Geometrically Complete Building Models

1

Geometrically Complete Building Models

Bernold Kraft, Wolfgang Huhnt Technische Universität Berlin

[email protected]

Abstract. The processes of building information modeling (BIM) focus on a digital model a representation of geometry and semantics. Any further use depends on the correctness and completeness of the elements of that model. Model validation attempts to check both aspects. As shown by Kraft and Huhnt (2013), semantics of building components depend on geometrical neighborhood. Geometrically complete models are needed as a basis for model validation. This paper focuses on completeness in geometry. A given geometry of a building forms the starting point. Algorithms check whether this geometry results in a correct decomposition of space. This includes the consideration of the exterior of a building as objects. A method based on a dual problem description is proposed. The Poincaré-Duality is utilized in order to derive a discrete topological space which can be handled by graph theory. A short description of a prototype software model setup will be provided. Finally, the state of the work is summarized.

1. Introduction

The modeling process of a digital building model consists of different activities. In general, a modeler transfers a planned system which he / she has in mind into a digital representation with certain simplifications under certain assumptions and intended use cases. We call such a digital representation a model. The process of Building Information Modeling in todays practice is based on two different approaches: modeling based on strict formal methods which are implemented as part of the used tools, and modeling based on human logic where used tools document this logic.

The effectiveness of a building information model (BIM) as a digital model is based on avoiding redundancy of information; and required dependent information must be generated from existing information. Semantics and geometry are bound as a state into a BI model. To be able to derive information, a BI model has to be seen as a computational model with algebraic properties – not as a database only. To be able to validate a BI model one has to be able to derive information. Every constraint validation is basically a verification of a query.

The paradigm of avoiding redundancy leads to a (new) future problem. If there’s information missing (i.e. an element of model) or it is not properly defined, derived information can either be wrong or even missing. Somehow missing information is a well-known bottleneck of BI modeling projects. Completeness in geometry is one major prerequisite for such a computational BI model (Kraft and Huhnt, 2013). In general, completeness is necessary in order to apply logic inference on any model. In this paper, we put a focus on geometry. We call a building model a geometrically complete building model where all locations in the three dimensional space are either occupied by building objects such as building components or rooms or by the exterior of a building. At present time, the question of completeness is answered by inspection only. The user is responsible for checking that there are no non-modeled spaces (voids) inside the model and that constructive components are connected.

The aim of this paper is to present a method based on existing work, which can help to validate the geometry of a BI model. In a second step, a validation of sematic can be applied. The second step relies on predicate logic. Only if all spatial relations are known, a validation can have a correct result. Therefore, completeness of spatial relations is required prior to any

2

reasoning process. A spatial graph is complete if the primal geometrical regions and their boundaries are known in completeness. Actually, the result of the first step is an input to the second step. In this paper, we concentrate on the first step.

2. Theoretical background

In order to verify a geometry model we need to define the mathematical background. In general, a representation of a solid geometrical object can be done by a polyhedron bounded by planar faces (b-rep). The mathematical model taken here is based on simplicial complexes. Starting from a set of -dimensional points , , … , we define an affine combination as ∑ . An affine hull is the set of all affine combinations. A convex combination implies that all are non-negative. The convex hull is the set of all convex combinations. Any two points of the set of points is affinely independent, if the supporting vectors

with1 are linearly independent.

A -simplex is the convex hull of 1 affinely independent points in -dimensional space with . For convenience the notation of is used to describe a -simplex. A simplicial complex is a finite set of simplices ∈ , such that the intersection of any two -simplices ∩ is empty or a common partial simplex with and , where is called a face of and . Actually, is a topological space with an embedding| | into the with . Since is decomposition any -simplex is a -cell of . A cell complex is a set of cells.

One major property of a topological space is the definition of neighbourhood . Furthermore, concerning a metrical space the neighborhood becomes an open -dimensional disk with , ∈ | , where … is the distance function according to the metric definition (i.e. euclidean) and some scalar value. If all points of a topological space have such an open disk as neighbourhood, is called -manifold. Each -manifold can be embedded onto a -sphere ∈ | 0, 1 . A compact cell complex forms up a compact 1 -manifold as boundary.

3. Problem description

Based on two examples we will give a short problem description. Model validation is part of the quality checking step within each project. This step has been performed exclusively by humans in pre-BIM projects with the help of check lists and workflow definitions. Nowadays digital models contain much data, especially in geometry models. Automation in model checking relies on a consistent logic based method to deal with spatial constraints.

