1. 1 91.427 Computer Graphics I, Fall 2008 Hierarchical Modeling I

2. 2 91.427 Computer Graphics I, Fall 2008 Objectives Examine limitations of linear modeling ­Symbols and instances Introduce hierarchical models ­Articulated models ­Robots Introduce Tree and DAG models

3. 3 91.427 Computer Graphics I, Fall 2008 Instance Transformation Start with prototype object (a symbol) Each appearance of object in model ==> instance ­Must scale, orient, position ­Defines instance transformation

4. 4 91.427 Computer Graphics I, Fall 2008 Symbol-Instance Table Can store model by assigning number to each symbol and storing parameters for instance transformation

5. 5 91.427 Computer Graphics I, Fall 2008 Relationships in Car Model Symbol-instance table does not show relationships between parts of model Consider model of car ­Chassis + 4 identical wheels ­Two symbols Rate of forward motion determined by rotational speed of wheels

6. 6 91.427 Computer Graphics I, Fall 2008 Structure Through Function Calls car(speed) { chassis() wheel(right_front); wheel(left_front); wheel(right_rear); wheel(left_rear); } Fails to show relationships well ==> Look at problem using graph

7. 7 91.427 Computer Graphics I, Fall 2008 Graphs Set of nodes and edges (links) Edge connects pair of nodes ­Directed or undirected Cycle: directed path that is a loop loop

8. 8 91.427 Computer Graphics I, Fall 2008 Tree Graph in which each node (except root) has exactly one parent node ­May have multiple children ­Leaf or terminal node: no children root node leaf node

9. 9 91.427 Computer Graphics I, Fall 2008 Tree Model of Car

10. 10 91.427 Computer Graphics I, Fall 2008 DAG Model If use fact that all wheels identical ==> directed acyclic graph ­Not much different than dealing with tree

11. 11 91.427 Computer Graphics I, Fall 2008 Modeling with Trees Must decide what information to place in nodes and what to put in edges Nodes ­What to draw ­Pointers to children Edges ­May have information on incremental changes to transformation matrices (can also store in nodes)

12. 12 91.427 Computer Graphics I, Fall 2008 Robot Arm robot arm parts in their own coordinate systems

13. 13 91.427 Computer Graphics I, Fall 2008 Articulated Models Robot arm = example of articulated model ­Parts connected at joints ­Can specify state of model by giving all joint angles

14. 14 91.427 Computer Graphics I, Fall 2008 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 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

15. 15 91.427 Computer Graphics I, Fall 2008 Required Matrices Rotation of base: R b ­Apply M = R b to base Translate lower arm relative to base: T lu Rotate lower arm around joint: R lu ­Apply M = R b T lu R lu to lower arm Translate upper arm relative to upper arm: T uu Rotate upper arm around joint: R uu ­Apply M = R b T lu R lu T uu R uu to upper arm

16. 16 91.427 Computer Graphics I, Fall 2008 OpenGL Code for Robot robot_arm() { glRotate(theta, 0.0, 1.0, 0.0); base(); glTranslate(0.0, h1, 0.0); glRotate(phi, 0.0, 1.0, 0.0); lower_arm(); glTranslate(0.0, h2, 0.0); glRotate(psi, 0.0, 1.0, 0.0); upper_arm(); }

17. 17 91.427 Computer Graphics I, Fall 2008 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 general node structure for nodes

18. 18 91.427 Computer Graphics I, Fall 2008 Possible Node Structure Code for drawing part or pointer to drawing function linked list of pointers to children matrix relating node to parent

19. 19 91.427 Computer Graphics I, Fall 2008 Generalizations Need to deal with multiple children ­How represent more general tree? ­How traverse such data structure? Animation ­How to use dynamically? ­Can create and delete nodes during execution?