CS 430/585 Computer Graphics I 3D Representation and Solid Modeling Week 8, Lecture 15

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CS 430/585Computer Graphics I3D 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

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Overview

• 3D Modeling
• Wireframes
• Surface Models
• Solids and Solid Modeling
• Sweep Representations
• Constructive Solid Geometry
• Boundary Representation

1994 Foley/VanDam/Finer/Huges/Phillips ICG
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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

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Representing 3D Objects

• Exact
• Wireframe
• Parametric Surface
• Solid Model
• CSG
• BRep
• Implicit Solid Modeling

• Approximate
• Facet / Mesh
• Just surfaces
• Voxel
• Volume info

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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

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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

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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

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Exact Representations

• Wireframe
• Parametric Surface
• Solid Model
• operations
• CSG, BRep, implicit geometry

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Wireframes

• Basic idea
• Represent the model as the set of all of its
  edges
• ExampleA simple cube
• 12 lines
• 8 vertices
• How about the faces?

Foley/VanDam, 1990/1994
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Wireframes Examples

• Cube with extra wires
• Question why would you want this?

Foley/VanDam, 1990/1994
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Issues with Wireframes

• Visually ambiguous
• No surfaces!
• Whats inside? Whats outside?
• Hidden line removal?
• What does validity entail?
• Dont 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!

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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

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Surface Models Examples

• Utah Teapot
• Original IGES standard
• Bezier patches

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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

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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
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Solid Object Definitions

• Boundary points
• Points where distance to the object and the
  objects complement is zero
• Interior points
• All the other points in the object
• Closure
• Union of interior points and boundary points

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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
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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 Mount _at_ UMD-CP
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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
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Boolean Operations
Foley/VanDam, 1990/1994
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Constructive Solid Geometry (CSG)

• A tree structure combining primitives via
  regularized boolean operations
• Primitives can be solids or half spaces

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A Sequence of Boolean Operations

• Boolean operations
• Rigid transformations

Pics/Math courtesy of Dave Mount _at_ UMD-CP
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The Induced CSG Tree
Pics/Math courtesy of Dave Mount _at_ UMD-CP
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The Induced CSG Tree

• Can also be represented as a directed acyclic
  graph (DAG)

Pics/Math courtesy of Dave Mount _at_ UMD-CP
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Issues with Constructive Solid Geometry

• Non-uniqueness
• Choice of primitives
• How to handle more complex modeling?
• Sculpted surfaces? Deformable objects?

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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

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Issues with CSG

• Minor changes in primitive objects greatly affect
  outcomes
• Shift up top solid face

Foley/VanDam, 1990/1994
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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

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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
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Sweep Representations

• Issue how to define intersection?

Foley/VanDam, 1990/1994
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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

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Parametric Modeling in CAD

• Feature relationships
• Constraints

Foley/VanDam, 1990/1994
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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.

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Lets Start SimplePolyhedral 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

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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
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Eulers Formula

• For simple polyhedra (no holes)Vertices -
  Edges Faces 2

Foley/VanDam, 1990/1994
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Eulers 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
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Euler Operators
Loop L ? H, Shell S ? C
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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

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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,

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BRep Data Structures

• Winged-Edge Data Structure (Weiler)
• Vertex
• n edges
• Edge
• 2 vertices
• 2 faces
• Face
• m edges

Pics/Math courtesy of Dave Mount _at_ UMD-CP
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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

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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

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Other IssuesNon-Manifold Solids

• There are cases where you may need to model
  entities that are not entirely 3D

Pics/Math courtesy of Dave Mount _at_ UMD-CP
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Other Issues in Boundary Representation Solid
Modeling

• Whats the surface?

Foley/VanDam, 1990/1994
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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