piece of geometry
No moving parts
We have animated these models in a way:
By moving and rotating them each frame
However, now we will focus on manipulating (animating) geometry that is:
Made of several rigid parts (this lecture)
Flexible with an underlying skeleton (next lecture)

4. 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…
A common assumption for such models is that the parts form a hierarchy
A tree structure defining how the parts are connected
We will call each part in a tree, a node

5. Limitations of Hierarchies
Most multi-part objects fit naturally into a hierarchical form
In particular it’s usually easy to determine a root node for the object
But consider a bicycle chain – which link is the root?
A hierarchical form also assumes that each node has only one parent that directly 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 a “solver” – part of a physics engine

6. Matrix Hierarchies In such a hierarchy:
Each node has a parent, or is the root of the tree
A node can have any number of children (including 0)
Each node in the hierarchy has a world matrix
Defining its position and orientation - just like a model
However, the world matrix for each node is stored relative to its parent
So each node is defined in the local space of its parent
Root is stored in absolute world space
Implies that child nodes inherit their parent’s movement

7. Matrix Hierarchy: Diagram
Such hierarchies are sometimes called Matrix Hierarchies or Transform Hierarchies

8. Building Hierarchies
The position of a child’s 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 node’s matrix defines the joint with its parent
Must ensure that we build the hierarchies and position matrices correctly
To allow required animation
Actually this is an issue for the 3D artist to resolve

9. Rendering Hierarchies
We want to render a hierarchy of model nodes
We need absolute world matrices for each node rather that the parent-relative world matrix that is stored
Can simply make the existing rendering code recursive – the code for each node is:
Get this node’s absolute world matrix by combining its relative matrix with the parent’s absolute world matrix
Render node with its absolute 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)

10. Rendering Hierarchies: Matrix Stack
Recursion may be inefficient for a real-world app that has many 100s or 1000s of models with many parts
A human model may have 50 or 60 parts
Each function call adds additional code to transfer control to the new function and return back again.
Not usually an issue, but games often need optimisation.
We can convert this recursive process to an iterative one
Instead of using recursion, we use a single loop
The key is to realise that the recursion here is only used to pass the parent’s matrix down to the children.
Instead, just have the nodes point to their parents.

11. Rendering Hierarchies Efficiently
Put nodes into a list (actually a vector or array)
Use depth-first order
Done in advance
Each part has its parent-relative matrix
In the example will use M0, M1 etc.
Also each node knows its parent

12. Rendering Hierarchies Example
Each part needs to know its absolute matrix for rendering
This is its own relative matrix * parent’s absolute matrix
Same as recursive method
But we can get the parent’s absolute matrix directly
Since we have a reference to the parent
And the parent has already been rendered earlier in the list
No need for recursion