Hierarchical Transformations and Models CSE 3541 Matt Boggus

Overview Previously – Modeling Transformations – Translate, Rotate, Scale (Fortress Quest)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 space B.Transformation matrices C.Both D.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 hierarchy a.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: R b – Apply M = R b to base Translate lower arm relative to base: T la Rotate lower arm around joint: R la – Apply M = R b T la R la to lower arm Translate upper arm relative to lower arm: T ua Rotate upper arm around joint: R ua – Apply M = R b T la R la T ua R ua 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 structure for nodes

A Chain of Transformations

Thinking of Transformations In a view of body-attached coordinate system

Thinking of Transformations In a view of body-attached coordinate system

Thinking of Transformations In a view of body-attached coordinate system

Thinking of Transformations In a view of body-attached coordinate system

Thinking of Transformations In a view of body-attached coordinate system

Thinking of Transformations In a view of global coordinate system

Thinking of Transformations In a view of global coordinate system

Thinking of Transformations In a view of global coordinate system

Thinking of Transformations In a view of global coordinate system

Thinking of Transformations In a view of global coordinate system

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 ( – Open Scene Graph ( – 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