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Hierarchical Transformations and Models
CSE 3541Matt Boggus
Overview• Previously – Modeling Transformations
– Translate, Rotate, Scale (Fortress Quest)
• Parenting in Unity• Scene Modeling – instances• Hierarchical modeling
– Place model pieces relative to other model pieces– Place objects relative to other objects– Tree representation and traversal
• Examples– Robot arm– Humanoid robot– Scene graphs
Unity parenting and lab2 demo
ParentingExample Unity Package
Modeling Objects
• A model has a default size, position, and orientation
• Modeling transformations set its appearance within the scene
Create a unit cylinder with its origin at (0,0,0)
Instance Transformation
• Start with an object – a symbol• Each appearance of the object in the scene is
an instance– Has its own scale, rotate, and translate
Symbol-Instance Table
Store a model by assigning a number to each symbol and storing the parameters for the instance transformation
Scene example
Critical Thinking – Symbol-Instance Table
• What does not have the be recomputed when using a symbol-instance table?
A. Vertex positions in local/model spaceB. Transformation matricesC. BothD. Neither
Symbol-Instance table efficiency
• 1 scale, 1 rotate, and 1 translate for each house
• Symbol-instance tables reuse data, but not the result of transformations
Draw a robot
Images by Denis Zorin
Draw a robot
Images by Denis Zorin
Draw a robot
• Positioning individual parts is hard• If the whole robot moves
– Reposition everything• If only the upper arm moves
– Reposition the upper and lower arm
• Divide the robot into pieces!– Combine as a hierarchy
Graphs
• Set of nodes and edges (links)• Edge connects a pair of nodes
– Directed or undirected• Cycle: directed path that is a loop
loop
Tree
• Graph in which each node (except the root) has exactly one parent node– May have multiple children– Leaf or terminal node: no children
root node
leaf node
Robot as a hierarchya.k.a. tree or directed acyclic graph
Images by Denis Zorin
Modeling with Trees/Graphs
• Nodes– What to draw– Pointers to children
• Edges– May have information on incremental changes to
transformation matrices (can also store in nodes)
Robot Arm
robot arm parts in their own coordinate systems
Articulated Models
• Robot arm is an example of an articulated model– Parts connected at joints– Can specify state of model by
giving all joint angles
Relationships in Robot Arm
• Base rotates independently– Single angle determines position
• Lower arm attached to base– Its position depends on rotation of
base– Must also translate relative
to base and rotate about connecting joint
Relationships in Robot Arm
• Upper arm attached to lower arm– Its position depends on both
base and lower arm– Must translate relative to
lower arm and rotate about joint connecting to lower arm
Required Matrices
• Rotation of base: Rb
– Apply M = Rb to base
• Translate lower arm relative to base: Tla
• Rotate lower arm around joint: Rla
– Apply M = Rb Tla Rla to lower arm
• Translate upper arm relative to lower arm: Tua
• Rotate upper arm around joint: Rua
– Apply M = Rb Tla Rla Tua Rua to upper arm
Tree Model of Robot
• Note code shows relationships between parts of model– Can change “look” of parts
easily without altering relationships
• Simple example of tree model• Want a general node structurefor nodes
A Chain of Transformations
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Thinking of Transformations• In a view of body-attached coordinate system
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Thinking of Transformations• In a view of body-attached coordinate system
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Thinking of Transformations• In a view of body-attached coordinate system
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X Y
Thinking of Transformations• In a view of body-attached coordinate system
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0cossin
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translrottranslrotT
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11L
X Y
2L
Thinking of Transformations• In a view of body-attached coordinate system
X
Y
100
010
01
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0cossin
0sincos
100
010
01
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0cossin
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translrottranslrotT
Thinking of Transformations• In a view of global coordinate system
X
Y
X Y
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0cossin
0sincos
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0cossin
0sincos 2
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translrottranslrotT
Thinking of Transformations• In a view of global coordinate system
X
Y
2L
X
Y
100
010
01
100
0cossin
0sincos
100
010
01
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0cossin
0sincos 2
22
221
11
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translrottranslrotT
2
X Y
2L
Thinking of Transformations• In a view of global coordinate system
X
Y
100
010
01
100
0cossin
0sincos
100
010
01
100
0cossin
0sincos 2
22
221
11
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2211
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translrottranslrotT
Thinking of Transformations• In a view of global coordinate system
21L
X Y
2L
X
Y
100
010
01
100
0cossin
0sincos
100
010
01
100
0cossin
0sincos 2
22
221
11
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2211
ll
translrottranslrotT
Thinking of Transformations• In a view of global coordinate system
2
11L
X Y
2L
X
Y
100
010
01
100
0cossin
0sincos
100
010
01
100
0cossin
0sincos 2
22
221
11
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translrottranslrotT
Robot as a hierarchy(save intermediate transformations)
Images by Denis Zorin
Scene Graphs
• Encoding the information in software lacks reusability
• Want a flexible, data-driven and extensible approach
• Scene-graphs– OpenInventor (http://www.coin3d.org/)
– Open Scene Graph (http://www.openscenegraph.com/)
– Many others
Scene Graph• Convenient Data structure for scene representation
– Transformations– Materials, color– Multiple instances
• Basic idea: Hierarchical Tree• Useful for manipulation/animation
– Especially for articulated figures• Useful for rendering too
– Multi-pass rendering, occlusion culling
Sample Scene
Hierarchical scene – chair 1
Hierarchical scene – chair 2
Hierarchical scene - chessboard
White chess pieces – child of chessboard
Rook piece – child of white chess pieces
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