1. CS 430/585 Computer Graphics I 3D Representation and Solid Modeling Week 8, Lecture 15
David Breen, William Regli and Maxim Peysakhov
Geometric and Intelligent Computing Laboratory
Department of Computer Science
Drexel University

2. Overview 3D Modeling Wireframes Surface Models
Solids and Solid Modeling
Sweep Representations
Constructive Solid Geometry
Boundary Representation
1994 Foley/VanDam/Finer/Huges/Phillips ICG

3. 3D Modeling 3D Representations Modeling in 3D Wireframe models
Surface Models
Solid Models
Meshes and Polygon soups
Voxel/Volume models
Decomposition-based
Octrees, voxels
Modeling in 3D
Constructive Solid Geometry (CSG), Breps and feature-based

4. Representing 3D Objects
Exact
Wireframe
Parametric Surface
Solid Model
CSG
BRep
Implicit Solid Modeling
Approximate
Facet / Mesh
Just surfaces
Voxel
Volume info

5. Representing 3D Objects
Exact
Precise model of object topology
Mathematically represent all geometry
Approximate
A discretization of the 3D object
Use simple primitives to model topology and geometry

6. Negatives when Representing 3D Objects
Exact
Complex data structures
Expensive algorithms
Wide variety of formats, each with subtle nuances
Hard to acquire data
Translation required for rendering
Approximate
Lossy
Data structure sizes can get HUGE, if you want good fidelity
Easy to break (i.e. cracks can appear)
Not good for certain applications
Lots of interpolation and guess work

7. Positives when Representing 3D Objects
Exact
Precision
Simulation, modeling, etc
Lots of modeling environments
Physical properties
Many applications (tool path generation, motion, etc.)
Compact
Approximate
Easy to implement
Easy to acquire
3D scanner, CT
Easy to render
Direct mapping to the graphics pipeline
Lots of algorithms

8. Exact Representations
Wireframe
Parametric Surface
Solid Model
operations
CSG, BRep, implicit geometry

9. Wireframes Basic idea: Example: A simple cube How about the faces?
Represent the model as the set of all of its edges
Example: A simple cube
12 lines
8 vertices
How about the faces?
Foley/VanDam, 1990/1994

10. Wireframes: Examples Cube with extra wires
Question: why would you want this?
Foley/VanDam, 1990/1994

11. Issues with Wireframes
Visually ambiguous
No surfaces!
What’s inside? What’s outside?
Hidden line removal?
What does validity entail?
Don’t we just have a bunch of wires?
Do they need to add up to something?
How to model wireframe shapes?
Wire by wire? Not very easy!

12. Surface Models Basic idea: Limitations: Used in many CAD applications
Represent a model as a set of faces/patches
Limitations:
Topological integrity; how do faces “line up”?; which way is ‘inside’/ ‘outside’?
Used in many CAD applications
Why? They are fine for drafting and rendering, not as good for creating true physical models

13. Surface Models: Examples
Utah Teapot
Original IGES standard
Bezier patches

14. Solids and Solid Modeling
Solid modeling introduces a mathematical theory of solid shape
Domain of objects
Set of operations on the domain of objects
Representation that is
Unambiguous
Accurate
Unique
Compact
Efficient

15. Solid Objects and Operations
Solids are point sets
Boundary and interior
Point sets can be operated on with boolean algebra (union, intersect, etc)
Foley/VanDam, 1990/1994

16. Solid Object Definitions
Boundary points
Points where distance to the object and the object’s complement is zero
Interior points
All the other points in the object
Closure
Union of interior points and boundary points

17. Issues with 3D Set Operations
Ops on 3D objects can create “non-3D objects” or objects with non-uniform dimensions
Objects need to be “Regularized”
Take the closure of the interior
Input set Closure Interior Regularized
Foley/VanDam, 1990/1994

18. Regularized Boolean Operations
3D Example
Two solids A and B
Intersection leaves a “dangling wall”
A 2D portion hanging off a 3D object
Closure of interior gives a uniform 3D result
Pics/Math courtesy of Dave UMD-CP

19. Boolean Operations Other Examples: (c) ordinary intersection
(d) regularized intersection
AB - objects on the same side
CD objects on different sides
Foley/VanDam, 1990/1994

20. Boolean Operations
Foley/VanDam, 1990/1994

21. Constructive Solid Geometry (CSG)
A tree structure combining primitives via regularized boolean operations
Primitives can be solids or half spaces

22. A Sequence of Boolean Operations
Rigid transformations
Pics/Math courtesy of Dave UMD-CP

23. The Induced CSG Tree
Pics/Math courtesy of Dave UMD-CP

24. The Induced CSG Tree
Can also be represented as a directed acyclic graph (DAG)
Pics/Math courtesy of Dave UMD-CP

25. Issues with Constructive Solid Geometry
Non-uniqueness
Choice of primitives
How to handle more complex modeling?
Sculpted surfaces? Deformable objects?

