Computer Graphics CS 543 Lecture 8 (Part 1) CS 543 Lecture 8 (Part - PowerPoint PPT Presentation

Computer Graphics CS 543 – Lecture 8 (Part 1) CS 543 Lecture 8 (Part 1) Hierarchical 3D Models Prof Emmanuel Agu Computer Science Dept. Worcester Polytechnic Institute (WPI)

Obj Objectives ti  Examine the limitations of linear modeling  Examine the limitations of linear modeling  Symbols and instances  Introduce hierarchical models I t d hi hi l d l  Articulated models  Robots R b  Introduce Tree and DAG models

I Instance Transformation t T f ti  Start with unique object (a symbol )  Start with unique object (a symbol )  Each appearance of object in model is an instance  Must scale, orient, position l  Defines instance transformation

S Symbol ‐ Instance Table b l I t T bl Can store a model by assigning number to each Can store a model by assigning number to each symbol and storing parameters for instance transformation

R l ti Relationships in Car Model hi i C M d l  Symbol instance table does not show  Symbol ‐ instance table does not show relationships between parts of model  Consider model of car  Consider model of car  Chassis + 4 identical wheels  Two symbols y  Rate of forward motion determined by rotational speed of wheels

Structure Through Function Calls St t Th h F ti C ll car(speed) car(speed) { chassis() wheel(right_front); ( ) wheel(left_front); wheel(right rear); g _ wheel(left_rear); }  Fails to show relationships well  Look at problem using a graph p g g p 6

G Graphs h  Set of nodes and edges (links)  Set of nodes and edges (links)  Edge connects a pair of nodes  Directed or undirected d d d  Cycle : directed path that is a loop loop loop 7

Tree  Graph in which each node (except the root) has  Graph in which each node (except the root) has exactly one parent node  May have multiple children  May have multiple children  Leaf or terminal node: no children root node root node leaf node 8

T Tree Model of Car M d l f C 9

R b t A Robot Arm parts in their own parts in their own robot arm coodinate systems 10

A ti Articulated Models l t d M d l  Robot arm is example of articulated model  Robot arm is example of articulated model  Parts connected at joints  Can specify state of model by C if f d l b giving all joint angles 11

R Required Matrices i d M t i  Rotation of base: R b Rotation of base: R b  Apply M = R b to base  Translate lower arm relative to base: T lu 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 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 Rotate upper arm around joint: uu  Apply M = R b T lu R lu T uu R uu to upper arm 12

Hi Hierarchical Transforms hi l T f  Robot arm: Many small parts  Robot arm: Many small parts  Attributes (position, orientation, etc) depend on each other each other hammer A Robot Hammer! lower arm base

Hi Hierarchical Transforms hi l T f  Object dependency description using tree  Object dependency description using tree structure Root node Root node Base Base Object position and orientation can be affected by its parent, grand-parent, grand-grand-parent Lower arm … nodes d Upper arm Hierarchical representation is known as Scene Graph Leaf node Hammer

T Transformations f ti  Two ways to specify transformations:  Two ways to specify transformations:  (1) Absolute transformation: each part of the object is transformed independently relative to the origin transformed independently relative to the origin Translate the base by (5,0,0); Translate the lower arm by (5,0,0); Translate the upper arm by (5,0,0); y … x z

R l ti Relative Transformation T f ti A better (and easier) way: A better (and easier) way: (2) Relative transformation: Specify the transformation for each object relative to its t f ti f h bj t l ti t it parent Step 1: Translate base and p its descendants by (5,0,0);

R l ti Relative Transformation T f ti Step 2: Rotate the lower arm and all its descendants relative to the base’s local y axis by -90 degree y y g y y z x x z

R l ti Relative Transformation T f ti  Represent relative transformation using scene  Represent relative transformation using scene graph Base Base T Translate (5,0,0) l t (5 0 0) Lower arm Rotate (-90) about its local y Upper arm Apply all the way pp y y down Apply all the way Hammer down

