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Lecture1Introductionandclassificationofgeometricmodeling forms1.1 MotivationGeometricmodelingdealswiththemathematicalrepresentationofcurves,surfaces,andsolidsnecessary inthedefinitionofcomplex physicalorengineering objects. Theassociated fieldofcomputational geometryisconcernedwiththedevelopment,analysis,andcomputerimplementationofalgorithmsencounteredingeometricmodeling. Theobjectsweareconcernedwithinengineeringrangefromthesimplemechanicalparts(machineelements)tocomplexsculpturedobjects suchasships,automobiles, airplanes, turbineandpropellerblades, etc. Similarly, forthe description of the physical environment we need to represent objects such as the oceanbottomaswellasthree-dimensionalscalarorvectorphysicalproperties,suchassalinity,temperature,velocities,chemicalconcentrations(possiblyasafunctionoftimeaswell).

Sculptured objects play a key role in engineering because the shape of such objects (e.g.foraircraft,shipsandunderwatervehicles) isdesigned inordertoreducedragor increasethethrust(eg. forpropellerblades). At thesametime these objects need tosatisfy otherdesignconstraintstopermitthemtofulfillcertaindesignrequirements(e.g. carryacertainpayload,be stable in perturbations, etc). Similarly, there are objects which have significant aestheticrequirements,eg. cars,yachts,consumerproducts.

Typically,engineersdealwiththedefinitionofcomplexshapessuchasengines,automobiles,aircraft, ships, submarines, underwater robots, offshore platforms, etc. The shape of theseobjects is usually not fully known in advance (except when a baseline design is available).Consequently,theusualdesignprocedureis iterative,involving: Shapecreationbasedoncertaindesignrequirements; Analysistoevaluatetheperformanceoftheobject;and, Shapemodification to improvetheshape,followedbyanalysis(andsoon inan iterative

loop) until a satisfactory (and in simple cases, an optimal) design is reached, whichsatisfiesallthedesignrequirementsandminimizesacertaincost function.

Geometric modeling attempts to provide a complete,flexible,andunambiguous representationoftheobject,sothattheshapeoftheobjectcanbe:

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Easilyvisualized(rendered) Easilymodified(manipulated) Increasedincomplexity Convertedtoamodelthatcanbeanalyzedcomputationally ManufacturedandtestedComputergraphicsisanimportanttoolinthisprocessasvisualizationandvisualinspection

oftheobjectarefundamentalpartsofthedesign iteration. Computergraphicsandgeometricmodelinghaveevolvedintocloselylinkedfieldswithinthelast30years,especiallyaftertheintroductionofhigh-resolutiongraphicsworkstations,whicharenowpervasiveintheengineeringenvironment.

The remainder of this lecture introduces many of the different approaches to geometricmodelingrepresentationsthathaveevolvedoverthelastfourdecades.1.2 Geometricmodeling formsSeveral different geometric modeling forms have evolved over the last forty years. For thedefinitionofmodel, wecansaythatanabstractentityM isa modelofanobjectO ifM canbeusedtoanswerspecificquestionsaboutO.

Different formsofgeometricmodelingcanbedistinguishedbasedonexactlywhat isbeingrepresented,theamountandtypeofinformationdirectlyavailable withoutderivation,andwhatother informationcanandcannotbederived.1.2.1 Wireframemodeling

Figure1.1: Wireframemodelofacube.Wireframemodeling,developedintheearly1960s,isoneoftheearliestgeometricmodeling

techniques. It represents objects by edge curves and vertices on the surface of the object,includingthegeometricequationsoftheseentities(andalsopossiblybutnotalwaysadjacencyinformation),asshowninFigure1.1. Thetraditionaldrawingsofashipslines(Figure1.2[4])isaformofawireframemodelofashiphull. Itiscreatedbyintersectingthehullsurfacewiththree sets of orthogonal planes. Usually the hull surface is taken as the molded hull surfacewhichistheinnersideofthehullplating. Intersectionsofthehullsurfacewithverticalplanes(from bow to stern) are called buttock lines. Intersections of the hull surface with horizontal

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planes (parallel to keel) are thewaterlines, while intersections withtransverse vertical planesarecalledsections. Wireframesarerather incompleteandpossiblyambiguousrepresentationsthatweresupersededbysurfacemodels.1.2.2 SurfacemodelingSurfacemodelingtechniques,developed inthe late1960s, goonestepfurtherthanwireframerepresentations by also providing mathematical descriptions of the shape of the surfaces ofobjects,asshowninFigure1.3.

