1. Hierarchical Modeling & Constructive Solid Geometry
COM337 COMPUTER GRAPHICS
Hierarchical Modeling &
Constructive Solid Geometry

2. Introduction When defining a complex object or system, it is easier to
first specify the subparts,
then, describe how the subparts come together
E.g., a bicycle can be described by
first describing a frame, wheels, fenders, handlebars, seat, chain, and pedals
then positioning these components to form the bicycle

3. Basic Modeling Concepts
Modeling is the creation and manipulation of a system representation
A representation of a system is called a model of the system
A model can be graphical or purely descriptive (such as a set of equations that describe relationships of system parameters)
Graphical models are often called geometric models
We will use the term “model” mostly to mean a geometric model

4. System Representations
The example graphical representation of a logic circuit on the right illustrates features common to many system models
Component parts are shown as geometric structures (symbols)
Relationships between symbols are represented with a network of connecting lines

5. System Representations
Repeated positioning of a few basic symbols (such as the two and gate symbols) is a common method for building complex models
Each such occurrence of a symbol in a model is called an instance of that symbol
In many cases symbols are dictated but in other cases symbols can be any convenient geometric pattern

6. System Representations
The information needed to construct and manipulate a model
can be stored in data structures, and/or
can be specified with procedures
In general, a model specification will contain both data structures and procedures, although some models are defined completely with data structures and others use only procedural specifications
In solid modeling, information from data structures (defining coordinates and other things) is used with few procedures
On the other hand, a weather model may need mostly procedures to calculate plots of temperature and pressure variations.

7. Symbol Hierarchies
Many models can be organized as a hierarchy of symbols
Basic elements for the model can be used to form composite objects (modules), which themselves can be grouped to form higher level objects, and so on …

8. Hierarchical Models
Suppose that we wish to build a model of an automobile that we can animate
The model can be composed of 5 parts (the chassis and the four wheels) each of which is described using standard primitives

9. Hierarchical Models
2 frames of a simple animation of the model are shown below
We can write a program to generate this animation by noting that if each wheel has a radius r, then a 360-degree rotation of a wheel corresponds to the car moving forward (or backward) a distance of 2πr
Program could contain one function to generate each wheel and another to generate the chassis
All functions could use the same input such as the desired speed and direction of the car

10. Hierarchical Models
This program is just the kind that we do NOT want to write
It is linear and does not show the relationships among the components of the car
There are 2 relationships that we want to exploit
We cannot separate the movement of the car from the movement of the wheels
All wheels are identical and merely located in different places with different orientations

11. Hierarchical Models

12. Hierarchical Models

13. Stack-Based Traversal in Application Code
Imagine that we will use a function called figure to do the drawing
The MV matrix, M, when this function is called will determine the position of the figure
The root node (first node) will draw the torso (using M)
Next node is the head: function head will be called with MV updated to MMh
Then, as we continue with left-upper arm, left-lower arm, right arm, left leg, and right leg, MV will be updated to MMlua, MMluaMlll, Mrua, …

14. Stack-Based Traversal in Application Code
After processing a node, if we go back up to its parent, we need to also roll back the transformation matrix.
This can easily be done by making use of a stack to store (push) copies of matrices and by recovering with popping.
Our code will have transformations mixed with pushes and pops of the MV matrix

15. Stack-Based Traversal in Application Code
var modelViewMatrix = mat4(); var mvStack = [ ]; function figure() { mvStack.push(modelViewMatrix); torso(); modelviewMatrix = mult(modelViewMatrix, translate); modelViewMatrix = mult(modelViewMatrix, rotate); head(); modelViewMatrix = mvStack.pop(); leftUpperArm();
modelViewMatrix = mvStack.pop(); mvStack.push(modelViewMatrix); modelviewMatrix = mult(modelViewMatrix, translate); modelViewMatrix = mult(modelViewMatrix, rotate); leftLowerArm(); rightUpperArm(); ⁞

16. Stack-Based Traversal in Application Code
function torso() { instanceMatrix = mult(modelViewMatrix, translate(0.0, 0.5*torsoHeight, 0.0)); mult(instanceMatrix, scale4(torsoWidth, torsoHeight, torsoWidth)); gl.uniformMatrix4fv( modelViewMatrixLoc, false, flatten(instance)); for (var i = 0; i < 6; ++i) { gl.drawArrays(gl.TRIANGLE_FAN, 4*i, 4); }
Functions for individual parts can be as on the right
The textbook first explains a simpler example for a robot arm with 3 parts
Have a look at that example first and try to complete this hierarchical model
If you can complete it, you already have the code for one of the bonuses in the 2nd assignment

17. Stack-Based Application Code vs. Alternative
The example approach has limitations
The code relies on programmer to deal with stacks (push and pops)
The code developed for the example object will not easily extend to another object
Building the model and rendering it is all together, we’d prefer to separate them
A more general and powerful method to work with hierarchies is to implement the tree as a data structure and to render it with a traversal algorithm (See textbook)

18. Hierarchical Models Summary
Combine smaller/simpler shapes to construct complex objects and scenes
Stored in trees or similar data structures
Operations are based on traversal of the tree
Can keep other information together with the geometry in tree such as transformations, attributes, bounding boxes, etc.

19. Scene Graphs
Directed Acyclic Graphs (DAGs) to represent scenes and complex objects
Nodes: Grouping nodes, Transform nodes, Level of Detail (LoD) nodes, Attribute nodes, State nodes
Leaves: Geometric object descriptions

20. Constructive Solid Geometry (CSG)
Combine multiple shapes with set operations (intersection, union, set difference) to construct new shapes

21. CSG

22. Implementing CSG
Ray casting methods are used for rendering and finding properties of volumes constructed with CSG
Rays are intersected with objects
Intersections are sorted according to the distance
Depending on the operation, extents of the new object can be found

23. Implementing CSG
Ray casting methods are used for rendering and finding properties of volumes constructed with CSG
Rays are intersected with objects
Intersections are sorted according to the distance
Depending on the operation, extents of the new object can be found