Hi Hierarchical Transforms Using OpenGL hi l T f U i O GL  Translate base and all its descendants by (5 0 0)  Translate base and all its descendants by (5,0,0)  Rotate lower arm and its descendants by ‐ 90 degree about local y ctm = LoadIdentity(); y(); Base … // setup your camera ctm * = Translatef(5 0 0); ctm * = Translatef(5,0,0); Lower arm Draw_base(); U Upper arm ctm * = Rotatef(-90, 0, 1, 0); Draw_lower _arm(); Hammer Hammer Draw upper arm(); Draw_upper_arm(); Draw_hammer();

O OpenGL Code for Robot GL C d f R b t mat4 ctm; mat4 ctm; robot_arm() { ctm = RotateY(theta); ( ) base(); ctm *= Translate(0.0, h1, 0.0); ctm *= RotateZ(phi); lower_arm(); ctm *= Translate(0 0 ctm = Translate(0.0, h2, 0.0); h2 0 0); ctm *= RotateZ(psi); upper_arm(); } } 20

H Humanoid Figure id Fi 21

B ildi Building the Model th M d l  Can build model using simple shapes  Can build model using simple shapes  Access parts through functions  torso() ()  left_upper_arm()  Matrices describe position of node with respect d b f d h to its parent  M lla positions left lower leg with respect to left upper l f l l h l f arm 22

T Tree with Matrices ith M t i 23

T Transformation Matrices f ti M t i  There are 10 relevant matrices  There are 10 relevant matrices  M positions and orients entire figure through the torso which is the root node torso which is the root node  M h positions head with respect to torso  M lua , M rua , M lul , M rul position arms and legs with l position arms and legs with  M l M M l l M respect to torso  M lla , M rla , M lll , M rll position lower parts of limbs with lla rla lll rll respect to corresponding upper limbs 24

glPushMatrix and glPopMatrix lP hM t i d lP M t i  Two important calls:  PushMatrix( ): Save current modelview matrix in stack  PopMatrix( ): restore transform matrix to what it was before PushMatrix( ) ( )

PopMatrix and PushMatrix Illustration i S in Stack k Ref: Computer Graphics Through OpenGL by Guha

St Stack ‐ based Traversal k b d T l  Set model ‐ view matrix to M and draw torso  Set model view matrix to M and draw torso  Set model ‐ view matrix to MM h and draw head  For left ‐ upper arm need MM lua and so on  For left upper arm need MM and so on  Rather than recomputing MM lua from scratch or using an inverse matrix, we can use the i i t i th matrix stack to store M and other matrices as we traverse the tree we traverse the tree 27

T Traversal Code l C d save present model view matrix save present model-view matrix figure() { figure() { PushMatrix() update model-view matrix for head torso(); Rotate (…); recover original model-view matrix head(); PopMatrix(); PopMatrix(); save it again sa e it again PushMatrix(); update model-view matrix Translate(…); for left upper arm for left upper arm Rotate(…); ( ) left_upper_arm(); recover and save original PopMatrix(); p (); model-view matrix again model-view matrix again PushMatrix(); rest of code 28

h Scene Graph G 29 S

P Preorder Traversal d T l PushAttrib PushAttrib PushMatrix Color Translate Rotate Object1 Translate Object2 Object2 PopMatrix PopAttrib … 30

I Inventor and Java3D t d J 3D  Inventor and Java3D provide a scene graph API  Inventor and Java3D provide a scene graph API  Scene graphs can also be described by a file (text or binary) binary)  Implementation independent way of transporting scenes  Supported by scene graph APIs  However, primitives supported should match capabilities of graphics systems  Hence most scene graph APIs are built on top of OpenGL or DirectX (for PCs) 31

VRML VRML  Want to have a scene graph that can be used  Want to have a scene graph that can be used over the World Wide Web  Need links to other sites to support distributed N d li k t th it t t di t ib t d data bases  Virtual Reality Markup Language Vi t l R lit M k L  Based on Inventor data base  Implemented with OpenGL I l d i h O GL 32

References  Angel and Shreiner, Interactive Computer Graphics (6 th edition), Chapter 8