Surfacemodelingtechniquesallowgraphicdisplayandnumericalcontrolmachiningofcare-fullyconstructedmodels,butusuallyofferfewintegritycheckingfeatures(e.g. closedvolumes).Thesurfacesarenotnecessarilyproperlyconnectedandthereisnoexplicitconnectivity informationstored. Thesetechniquesarestillusedinareaswhereonlythevisualdisplayisrequired,e.g. flightsimulators.1.2.3 SolidmodelingSolidmodeling,firstintroducedintheearly1970s,explicitlyorimplicitlycontainsinformationabout

the

closure

and

connectivity

of

the

volumes

of

solid

shapes.

Solid

modeling

offers

a

numberofadvantages over previouswireframeandsurfacemodelingtechniques. Inprinciple,itguaranteesclosedandboundedobjectsandprovidesafairlycompletedescriptionofanobjectmodelledasarigidsolid in3Dspace [7,6,8].

Figure1.4illustratesthatforaboundarybasedsolidmodelofasinglehomogeneousobject,everysurfaceboundaryisalwaysdirectlyadjacenttooneothersurfaceboundary,guaranteeingaclosedvolume. Solidmodels,unlikesurfacemodels,enableamodelingsystemtodistinguishthe outside of a volume from the inside. This capability, in turn, allows integralpropertyanalysis forthedeterminationofvolume,centerofvolumeorgravity, momentsof inertia,etc.

Anexample isBaumgartswingededgedatastructure [1,2],whereeveryedgehasastartand end point, a face on either side, and at least two edges from each vertex bounding thefaces. Thisinformationcanbeputintabularform(perhapsusingarelationaldatabase)orinagraph likedatastructureandusedtoensureadjacency.

Typicalsolidmodelingsystemsalsooffertoolsforthecreationandmanipulationofcompletesolidshapes,whilemaintainingthe integrityoftherepresentations.

Solidmodelingtechniquesexcludethetwopreviousmodelingforms(wireframeandsurfacemodeling). The reason isthat the solidmodeling forms are traditionally constrainedto workonlywithtwo-manifoldsolids.

In a two-manifold solid representation, every point on the surface has a neighborhood onthe surface which is topologically equivalent to a two-dimensional disk. In other words, eventhoughthesurfaceexistsinthreedimensionalspace, itistopologicallyflatwhenthesurfaceisexaminedclosely inasmallenoughareaaroundanygivenpoint,as illustratedbythecubeinFigure1.5.1.2.4 Non-two-manifoldmodelingNon-two-manifold modeling [1, 9, 5, 10] is a new modeling form which removes constraintsassociated with two-manifold solid modeling forms by embodying all of the capabilities of

Figure1.2: Wireframemodelofashiphull.





















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Figure1.3: Surfacemodelofacube.

Figure1.4: Solidmodelofacube.the previous three modeling forms in a unified representation. The following diagrams (inFigure1.6)demonstratenon-two-manifoldsituations.

In an environment which allows non-two manifold situations, the surface area around agiven point on a surface might not beflat in the sense that the neighborhood of the pointneed not be equivalent to a simple two-dimensional disk. This allows topological conditionssuchasaconetouching another surfaceatasingle point,morethantwo facesmeeting alonga common edge, and wire edges emanating from a point on a surface. A non-two-manifold representationthereforeallowsageneralwireframemeshwithsurfacesandenclosedvolumesembedded in space. Overall, non-two-manifold representations have superior flexibility, canrepresent a larger variety of objects, and can support a wider variety of applications thantwo-manifoldrepresentations,butatacostofa largersizeandmorecomplexdatastructure.

Applicationsofthenon-two-manifoldrepresentation include: Distinguishbetweentwodifferentsolids,suchasabeamweldedtoaplate(Figure1.7). Representasolidvolumewithacutoutandthevolumethatwascutout(Figure1.8). Distinguishbetweenthecomponentsofacompositeplate(Figure1.9). Representafiniteelementmeshembeddedinasolidobject(Figure1.10). Representdifferentdimensionssimultaneously,suchasavolumewithacutplaneandan

axisofrevolution(Figure1.11).