Spatial logic and Qualitative Spatial Reasoning (QSR) provide a foundation to reason for a valid geometry of a BI model (Borrmann and Beetz, 2010). Qualitative reasoning answers primarily whether a spatial relation exists. It cannot directly help to verify whether a complete partial region of a component has a correct neighbourhood. Nevertheless, such quantification relies on a consistent qualitative spatial evaluation of the geometry model.

Figure (1) refers to a use case found in structural engineering. The structural component partition into vertical and horizontal objects may lead to an inconsistent geometry configuration. The classic clash detection doesn’t cover such a situation, since it is no clash. A qualitative spatial evaluation doesn’t answer the question whether the whole top of the column is covered by an adjacent structural component.

Figu

Considetask. Ninterconroom.

Both eprocessiinvestigbeyond validatioresult of

4. Dua

Duality betweenHenri Pan isommanifol

Each 3-Any twface is exterioron the b

Figu

Rememedges. Ttop faceThe dua

ure 1 – A str

ering figureNon-materiannection bet

examples aing is one

gation on vothe scope

on, it is cruf any compu

al problem

is a genern two (topo

Poincaré (18morphism beld when usin

-cell (polyhewo adjacent

penetrated r 3-cell, a dboundary of

ure 3 – A str

mbering the The invalide. Since it ial problem c

ructural usag

e (2), a diffal componetween struc

are not adde solution oids which ie of classiccial to be abutational ru

description

ral principlological) pr893). Althouetween a ng the math

edron) is repolyhedronby an undi

dual exteriorf the 3-cell c

ructural usag

two exampd configurats a cellular can be expr

e of a model

ferent questents i.e. s

ctural compo

dressed bywhich cau

is part of thc clash deble to comp

un is identifi

n

e in matheroblem spacugh the mat-simplex an

hematical m

elated to a dns have at lirected edger vertex is ccomplex.

e of a model

ples, figureion has twodecomposi

essed by fol

3

l Figure

tion arises space objeonents whic

y classic cuses a highe verificatioetection mepare any twoiable within

ematics. Baces. A specthematical dnd -

model of sim

dual 0-cell (least one une. By introdconnected t

l Fig

e (3) showo edges conition, the tollowing con

2 – Intended

within the ects are tach share a f

clash detecgh computaon process oethods. Froo verification a model st

asically, it icial kind ofdefinition issimplex of

mplicial com

dual vertexnoriented faduction of to any dual

gure 4 – Usa

s the dual nnecting theop face is senstraint:

d space versu

subsequentaken to pface of a com

tion methoational effoof geometrim the per

on results. Tate.

is an equivf duality has complex, ian oriented

mplexes.

) in a three ace as 2-celthe conceptvertex of a

age of a futur

vertices ane column weparated int

us missing sp

nt process mprovide topommon arch

ods. Geomort. In speical complerspective oThat implies

valence relaas been defit states thatd and unbou

dimensionall in commt of the unb

a polyhedron

re process mo

nd their unwith exteriorto two parti

pace

modeling pological hitectural

metrically ecial the teness is f model s that the

ationship fined by t there is unded -

al space. mon. This

bounded n placed

odeling

ndirected r via the ial faces.

4

“The configuration figure (1) is valid, if all related dual vertices through the top face of a structural column are dual vertices of a slab.”

To get complete, we need the definition of a void cell complex.

“The configuration figure (2) is valid, if the non-modelled space (void) is filled by an object.”

A void cell represents a region which is not part of any component of the initial geometry model. Furthermore, it is not a part of the exterior nor connected to it. A void 3-cell is represented by a compact closed 2-manifold at its boundary. Hence, a compact void cell complex has the same property. Concerning the dual description (figure 4), a void cell must become an explicit element of the duality model. To summarize the model, the overall complex of 3-cells including void cells and the exterior cell forms up a 3-manifold.

5. Related work

Three general modeling methods can be distinguished (Mäntylä 1988). The approach of Constructive Solid Geometry (CSG) utilizes point set theory and Boolean operations; the approach of Boundary Representation (b-rep) is related to Euler operation on simplicial complexes. Finally, the Cellular Decomposition (CD) relies on the type definition of cells.

Modeling CD can be achieved by construction of rooted spatial decomposition trees (i.e. octrees) which can be queried and transformed by graph theoretical methods (Borrmann 2007). Any geometric object is representable by a cell complex with approximated boundary. The shape of a cell is given implicitly by the overall shape (i.e. an equally sized box). A topology is induced by the definition of neighbourhood given by the octree cell structure. In contrast to the implicit cell definition strong typed explicit cell definitions (i.e. a tetrahedron) are bound to a special use case (i.e. Delaunay tetrahedralisation) (Pion and Teillaud, CGAL 4.4 2014).

