Download Template - Department of Computer Science

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
http://gicl.cs.drexel.edu
1
Overview
•
•
•
•
3D Modeling
Wireframes
Surface Models
Solids and Solid Modeling
– Sweep Representations
– Constructive Solid Geometry
– Boundary Representation
2
1994 Foley/VanDam/Finer/Huges/Phillips ICG
3D Modeling
• 3D Representations
–
–
–
–
–
–
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
3
Representing 3D Objects
• Exact
– Wireframe
– Parametric
Surface
– Solid Model
• Approximate
– Facet / Mesh
• Just surfaces
– Voxel
• Volume info
• CSG
• BRep
• Implicit Solid
Modeling
4
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
5
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
6
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
7
Exact Representations
• Wireframe
• Parametric Surface
• Solid Model
– operations
– CSG, BRep, implicit geometry
8
Wireframes
• Basic idea:
– Represent the model
as the set of all of its
edges
• Example:
A simple cube
– 12 lines
– 8 vertices
• How about the
faces?
9
Foley/VanDam, 1990/1994
Wireframes: Examples
• Cube with extra
wires
• Question: why
would you want
this?
10
Foley/VanDam, 1990/1994
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!
11
Surface Models
• Basic idea:
– 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
12
Surface Models: Examples
• Utah Teapot
• Original IGES
standard
• Bezier patches
13
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
14
Solid Objects and Operations
• Solids are point sets
– Boundary and interior
• Point sets can be operated on with
boolean algebra (union, intersect, etc)
15
Foley/VanDam, 1990/1994
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
16
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
17
Foley/VanDam, 1990/1994
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
18
Pics/Math courtesy of Dave Mount @ UMD-CP
Boolean Operations
• Other Examples:
• (c) ordinary
intersection
• (d) regularized
intersection
– AB - objects on the
same side
– CD objects on
different sides
19
Foley/VanDam, 1990/1994
Boolean Operations
20
Foley/VanDam, 1990/1994
Constructive Solid Geometry
(CSG)
• A tree structure
combining primitives
via regularized
boolean operations
• Primitives can be
solids or half spaces
21
A Sequence of Boolean
Operations
• Boolean operations
• Rigid transformations
22
Pics/Math courtesy of Dave Mount @ UMD-CP
The Induced CSG Tree
23
Pics/Math courtesy of Dave Mount @ UMD-CP
The Induced CSG Tree
• Can also be
represented as a
directed acyclic
graph (DAG)
24
Pics/Math courtesy of Dave Mount @ UMD-CP
Issues with
Constructive Solid Geometry
• Non-uniqueness
• Choice of primitives
• How to handle more complex modeling?
– Sculpted surfaces? Deformable objects?
25
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
26
Issues with CSG
• Minor changes
in primitive
objects greatly
affect outcomes
• Shift up top solid
face
27
Foley/VanDam, 1990/1994
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
28
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
29
Foley/VanDam, 1990/1994
Sweep Representations
• Issue: how to define
intersection?
30
Foley/VanDam, 1990/1994
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…
31
Parametric Modeling in CAD
• Feature relationships
• Constraints
32
Foley/VanDam, 1990/1994
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.
33
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
34
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
35
Foley/VanDam, 1990/1994
Euler’s Formula
• For simple polyhedra (no holes):
#Vertices - #Edges + #Faces = 2
36
Foley/VanDam, 1990/1994
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.)
37
Foley/VanDam, 1990/1994
Euler Operators
Loop L  H, Shell S  C
38
Steps to Creating a Polyhedral
Solid Modeler
• Representation
– Points, Lines/Edges, Polygons
• Modeling
– Generalization of 3D clipping to nonconvex polyhedra, enables implementation
of booleans
39
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, …
40
BRep Data Structures
• Winged-Edge Data
Structure (Weiler)
• Vertex
– n edges
• Edge
– 2 vertices
– 2 faces
• Face
– m edges
41
Pics/Math courtesy of Dave Mount @ UMD-CP
BRep Data Structure
• Vertex 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
42
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
43
Other Issues:
Non-Manifold Solids
• There are cases where you may need to
model entities that are not entirely 3D
44
Pics/Math courtesy of Dave Mount @ UMD-CP
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
46