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P2

P1

Figure1.5: Thecubeisatwomanifoldobject.1.3 BasicclassificationofsolidmodelingmethodsCurrentcomputer-aideddesignandmanufacturing(CAD/CAM)systemsusedforsolidobjectrepresentationaregenerallybasedonthreedifferenttypesofmodelingmethods:

1. Decompositionmodelsthatrepresentsolids intermsofasubdivisionofspace. - p.72. ConstructivemodelsthatrepresentsolidsbyBoolean(set)operationsonprimitivesolids

suchasrectangularboxes,cylinders,spheres,cones,torii(appropriatelysized,positionedandoriented). - p.14

3. Boundarymodelsthatrepresentsolidsintermsoftheirboundingfaces,whicharethem-selvesboundedbyedgesandtheedgesbyvertices. - p.16

Amoredetaileddescriptionofthesemodelsfollows.

1.3.1

Decompositionmodels

ExhaustiveenumerationExhaustiveenumerationisarepresentationbymeansofcubesofuniformsize,orientation,andwhich are nonoverlapping, see Figure 1.12. An object is represented by a three dimensionalBoolean array. Each cell represents a cubic volume of space. If a cell intersects with theregion of interest it has a true value. Otherwise, the value is false. This can be pictured asa box divided into 3D cubical pixels, with 0 assigned if empty and 1 assigned if full. Thisrepresentation involves: Regularsubdivisionofspace.

Itstores

just

one

corner

of

each

cube.

For fixed space of interest we needjust a 3-D array, Cijk of binary data, and overall

box/spacecoordinates:1 ifthecubei,j,kintersectsthesolid

Cijk = 0 ifthecubei,j,kisempty (1.1)


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common edge

face

Figure1.6: Examplesofnon-twomanifoldmodels.


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beam

plate

Figure1.7: Beamweldedtoaplate.cutout

block

Figure1.8: Blockwithcutout.Applicationsofexhaustiveenumerationmethodsinclude: Underwaterenvironmentrepresentation. Finiteelementsmeshing(firststep inanalgorithmtobuildsuchamesh). Medical3Ddatarepresentation. Preprocessing representation for speeding up operations on other representations (eg.

approximating integral propertiessuchasvolume,centerofgravity, momentsof inertia,distancetransforms).

Propertiesofexhaustiveenumerationmethodsinclude:A

composite plate

B

C

Figure1.9: Compositeplate.


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Figure1.10: Finiteelementmesh.

center line

center line

cutting plane

solid volume

Figure1.11: Representation ofdimensions.

Figure1.12: Exhaustiveenumeration.


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1 2

3

4

1

2 3

4

full

empty

partially full

Figure1.13: Quadtreerepresentation. Expressivepower: approximationscheme. Unambiguous and unique for fixed space and resolution. There do not exist different

representations forthesameobject. Memory intensive: eg. 2563 16M bitsandthis isabareminimum. Closure1 ofoperations(eg.Booleans). Computationaleaseforalgorithms: VLSIimplementationforvolumerendering. However,

forhighresolutionthealgorithmslowsdown.BoundarycellenumerationThis

is

aboundary

based

version

of

the

above

technique.

Only

the

cells

that

intersect

region

boundarieshavetruevalues.SpacesubdivisionSomeofthemotivationsbehindspacesubdivisionmethods include: Smallermemoryrequirements ifadaptivesubdivisionisused; Octree/quadtreerepresentationsleadtoarecursivesubdivisioninto8octants(or4quad-

rants)thatcanberepresentedasan8-arytree(or4-arytree).In an octree representation a solid region is represented by hierarchically decomposing a

usuallycubicvolumeofspaceintosuccessivelysmallercubes(8ofthem). Hierarchicaldivisionand cube orientation usually follows the spatial coordinate system. An example of quadtree,thetwodimensionalanalogue, isshownFigure1.13.

1ClosuremeansthatanoperationsuchasBooleanresultsinanobjectofthesametopologicaltypethatcanberepresentedbythesametypeofdatastructure.


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Unambiguousandunique: forafixedresolutionthere isonlyonecompacted2 octree; Memory: not as large as Exhaustive Enumerations, yet much larger than Boundary

Representation andConstructiveSolidGeometrymodels; Closureofoperations: forexampleBooleanoperationsandtransformations;

Computationalease:

somewhat

more

complex

than

exhaustive

enumeration.

CelldecompositionsThemotivation forcelldecompositionmethods is: Useofelementsotherthancubes,seeFigure1.15 foranexample. Application: finiteelementmethod,scientificvisualization. Cells are parametrized instances of a generic cell type, eg. a cell boundedby quadratic

curvesandsurfaces. Cellsarehomeomorphictospheres. Cellsmeetatavertex,edge,faceotherwisetherepresentation is invalid. Cellsaredisjointandnon-overlapping. Cellsmaybelongtodifferentcelltypes,eg. box-like,tetrahedra-like,etc.