Several techniques have been found for modeling simplicial complexes since the recent three decades (Asghari et al. 2013). In principle, the models differ in how a -simplex is expressed, either explicit or implicit by the derivation through existing sub simplices. The purpose of the model has an additional influence on the initial assumptions. An oriented manifold benefits from deriving redundant topological information (i.e. face and shells within the Half Edge). A non-manifold has to map a discontinuous topological disk into the data structure. In general, the most apparent use cases considering non-manifolds are given when mixing 2D and 3D geometry information. The Combinatorial Map or Generalized Map (G-MAP) technique represents an approach which utilizes graph theory for this purpose (Lienhardt 1994). Meanwhile edge based techniques only hold topological needed entities explicitly, the G-Map approach sets up a -dimensional polytope by all its 1 faces. The advantage in application in Geographic Information Systems (GIS) relies on the capability to combine faces of different polytopes with different dimensions to each other (Ohori et al. 2013). To use that capability, an a-priori knowledge about decomposition of the embedding -dimensional space is needed.

The dual problem description relies on a decomposition of 3-dimensional space into 3-cells. By generalizing a 3-manifold into a general cell description with the help of simplicial complexes, the result will not be bound to a special use case. The first attempt to connect adjacent cells of non-manifolds has been introduced by the Radial Edge data structure (Weiler 1988). In order to directly model a 3-manifold, Lopes and Tavares (1997) extend the Half Edge (HE) definition by Half Faces. The principle construction is achieved by Morse

operatorof spaceRadial EtechniqustructurtriangulQE intrdimensiintroducHE and

Both exAutoma(Eastmaresultingtouch iscomputacomponconstraiautomat(Dimya

The duamatter obeen fomodels

6. Glu

The duaof simpsimple vembedd(i.e. usinedge cocompris

BoguslaHalf Ed(5) uses

rs gluing He into convEdge and thue stores thre (QE) halation at theroduces addional space ced the conthe QE dua

xamples givatically chean et al. 20g in a set os no clash.ation. Enha

nent) in comints (Solibrtism on pe

adi and Amo

al problem of spatial loound there w

have shown

uing primal

al problem dlicial compvertex as it

ding intong the centronnecting bsed of two m

awski (2011dge (DHE) s the examp

alf Faces ofvex regionshe Half Fache duality aas been to e same time ditional ope(Boguslasw

ncept of theality approa

ven in sectiecking a m009). Existinof pairs of c. To identifanced methombination wri 2014). Reerformance or 2013).

descriptionogic and Quwill be a fn the princip

l and dual p

description lexes each ts dual repr has no mere of circumboth dual

models at th

Figure

1, et al. 201which relatle of two to

f adjacent cfrom the v

ce approachat the same

bind theby simple o

erators meawski and Goe Dual Halfach into a sin

ion (3) are model requir

ng clash decomponentsfy a clash, ods are usinwith the exiecent review

indicators;

n reveals anualitative Spfield of appple utilizati

problem de

states, that 3-cell has aresentation. eaning, as lomcircle). Ea

vertices ofhe same time

e 5 – Two tetface (a,b,c)

0) introducetes the primouching tetra

5

ells. Bilchuvery first sc

h, although time. The derived Vooperators (Gant to suppold 2007). Rf Edge (DHngle DHE s

typical useres that foretection onls which int the bound

ng semanticistence of aws on this ; meanwhil

an associatiopatial Reaso

pliance of Qon (Borrma

escription

if we use a as its dual a

The dual song the deteach face of f adjacent e at one poi

trahedra (a,b) and (a,c,b)

es a new hamal space of

ahedra whic

uk and Pahl cratch. Theit doesn’t moriginal intoronoi diagGuibas and port traversRecently, B

HE) which tstructure of

e cases for rmal rules ly searches tersect eachdary interse classificati

a touch or itopic indic

le principle

on to spatiaoning (QSRQSR. Detaiann and Bee

decomposi0-cell. Geo

space has termination a tetrahedrotetrahedra.

int in time g

,c,d) and (a,c) respectively

alfedge-basef to its ch share a c

(2012) intene model hasmention bottention of thgram and tStolfi 1985

sal over celoguslawskiakes advanan underlyi

BI model are performfor interse

h other. By ection has tion of compintersectioncate an incre methods

al relationsR). Once thaled reviews

etz, 2010).