26. Issues with Constructive Solid Geometry
Non-Uniqueness
There is more than one way to model the same artifact
Hard to tell if A and B are identical

27. Issues with CSG
Minor changes in primitive objects greatly affect outcomes
Shift up top solid face
Foley/VanDam, 1990/1994

28. Uses of CSG Constructive Solid Geometry
Found (basically) in every CAD system
Elegant, conceptually and algorithmically appealing
Good for
Rendering, ray tracing, simulation
BRL CAD

29. Sweep Representations
An alternative way to represent 3D obj.
Idea
Given a primitive (e.g. polygon,sphere )
And a sweep (e.g. vector, curve…)
Define solid as space swept out by primitive
Foley/VanDam, 1990/1994

30. Sweep Representations
Issue: how to define intersection?
Foley/VanDam, 1990/1994

31. CAD: Feature-Based Design
CSG is the basic machinery behind CAD features
Features are
Local modifications to object geom/topo with engineering significance
Often are additive or subtractive mods to shape
Hole, pocket, etc…

32. Parametric Modeling in CAD
Feature relationships
Constraints
Foley/VanDam, 1990/1994

33. Boundary Representation Solid Modeling
The de facto standard for CAD since ~1987
BReps integrated into CAGD surfaces + analytic surfaces + boolean modeling
Models are defined by their boundaries
Topological and geometric integrity constraints are enforced for the boundaries
Faces meet at shared edges, vertices are shared, etc.

34. Let’s Start Simple: Polyhedral Solid Modeling
Definition
Solid bounded by polygons whose edges are each a member of an even number of polygons
A 2-manifold: edges members of 2 polygons

35. Properties of 2-Manifolds
For any point on the boundary, its neighborhood is a topological 2D disk
If not a 2-manifold, neighborhood not a disk
Foley/VanDam, 1990/1994

36. Euler’s Formula
For simple polyhedra (no holes): #Vertices - #Edges + #Faces = 2
Foley/VanDam, 1990/1994

37. Euler’s Formula (Generalized)
#Vertices - #Edges + #Faces - #Holes_in_faces = 2 (#Components – Genus)
Genus is the # holes through the object
Euler Operators have been the basis of several modeling systems (Mantyla et al.)
Foley/VanDam, 1990/1994

38. Euler Operators
Loop L  H, Shell S  C

39. Steps to Creating a Polyhedral Solid Modeler
Representation
Points, Lines/Edges, Polygons
Modeling
Generalization of 3D clipping to non-convex polyhedra, enables implementation of booleans

40. State of the Art: BRep Solid Modeling
… but much more than polyhedra
Two main (commercial) alternatives
All NURBS, all the time
Pro/E, SDRC, …
Analytic surfaces + parametric surfaces + NURBS + …. all stitched together at edges
Parasolid, ACIS, …

41. BRep Data Structures Winged-Edge Data Structure (Weiler) Vertex Edge
n edges
Edge
2 vertices
2 faces
Face
m edges
Pics/Math courtesy of Dave UMD-CP

42. BRep Data Structure Vertex structure Edge structure Face structure
X,Y,Z point
Pointers to n coincident edges
Edge structure
2 pointers to end-point vertices
2 pointers to adjacent faces
Pointer to next edge
Pointer to previous edge
Face structure
Pointers to m edges

43. Issues in Boundary Representation Solid Modeling
Very complex data structures
NURBS-based winged-edges, etc
Complex algorithms
manipulation, booleans, collision detection
Robustness
Integrity
Translation
Features
Constraints and Parametrics

44. Other Issues: Non-Manifold Solids
There are cases where you may need to model entities that are not entirely 3D
Pics/Math courtesy of Dave UMD-CP

45. Other Issues in Boundary Representation Solid Modeling
What’s the surface?
Foley/VanDam, 1990/1994

46. Programming Assignment 5
Extend XPM to 60 different RGB colors
Read 3 models and assign each a color
Implement Z-buffer rendering
Implement front & back cutting planes
Only render parts of models between planes
Implement linear depth-cueing
Color = base_color(z-far)/(near-far)
Re-use and extend 2D polygon filling