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

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CS 430/585Computer 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 Universityhttp://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

• Example:A 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?


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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!

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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)


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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

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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

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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 @ 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


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Boolean Operations


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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


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The Induced CSG Tree


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The Induced CSG Tree

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


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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


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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


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Sweep Representations

• Issue: how to define intersection?


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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

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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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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

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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


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Euler’s Formula

• For simple polyhedra (no holes):#Vertices - #Edges + #Faces = 2


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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.)


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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


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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 Issues:Non-Manifold Solids

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


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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

Documents

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