tion model ometrically, opological of coordinaon is penetr. Actually, glued by mo

c,b,e) sharingy

ed geometrydual topolo

common fac

end a decoms similaritieth of them. he Quad Edthe dual D

5). The Augll complexi (2011, et a

ntages of theing 3D spac

checking mmed by algections of g

definition ato be classponents (i.en to build ucreasing res

have not

s which areat spatial grs with focu

of with a tetrahedrproperties,

ates has no mrated by an

such a model constra

g the

y model calogical spacece comprise

mposition es to the Another dge data

Delauney gmented-es in 3-al. 2010) e simple ce.

methods. gorithms geometry a simple sified by . type of

up model earch in evolved

e a main raph has

us on BI

the help ron has a

too. An meaning oriented

model is aints.

led Dual e. Figure ed of two

half facproved

Both spconstrai

Each duto the inedge (adjacenfeature compliaand in inver

The initvery firwhich acells (idecompcomplet

A key overlappevolvescomponhomeom

Figu

The bouduality an excland 2equally returns

es. Only thesimultaneou

paces, the gints in conte

ual halfedgenverse orien

) connent face. An a

is defined ant with a g

provrse orientati

tial intentiorst scratch. allows over.e. two equ

position conte, it has to

property oping situati into a set

nents. The morphic to a

ure 6 – A co

undary of and the knolusive Or-o) is omitteoriented edthe next inv

e topologicaus construct

geometrical ext of a sing

.

.

.

.

.

e has four nted edge (ecting the dadditional re

in contrastgeneral halfevide the abon, respecti

on of the DHModern c

rlapping celual-sized c

ntradicts thabe transform

f CD is thon leads tot of cells. F

base coma torus (gen

lumn penetra

the columnowledge of operation ofed. Concerndge on somverse orient

al necessarytion of both

primal spacgle DHE .

. .

. .. .

. .

. .

references:), to th

ual verticeseference cont to the oriedge based ility to travively.

HE model hconstructionlls (i.e. a cocolumns plaat proceduremed into a c

hat no two a subdivisi

Figure (6) mponent isnus is one).

ating a base

n componenthe exteriorf both partning the DHme adjacent ted edge wh

6

y 0- and 1-ch topologies

ce and the

. . .

... .

: A referenche next edgs of the 3-cennects the eiginal modegraph mod

verse the adj

has been ton software olumn withiaced at thee. In order cellular dec

cells can ion of the oprovides a

s penetrate

and its dual

nt 1, 2r. Basicallytial cells wHE structurcell having

hich is a bo

cells are defs with the he

dual topolo

.. ..

ce to the nexe leaving thell in focus edge with itel of Bogusdel. Two derdjacent edge

o construct ais based oin a wall) ae same plato verify a omposition

occupy theoriginal coma theoreticad complete

description a

can be rey, the set of where any ire theg the oppos

oundary edg

fined explicelp of Euler

ogical space

. . .

xt edge on she origin (and the adjs target vertslawski (20rived relatio

es either in

a 3-manifolon scenegraand even geace). The mgeometry m

n.

e same spamponent. Thal example ely such t

according to

econstructedboundary f

internal fac referenc

site face. Thge of a neigh

citly. The aur operators.

e are bound

same face () and

jacent cell gtex. This la

011) in ordons, namelysame orient

ld starting faph transfoeometrical method of model whet

ace. Therefohe componeof two cothat the r

Poincaré du

d with the faces is the ce (i.e. betwce returns the given prhbor face u

uthor has

d by five

), the dual

given by st model

der to be y tation or

from the ormation, identical cellular

ther it is

fore, any ent itself nflicting

result is

ality

help of result of ween 1 the next rocedure

under the

assumptcurrent

, ∶

It is obvperspecsplittingproducepoints. correct t

7. Prin

As statedecompworkflobeen syfilled w

Since thincremebecomeis the cAdditiowithin a

The abimodel. capturedfull extethis cauknown, the leas

tion of the boundary fa

. . ..

vious, that ttive, a 3-sim

g an existines cell decoUsing Conthem) has b

nciple meth

ed, the BI mposition appow is more ynchronizedwith geometr

he least comentally intoes a tetrahedcapability tonally, the v

a black box

lity to repreBasically, d and the inent of the muses a drawa hierarchi

st accepted

existence oface and the

∈

the most crumplex is th

ng cell whicomposition bnstrained Dbeen shown

hod and mo

modeling teproach. Coan advanta

d, every partry. Finally,

mmon modo a cellulardron havingo access a vertices are of the BIM

Figure 7 –

esent 2-manthere are twncident vert

model is unkwback in secal clusterindistance be

of a closed set of du

ucial point ihe simplestch containsbut does noelaunay tetapplicable

odel setup

echniques inonsidering tage than a nticipant haswhen valida

del are -simr decompos

four adjacegeometry massumed to

M platform.