Figure1.15: Acelldecomposition(finiteelementmesh).Acelldecompositioncanberepresentedusingthecell-tupledata structure [3]. SeeFigure

1.16 fora2Dexample.Thepropertiesofcelldecompositionmethodsare: Expressivepower: verygeneralandaccurate;

2Algorithmssuchas set operations cancreate octrees with unnecessary nodes(eg. an internalnodes whosechildrenareallblack). Suchnodescanberemovedwitharelativelysimpletreetraversalalgorithm.


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Figure1.16: Celldatastructure. Validity: requiresan intersectiontestforverification; Unambiguousrepresentation; Nonunique: Similar to the Constructive Solid Geometry method we will see below, the

same object can be represented at different resolutions or with different types of mesh(eg. hexahedral,tetrahedral,etc.);

Generation: byconversion fromotherrepresentations; Concise: memoryutilizationislessthanoctrees,yetmorethanBoundaryRepresentation; Applicability: finiteelementmeshing,multimaterialnon-homogeneousobjects,visualiza

tionoffields,etc.1.3.2 Constructivesolidgeometry(CSG)Constructive Solid Geometry (CSG) is the Boolean combination of primitive volumes thatincludethesurfaceandtheinterior. Forexample,primitivesincludingrectangularbox,sphere,cylinder,coneandtoruscanbecombinedusingintersection,unionanddifferenceoperatorstoformcomplexsolids. Positioningoperators(position,orientation)andsizeoperatorsareappliedtotheprimitivesbeforetheBooleanoperatorsare invoked,seeFigure1.17 foranexample.Terminalnodesonthebinarytreeareprimitivevolumes;othernodesareBooleanoperators.Thisrepresentationhasadirectmanufacturinganalogue,wheredifference indicatesdrillingormachiningandunionindicates forexamplewelding.

Another example of a related representation issweeps, where more general primitives areobtainedbysweepingasolidalongaspacecurveorsweepingaplanarcurvethrougharevolution


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(U)

(-)

P(sphere)

P(box) P(cylinder)

Figure1.17: Booleanoperationsandprimitives.


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aboutanaxisinitsplane. Sweepsareusefulintherepresentationofblends,volumessweptbymachinetools,and inrobotics.

Inasurveyofmachineelements,90to95%ofpartscouldberepresentedaccuratelyusingtheCSGmethodwiththeabovesimpleprimitivesolids.1.3.3 Boundaryrepresentation (B-Rep)Objectsarerepresentedintermsoftheirboundaryelements(e.g. vertices,edges,faces)whicharerelatedthroughadjacency. Thisisthemostgenerallyusedrepresentationtodayduetoitsflexibility. Inthese notes, we develop the theoryof curvesandsurfaceswhich formtheedgesandfacesofB-Repmodels.

Figures1.18and1.19showanexampleofatetrahedronanditsB-Repmodel.

V1f2

V4

V3

e1e2

e3

e5

e6

e4

f4 f3

f2

V2

V4

V1

V2

e1 e3

e2 f3

f4

e4

e5e6

V3

Top View

f1

Figure1.18: Atetrahedron

Twoboundaryelements,whichareboundedbynextlowerdimensionboundaryelements,arecalledadjacent,iftheyshareonecommonnextlowerdimensionboundaryelement. Forinstance,twosurfaceshavingacommonedgeareadjacent,ortwoedgeshavingacommonvertexareadjacent.


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Object definition (region)

Boundary definition

Faces:f1

f2 f3 f4

Loops:

Edges:

Vertices:

Topological Information

L1 L2 L3 L4

e1 e2e3 e4

e5 e6

V1 V2 V3 V4

Vertex assignment(x1,y1,z1) (x2,y2,z2) (x3,y3,z3) (x4,y4,z4)

(Geometry)

Figure1.19: Boundaryrepresentationmodelforatetrahedron


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1.4 Alternateclassificationofgeometricmodeling forms1.4.1 IntroductionThewidevarietyofrepresentationtechniquesdeveloped(manyofwhichwereidentifiedabove)can bedifferentiated on the basis of at least three independent criteria concerning the representation: boundarybasedorvolumebased objectbasedorspatiallybased evaluatedorunevaluatedinformArepresentation isboundarybased ifthesolidvolume isspecifiedby itssurfaceboundary.

Ifthesolid isspecifieddirectlybyitsvolume it isvolumebased.A representation isobject based if it is fundamentally organized according to the charac

teristics of the actual geometric shape itself. It is spatially based when the representation isorganizedaroundthecharacteristicsofthespatialcoordinatesystem ituses.

