Hierarchical Modeling Tree of Transformations Display Lists and

Hierarchical Modeling: Tree of Transformations, Display Lists and Functions, Matrix and Attribute Stacks, Hierarchical Modeling Hofstra University 1

Modeling complex objects/motion n n Decompose object hierarchically into parts Model each part in local frame, and use affine transformations to position it at its proper place in the part above it in the hierarchy Hierarchical Modeling Hofstra University 2

The pipeline conceptually n Construct shapes from geometric primitives n Arrange the objects in three dimensional space and select the desired vantage point for viewing the composed scene. n n n Calculate the color of all the objects. The color might be explicitly assigned by the application, determined from specified lighting conditions, or obtained by pasting a texture onto the objects. Convert the mathematical description of objects and their associated color information to pixels on the screen. During these stages, an API might perform other operations, such as eliminating parts of objects that are hidden by other objects. Hierarchical Modeling Hofstra University 3

The Rendering Hierarchical Modeling Hofstra University 4

World Coordinates and Model Coordinates Hierarchical Modeling Hofstra University 5

World Coordinates and Model Coordinates Hierarchical Modeling Hofstra University 6

Viewing the World Hierarchical Modeling Hofstra University 7

Viewing Transformation Hierarchical Modeling Hofstra University 8

MVT, Lights, Material, Shading, … Rendering Hierarchical Modeling Hofstra University 9

Coordinate Systems n n n Object/Model World Camera/eye Image/Normalized View Volume Screen, Pixel Hierarchical Modeling Hofstra University 10

Possible approaches n n Symbols and instancing, instancing transforms: can be represented by a table. Hierarchical modeling. Scene graphs (SG). n n (Scene) Trees DAGs Hierarchical Modeling Hofstra University 12

Instancing in Open. GL n In Open. GL, instancing is created by modifying the model-view matrix: gl. Matrix. Mode( GL_MODELVIEW ); gl. Load. Identity(); gl. Push. Matrix( ); gl. Translatef(. . . ); gl. Rotatef(. . . ); gl. Scalef(. . . ); symbol(); gl. Pop. Matrix( ); Hierarchical Modeling Hofstra University 13

Symbols & Instances Hierarchical Modeling Hofstra University 14

Village code gl. Matrix. Mode(GL_MODELVIEW); gl. Load. Identity(); setview(); //instance 1 Basic simple house gl. Push. Matrix(); house(); gl. Pop. Matrix(); //simple house instance 2 Big house Translated gl. Push. Matrix(); gl. Translatef(6, 0, 0); gl. Scalef(2, 2, 2); house(); gl. Pop. Matrix(); //simple house instance 3 Intermed. house Rotated 15 deg and Translated gl. Push. Matrix(); gl. Translatef(0, 6, 0); gl. Rotated(15, 0, 0, 1); gl. Scalef(2, 1, 1); house(); gl. Pop. Matrix(); //david house Instance 1 gl. Push. Matrix(); gl. Translatef(-6, 0, 0); gl. Scalef(2, 2, 2); david_house(); gl. Pop. Matrix(); Hierarchical Modeling Hofstra University 15

Hierarchical Modeling n DAG allow high-level sharing of objects but often make the memory costs and code complexity too high. DAGs are natural in many applications, e. g. , modeling polygonal meshes where many edges and vertices are shared by higher level objects. Hierarchical Modeling Hofstra University 16

DAG example: polygonal mesh surface Polygon 1 Edge 1 Vertex 1 Edge 2 Vertex 2 Hierarchical Modeling Polygon 2 Polygon 3 Polygon n Edge 3 Vertex 3 Hofstra University 17

Hierarchical Modeling using Trees n n Primitives that can be drawn directly, at leaves Internal nodes become instances Transformations that position instances of objects label the edges Traverse in pre-order (visit parent, visit children left to right). n n Stack based implementation Model independent traversal algorithm Hierarchical Modeling Hofstra University 18

Example: simple 2 D drawing Hierarchical Modeling Hofstra University 19

Hierarchical modeling: tree of transformations T 1. T 2. T 5. T 7. T 9(LINE) Final picture, obtained by traversing in pre-order the tree Hierarchical Modeling Hofstra University 20

