1

Advanced Computer Graphics

Lecture 3: Modeling (2): Solids

Bernhard JungTU-BAF, Summer 2007

Overview

SolidsMotivationB-RepsSweepsVoxels & OcttreesConstructive Solid Geometry

Advanced MethodsBlobby ObjectsFractals…

Further readingA. Watt. 3D Computer Graphics. 3rd edition. Addison-Wesley. 2000. Chapter 2.Foley, van Dam, Feiner, Hughes, Phillips. Introduction to Computer Graphics. Addison-Wesley.

1997. Chapter 10.

2

Solids

Idea: Represent solid interiors of objects

surface may not be described explicitlyTypes

Boundary Representation (B-Rep)PrimitivesSweepsVoxelOctreeCSG…

Visible Human

SolidsMotivation

Some acquisition methods generate solidsexample: CAT (or CT) scan

"Computed Axial Tomography" is the process of using computers to

generate a three-dimensional image from flat (i.e, two-dimensional) x-ray

pictures, one slice at a time

3

SolidsMotivation

Some applications require solidsexample: CAD / CAM

Solids – Desirable Properties of Representations [Requicha, 1980]

domainit should be possible to represent a large number of shapes

unambiguousa representation should correspond to only one solid

uniquecan encode any given solid in only one way

accurateallows an object to be represented without approximation

impossibility of creating invalid representationseasy to create valid representationsclosure under rotation, translation etc

performing these operations on valid solids yields valid solidscompact

save space, therefore efficient transmission over a networksupport of efficient algorithms

efficient rendering, calculation of physical properties

4

Solids

Boundary Representation (B-Rep)

Solid represented by its surface boundariesnot every set of surfaces represents a solid !

Solids

Boundary Representation (B-Rep)Euler-Poincaré Formula (for objects with no holes)

v – e + f = 2 where v is number of vertices, e is number if edges, and f is number of faces

cube with 6 faces, 8 vertices, and 12 edges

cube with 7 faces, 8 vertices, and 13 edges

5

Solids

Boundary Representation (B-Rep)Generalized Euler-Poincaré Formula (for objects with holes)

v – e + f = 2 ( 1 - g )where v is number of vertices, e is number if edges, and f is number of faces, g is gender (≡ number of holes)

cube with 14 faces, 24 vertices, 40

edges, and gender 2

SolidsPrimitives (Primitive Instancing)

3D modeling systems typically provide a set of primitive shapes

e.g. boxes, spheres, …sometimes also more complex objects

Primitives are parametric, to instantiate user supplies dimensional parametersAdvantages

easy to modelcompact description 3D viewer can decide rendering detail

VRML: Box, Cone, Cylinder, Sphere

6

SolidsSweeps

2D cross sectiontranslational sweeprotational sweep

common in CAD systems

VRML: Extrusion2D cross section

translationalsweep

rotational sweep

SolidsSweeps

may sweep along curves

… even helical ones

7

Solids

Sweeps – VRML Extrusionsdefine 2D cross sectionextrude along spine

defined by sequence of 3D pointslinear interpolation in-between

May define a scale factor for spine definition points

Extrusion {crossSection [1 1, 1 -1, -1 -1, -1 1, 1 1]spine [0 0 0, 0 2 0, 0 4 0]scale [1 1, 0.5 0.5, 1 1]}Extrusion {crossSection [1 1, 1 -1, -1 -1, -1 1, 1 1]spine [0 0 0, 0 2 0, 1 3 0, 2 3 0]scale [1 1, 1 0.5, 0.5 1, 0.5 0.5]}

SolidsVoxels

Partition space into uniform gridGrid cells are called a voxels (like pixels)

Store properties of solid object with each voxelOccupancyColorDensityTemperatureetc.

8

SolidsVoxel Acquisition

Scanning devicesMRI (magnetic resonance imaging)

does not involve X-raysCAT / CT (Computed Axial Tomography)

SimulationFEM (Finite Element Method)

SolidsVoxel Storage

O(n3) storage for n x n x n grid1 billion voxels for 1000 x 1000 x 1000

9

SolidsVoxel Boolean Operations

union, intersection, difference of two objectscompare objects voxel by voxel

trivial

SolidsVoxels

AdvantagesSimple, intuitive, unambiguousSame complexity for all objectsNatural acquisition for some applicationsTrivial Boolean operations

DisadvantagesApproximateNot affine invariant

affine transformations: translation, rotation, scaling, …not affine: perspective transformations