Evaluated or unevaluated characterization is roughly a measure of the amount of worknecessarytoobtain informationabouttheobjectsbeingrepresentedwithrespecttoaspecificgoal.

What is best depends on the application! A good system should support multiple representational techniquestoensuretheirefficiencyoverabroadrangeofapplications.

We have threedifferentcriteria withtwo choices, soeight categories result. The followingTable1.1givesexamplesineachcategory:

UnevaluatedClassboundarybased volumebased

spatialbased HalfSpace Octreeobjectbased EulerOperators CSG

EvaluatedClassboundarybased volumebased

spatialbased BoundaryCellEnumeration CellEnumerationobjectbased BoundaryRepresentation Non-parametricPrimitives

Table1.1: Classificationofgeometricmodelingforms

1.4.2

Unevaluatedrepresentation

systems

Unevaluatedrepresentation systemsrequiresomeformofproceduralinterpretationtobeusedwithrespecttothespecifiedapplication.


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Spatial, boundary based: half space technique A solid is represented by successivelydividingspace inhalfwithusually infinitesurfacedescriptionsandselectingthehalfspaceonaspecifiedsideofthesurface,eventuallyenclosingthesolidregion. Theintersectionofthehalfspaces representsthe solid. Onlyconvex regionscanthus bedescribedunlessunionsarealsoemployed. Figure1.20demonstratesthehalfspacetechnique inatwodimensionalformat.

Figure1.20: Halfspacetechniqueofmodelrepresentation.Thistechnique isclassifiedasspatialbasedbecausethesurfacedescriptionsarepositioned

inspatialcoordinatespaceratherthanbeingrelativetotheobject.Spatial,volumebased: octreerepresentation Asolidregionisrepresentedbyhierarchi-callydecomposingausuallycubicvolumeofspaceintosuccessivelysmallercubes(8ofthem).Hierarchicaldivisionandcubeorientationusuallyfollowsthespatialcoordinatesystem.Object, boundary based: Euler operators The object is procedurally described as asequenceofEulerOperators,as inFigure1.21. An(amorphous)topological sphere istopo-logicallymodifiedusingtheEulerOperatorssuchas:

msv=makeshell,vertex mev=makeedge,vertex mef=makeedge, face

v1

v2

v3

v4 f1

e1

e2

e3

e4

f2

Figure1.21: BoundarybasedrepresentationUsingEuleroperations.


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TheseoperatorsensurethatEulersFormula isalwayssatisfied:VE+FLi =2(SG)

where: V =numberofvertices E =numberofedges F =numberoffaces Li =numberof internal loops S =numberofsurfaces(shells) G=genus(numberofhandlesorthroughholes)

Object,volumebased: constructivesolidgeometry(CSG) ConstructiveSolidGeometry(CSG) is theBoolean combination ofprimitive volumesthat includethesurfaceandtheinterior. Forexample,abox,sphere,cylinder,torusandconecanbecombinedusingintersection,unionanddifferenceoperatorsto formmanysurfaces. Inaddition,positioningoperatorssuchasposition,orientationandsizeareappliedtotheprimitivesbeforetheBooleanoperatorsareapplied.

Anotherexampleofanobject,volumebasedrepresentation issweeps,wheremoregeneralprimitives are obtained by sweeping a solid along a space curve or sweeping a planar curvethrougharevolution.1.4.3 EvaluatedrepresentationsystemsEvaluatedrepresentationsystemsusuallyrequiresubstantiallylesscomputationtobeusefulinspecific

applications.

Spatial,volumebased: cellenumeration AnobjectisrepresentedbyathreedimensionalBooleanarray. Eachcellrepresentsacubicvolumeofspace. Ifacellintersectswiththeregionof interest it has a true value. Otherwise, the value is false. This can be pictured as a boxdividedintopixels,with0assigned ifemptyand1assigned iffull.Spatial,boundarybased: boundarycellenumeration Thisisaboundarybasedversionoftheabovetechnique. Onlythecellswhich intersectregionboundarieshavetruevalues.Object,BoundaryBased: BoundaryRepresentation(b-rep) Objectsarerepresentedin

terms

of

their

boundary

elements

(e.g.

vertices,

edges,

faces)

which

are

related

through

incidence and adjacency. This is the most generally used representation today, and will bediscussed indetail infurtherlectures.Object,volumebased: non-parametricprimitives Simplefixedpositionobjects. Thisisnotaparticularlyflexiblerepresentation.


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