Compress the tree n n n If an edge (A, B) is labeled with identity transformation, remove it , and move any drawings from node B into A If edges (A, B) and (A, C) have the same labels, merge the nodes B and C Example: in the previous tree T 8 will be removed, and the unit square moved to the parent node Hierarchical Modeling Hofstra University 21

From the tree model to Open. GL code n n Start from the root, traverse tree in pre-order When visiting a node, if there is a primitive at the node that can be drawn, draw it If going down from a node which has more then one unvisited child, store(push) CTM and attributes before continuing down If coming back up an edge to a node which has unvisited children, pop CTM and attributes Hierarchical Modeling Hofstra University 22

Stack implementation of the compressed tree gl. Scalef(1. 0, -1. 0, 1. 0); //T 1 gl. Push. Matrix(); gl. Translatef(1. 5, 3. 0, 0. 0); //T 2 gl. Scalef(1. 0, 2. 0, 1. 0); //T 5 gl. Begin(GL_LINE_LOOP); //BOX gl. Vertex 3 f(0. 0, 0. 0); gl. Vertex 3 f(1. 0, 0. 0); gl. Vertex 3 f(0. 0, 1. 0, 0. 0); gl. End(); gl. Translatef(0. 5, 0. 0); //T 7 gl. Rotatef(90. 0, 1. 0); //T 9 gl. Begin(GL_LINES); //LINE gl. Vertex 3 f(0. 0, 0. 0); gl. Vertex 3 f(1. 0, 0. 0); gl. End(); Hierarchical Modeling gl. Pop. Matrix(); gl. Push. Matrix(); gl. Translatef(2. 0, 1. 0, 0. 0); //T 3 gl. Scalef(0. 5, 1. 0); //T 6 draw. Unit. Circle(NUM_SLICES); gl. Pop. Matrix(); gl. Scalef(4. 0, 5. 0, 1. 0); // T 4 gl. Begin(GL_LINE_LOOP); // BOX gl. Vertex 3 f(0. 0, 0. 0); gl. Vertex 3 f(1. 0, 0. 0); gl. Vertex 3 f(0. 0, 1. 0, 0. 0); gl. End(); gl. Flush(); Hofstra University 23

Display Lists n Need to draw the display at a rate sufficient to avoid noticeable flicker Hierarchical Modeling Hofstra University 24

Display Processor n n n Display processor – special purpose, limited instruction set, oriented toward drawing graphics primitives Display memory holds lists of instructions called display lists Display list executed repeatedly at a rate sufficient to avoid flicker Hierarchical Modeling Hofstra University 25

Display Modes n n n Immediate Mode – as soon as the program executes a statement that defines a primitive, it is sent to the server Retained Mode – define the object once, then put its description in a display list. Redisplayed with a simple function call Reduced network traffic Hierarchical Modeling Hofstra University 26

Open. GL Display Lists #define BOX 1 gl. New. List(BOX, GL_COMPILE); gl. Begin(GL_POLYGON); gl. Color 3 f(0. 0, 1. 0, 0. 0); gl. Vertex 2 f(-1. 0, -1. 0); gl. Vertex 2 f( 1. 0, 1. 0); gl. Vertex 2 f(-1. 0, 1. 0); gl. End( ); gl. End. List( ); Send list to server but don’t display contents GL_COMPILE_AND_EXECUTE This can change the original state and leave it altered gl. Call. List(BOX); Hierarchical Modeling Hofstra University 27

Definition and Execution #define BOX 1 gl. New. List(BOX, GL_COMPILE); gl. Push. Attrib(GL_ALL_ATTRIB_BITS); gl. Push. Matrix( ); gl. Begin(GL_POLYGON); gl. Color 3 f(0. 0, 1. 0, 0. 0); gl. Vertex 2 f(-1. 0, -1. 0); Push and pop gl. Vertex 2 f( 1. 0, -1. 0); attributes and coordinates gl. Vertex 2 f( 1. 0, 1. 0); to the stacks gl. Vertex 2 f(-1. 0, 1. 0); gl. End( ); gl. Pop. Attrib( ); gl. Pop. Matrix( ); gl. End. List( ); Hierarchical Modeling Hofstra University 28