Large storage requirementsExpensive display

10

SolidsQuadtrees and Octrees

Refine resolution of voxels hierarchically

More concise and efficient for non-uniform objects

Octree

SolidsQuadtree Boolean Operations

11

SolidsConstructive Solid Geometry

Represent solid object as hierarchy of booleanoperations

unionintersectiondifference

CSG Treeleaves: primitiveinterior nodes: booleanoperation or affine transformation

SolidsConstructive Solid Geometry

DrawbacksExhibits no explicit geometric information. Requires expensive tree evaluations Imposes limitations on operations available to create and modify a solid

12

Advanced Modeling Utilities

Many high level modeling tools provide:SkinningSkin of Virtual ActorsBlobby SurfacesBeveling, Rounding, Fillets

A procedural modeling technique:Fractals

Advanced ModelingSkinning

• user defines segments of anarticulated body

• animation system stitchesthe segments together to create a flexible skin

13

Advanced Modeling

Automated Generation of a Skin

• user defines body skeleton

• animation system wraps a flexible skin around the skeleton

skeleton automatic skinless detail

manual skinmore detail

Advanced Modeling

Blobby Surfaces (aka Metaballs)

• User defines a set of metaballs

• Each metaball (blue) has its ownsphere of influence (light blue)

• Where these spheres of influencecross determines how the objectsfuse

metaballs blobbed togetherunblobbed metaballs

14

Advanced Modeling

Blobby Surfaces (aka Metaballs)

from http://www.robinwood.com/Catalog/Technical/BryceTuts/BryceTutSet.html

Advanced Modeling

Blobby Surfaces (aka Metaballs)

Modeling: Vikki Dawsonusing Bryce 5

from: http://www.3dtoday.com/3dtutorials/2003/May/B5Character/

15

Advanced ModelingBlobby Surfaces (aka Metaballs)

defined via implicit functions: F(x,y,z) = 0e.g. unit sphere: F(x,y,z) = x2 + y2 + z2 – 1 = 0

restricted to simple base shapese.g. spheres, ellipsoids, (quadrics)

add, subtract, other forms of blending

easy collision detection

relatively hard to render (need to ray-trace or convert to polygon, both slow)

iso-line of equal temperaturearound two heat sources

Advanced Modeling

Blobby Surfaces (aka Metaballs)

http://www.corelmag.com/1103/overview/Metaball_Modeling.html

16

Advanced ModelingBeveling, Rounding, Fillets

Customize edges between adjacent surfaces

filletsbeveling & rounding

Advanced ModelingFractals

Useful for describing natural 3D phenomenonTerrainPlantsCloudsWaterFeathersFuretc.

A kind of “procedural modeling”

17

Advanced ModelingFractalsFractal Generation

Deterministically self-similar fractalsParts are scaled copies of original

Statistically self-similar fractalsParts have same statistical properties as original

Advanced Modeling – Fractals

Deterministic Fractal Generation

General procedure:Initiator: start with a shapeGenerator: replace subparts with scaled copy of original

Apply generator repeatedly

18

Advanced Modeling – Fractals

Deterministic Fractal Generation

Useful for creating interesting shapes

Advanced Modeling – Fractals

Deterministic Fractal Generation

Useful for creating interesting shapes

19

Advanced Modeling – FractalsStatistical Fractal Generation

General procedure:Initiator: start with a shapeGenerator: replace subparts with a self-similar random pattern

Random Midpoint Displacement

Advanced Modeling – FractalsStatistical Fractal Generation

Example: Terrain

20

Advanced Modeling – FractalsStatistical Fractal Generation

Useful for creating 3D plants

Advanced Modeling – FractalsStatistical Fractal Generation

Useful for creating 3D plants

21

Comparison of Solid RepresentationsAccuracy

Polygonal b-reps and voxels / octtrees: can only approximate shapes (often good enough)better: CSG, b-rep with curved surfaces, primitives

Domainlimited for primitive instancing, sweepsvoxels / octtrees and b-reps can represent a wide class of objects

Uniquenessonly voxels / octrees guarantee uniqueness (an object can only be represented in one way)

Validity b-reps are the most difficult to validate

Closure under boolean operationsprimitives cannot be combined at all; sweeps not closed under boolean operations

Efficient renderingeasy for (polygonal) b-repsharder for voxels / octrees

Compactnesshigh memory requirements for voxels / octtrees; better: CSG