Stack implementation of the tree of transformations gl. Scalef(1. 0, -1. 0, 1. 0); gl. Push. Matrix(); gl. Translatef(1. 5, 3. 0, 0. 0); gl. Scalef(1. 0, 2. 0, 1. 0); gl. Call. List(BOX); gl. Translatef(0. 5, 0. 0); gl. Rotatef(90. 0, 1. 0); gl. Begin(GL_LINES); gl. Vertex 3 f(0. 0, 0. 0); gl. Vertex 3 f(1. 0, 0. 0); gl. End(); Hierarchical Modeling gl. Pop. Matrix(); gl. Push. Matrix(); gl. Translatef(2. 0, 1. 0, 0. 0); gl. Scalef(0. 5, 1. 0); draw. Unit. Circle(NUM_SLICES); gl. Pop. Matrix(); gl. Scalef(4. 0, 5. 0, 1. 0); gl. Call. List(BOX); gl. Flush(); Hofstra University 29

Example: cart with two wheels Hierarchical Modeling Hofstra University 30

Example: wheel Model a red wheel of radius WHEEL_RAD, consisting of a circle and two crossed spikes. Primitive components are unit circle and crossed orthogonal spikes of length 2. gl. New. List(WHEEL, GL_COMPILE); gl. Push. Matrix(); gl. Push. Attrib(GL_COLOR); // red wheel with WHEEL_RAD gl. Color 3 f(1. 0, 0. 0); // red gl. Scalef(WHEEL_RAD, 1. 0); // scale to wheel radius gl. Begin(GL_LINES); gl. Vertex 2 f( -1. 0, 0. 0); gl. Vertex 2 f( 0. 0, -1. 0); gl. Vertex 2 f( 0. 0, 1. 0); gl. End(); draw. Unit. Circle(NUM_SLICES); gl. Pop. Attrib(); gl. Pop. Matrix(); gl. End. List(); Hierarchical Modeling // cross, each bar length 2 // unit circle Hofstra University 31

Example: rod with 2 fixed wheels Model the cart consisting of two red wheels of radius WHEEL_RAD, positioned at the ends of a green rod of length 2*HALF_AXIS_LENGTH. The two wheels are rotated at different angles. gl. New. List(CART, GL_COMPILE); gl. Push. Matrix(); gl. Push. Attrib(GL_COLOR); gl. Color 3 f(0. 0, 1. 0, 0. 0); // green gl. Begin(GL_LINES); // rod gl. Vertex 2 f(-HALF_AXIS_LENGTH, 0. 0); gl. Vertex 2 f( HALF_AXIS_LENGTH, 0. 0); gl. End(); gl. Push. Matrix(); gl. Translatef(-HALF_AXIS_LENGTH, 0. 0); //position front WHEEL at axis gl. Rotatef(TH, 0. 0, 1. 0); // rotate a WHEEL by TH gl. Call. List(WHEEL); gl. Pop. Matrix(); gl. Translatef(HALF_AXIS_LENGTH, 0. 0); //position back WHEEL at axis gl. Rotatef(TH+INCR, 0. 0, 1. 0); // rotate a WHEEL by TH+INCR gl. Call. List(WHEEL); gl. Pop. Attrib(); gl. Pop. Matrix(); gl. End. List(); Hierarchical Modeling Hofstra University 32

Example: cart with 2 wheels void display() { gl. Clear(GL_COLOR_BUFFER_BIT); gl. Call. List(CART); gl. Flush(); } void myinit() { gl. Clear. Color(1. 0, 0. 0); gl. Matrix. Mode(GL_PROJECTION); gl. Load. Identity(); glu. Ortho 2 D(-225. 0, -225, 225. 0); gl. Matrix. Mode(GL_MODELVIEW); } define. Parts(); // WHEEL and CART are display lists defined here, main() …. As usual, negotiate with Windows Sys, register callbacks, run loop to process events Hierarchical Modeling Hofstra University 33

Example: spin the wheels n n Limitations of display lists Use of functions in drawing instances of objects at tree nodes Continuously keep changing the value of rotation angle for turning a wheel Need of double buffering Hierarchical Modeling Hofstra University 34

Example: spin the wheels, 2 void draw_cart() { gl. Push. Matrix(); gl. Push. Attrib(GL_COLOR); gl. Color 3 f(0. 0, 1. 0, 0. 0); // draw axis in green gl. Begin(GL_LINES); gl. Vertex 2 f(-HALF_AXIS_LENGTH, 0. 0); gl. Vertex 2 f(HALF_AXIS_LENGTH, 0. 0); gl. End(); gl. Push. Matrix(); gl. Translatef(-HALF_AXIS_LENGTH, 0. 0); // position front wheel at axis, T 1 gl. Rotatef(theta, 0. 0, 1. 0); // spin front WHEEL by theta gl. Call. List(WHEEL); // draw wheel gl. Pop. Matrix(); gl. Translatef(HALF_AXIS_LENGTH, 0. 0); // position back wheel at axis, T 2 gl. Rotatef(theta, 0. 0, 1. 0); // spin back WHEEL by theta gl. Call. List(WHEEL); // draw wheel gl. Pop. Attrib(); gl. Pop. Matrix(); } Hierarchical Modeling Hofstra University 35

Open GL interactive program structure int main(int argc, char** argv) { glut. Init(&argc, argv); glut. Init. Display. Mode (GLUT_SINGLE | GLUT_RGB); glut. Create. Window(”generic example"); myinit (); glut. Reshape. Func (my. Reshape); glut. Mouse. Func (mouse); glut. Motion. Func(do_on_motion); glut. Idle. Function(idle); glut. Display. Func(display); glut. Main. Loop(); } Hierarchical Modeling Hofstra University 36

Example: spin the wheels, 3, the Idle Function void idle(void) { theta += INCR_ANGLE; glut. Post. Redisplay(); } void mouse. Motion(int x, int y) { transx = x; transy = y; glut. Post. Redisplay(); } void display() { gl. Clear(GL_COLOR_BUFFER_BIT); // IDLE callback function updates // rotation angle that spins a wheel // MOTION callback function updates // translation vector that moves whole // DISPLAY callback draws whole object // in back buffer then swaps buffers // translate whole object axis with spinning wheels gl. Translatef( (float)transx, (float)(WINDOWH-transy), 0. 0); draw_cart(); } glut. Swap. Buffers(); Hierarchical Modeling // SWAP BUFFERS Hofstra University 37

Example: spin the wheels, 4, the Idle Function int main(…) { glut. Init. Display. Mode(GLUT_DOUBLE|GLUT_RGB); // double buffering. . . glut. Display. Func(display); glut. Motion. Func(mouse. Motion); glut. Idle. Func(idle); glut. Main. Loop(); … } Hierarchical Modeling Hofstra University 38

n HW: Model hierarchically a 3 D scene consisting of a circular road, a car, at least 2 trees along the road. Each component has a different color than the rest. n Allow for interactively changing the camera position, allowing the camera to rotate around the horizontal and vertical axis of the coordinate system attached to the scene, and also for interactively choosing an option to move the car forward or backward n A car has a rectangular body and 4 wheels n A tree has a trunk (cylinder) and a crown (sphere) n The road is a circular loop n Model each object (object part) in its own local coordinate system; use transformations to position all objects in the scene. n Trees of transformations should help in modeling the individual objects (tree, car), and also in putting the whole scene together. n Use functions or display lists to model the components n HW is due in 2 weeks. Pace yourself well: complete the modeling in a week, so the week after you could deal with the interaction and the animation. Model and test each object individually , first. Hierarchical Modeling Hofstra University 39

Example: model circular road // ROAD_RAD 1, ROAD_RAD 2, ROAD_SL, ROAD_RINGS defined globally void draw_road() { GLUquadric* q; q = glu. New. Quadric(); // dynamic array to hold vertices of road // creates and returns a ptr to new quadric object gl. Push. Matrix(); gl. Push. Attrib(GL_ALL_ATTRIB_BITS); gl. Color 3 f(0. 5, 0. 5); // road is gray glu. Disk(q, ROAD_RAD 1, ROAD_RAD 2, ROAD_SL, ROAD_RINGS); gl. Pop. Matrix(); gl. Pop. Attrib(); } glu. Delete. Quadric(q); Hierarchical Modeling Hofstra University 40

Example: model circular road void display() { gl. Clear(GL_COLOR_BUFFER_BIT); // gl. Rotatef(45, 1. 0, 0. 0); //glu. Look. At(0. 0, -1. 0, 0. 0, 1. 0); draw_road(); gl. Flush(); } Hierarchical Modeling Hofstra University 41