A model set

nifolds as a wo phases. tices are coknown. The

earch perforng method between any

7

2-manifoldual vertices

is the searchcell. Each

s that pointot guaranteetrahedralisatby Verbree

n geometrythe distribuneed for chs to know, wating a mod

mplices, a gsition modeent 3-cells. model by io be not un

tup (UML si

pre-stage oWithin the

ollected intoe k-d-tree crmance. Onbuilds a reptwo points.

d representaof cell com

h of the set insertion of

t. An increme the existetion to vali

e and Si (200

y deal with uted projechange. On twhether herdel, a consis

given geomel (CDM) The startingits vertices

nique since

mplified clas

f 3-manifole first phaso a k-d-treecan be extennce all poinpresentative . Afterward

ation. Initiamplex have t

of intersectf a single nmental Delance of edgeidate 2-man08).

methods coct work in the other har conceptionstent geome

metry modelby triangul

g point of thand its tri

the triangul

ss diagram)

lds is an advse the existi (k-dimensi

nded with nnts of a sincluster vert

ds the prima

ally a DHEto be define

tions. Fromnew point raunay trianes between nifolds (or

ontrary to thBI model

and, if a mnal space is

etry is neces

l has to be ulation. Eache principleiangulated slation is oft

vantage of tting trianguional). Initi

new points angle compotex given soal shell is g

of the d.

m another esults in

ngulation specific even to

he space ling this odel has s already ssary.

inserted ch 3-cell e method surfaces. ten done

the DHE ulation is ially, the although nent are ome as enerated

by first referencedges fo

The duathe DHvalidatioSince thedge is has to ethe captriangulset is no2010). A(Sallee

Figure consistifigure (only onfinally t

8. Firs

As a firmaps a Modelinobject iassumedprinciplmay exvertex coordin

Once a which hcoordin

1 Autodes2 http://w

building fce). At this orming self-

al constructE model coon is run where is no ua problem.

exist. An ocability to slated axis-aot unique anA triangula1982).

(8) summang of six t

(8). The duance. Two tetwo have fo

st results

rst examplesubset of thng Framews transformd to be oriele, any vertist times has to be

nates with a

polyhedronhas at least nates. If ther

sk Revit®, htt

www.eclipse.or

face loops (time the du

f-connecting

tion is prodonstraints. T

which also cunique relati

To be able ctree structuspeed up seligned box nd can evention of a re

F

arizes the ptetrahedra. al graph haetrahedra h

our adjacent

e a Revit1 Mhe software

work2 (EMFmed into a p

ented counttex is identidue to thatunique. A

given in p

n has been a single initre’s a match

tp://www.autorg/modeling/e

( refeual represeng cycles.

duced reversThe dual facloses the eion of a singto build up ure has beeearch whendecompose

n cause degectangular b

Figure 8 – A

primal and For conven

as distinct ohave only tw

cells. The e

Model has bmodel com

F). To adapprimal DHEter-clockwiified by its t fact. In o

As describedprior.

adapted, ittial cube foh within the

odesk.com/revemf/

8

ference) follntation cons

sely. First tace loops aedge cycle gle cell to a a cell comp

en chosen wnever a newed into six generated cebox can eith

triangulated

dual represnience, onlyorientations wo adjacentexterior cell

been investmponents topt geometryE model. Atse. Each trcoordinate

order to prod earlier, i

ts unique vormed by sixe CDM, the

vit

lowed by gsists of only

he dual edgare closed i

around evea cell complplex increm

which proviw point hastetrahedra. ells (i.e. neaher be comp

d Six-Tet cub

sentation ofy one tetraon each edt cells, twol is connect

tigated. A pan Ecore –

y, the triangt the curreniangle is fo

es. Once eacoceed with identificatio

ertices are x tetrahedrae vertex wil

gluing the py a single d

ges are pairn a last steery vertex (lex, the con

mentally, an des a contas to be insTriangulati

ar-zero voluprised of fiv

be

f an initial hedron is f

dge. Identicahave three

ed to all six

plugin has b– Model utilgulated surfnt developmormed up bych triangle gluing the

on is done

inserted inta. Each vertll not be ins

primal facedual vertex a

red with theep. Finally ( refntainment oinitial cell

ainment feaserted. A cuion of regulume cells) (ve or six te

scalable ufilled (Id: 1al edges ar

e adjacent cx cells.

been writtelized by the

rface of a 3ment, any triby three ver

is known, primal fac

by cluste

to the CDMtex is querieserted, othe

es ( and dual

e help of a model ference). f a DHE complex

ature and ube is a lar point (Verbree etrahedra

nit cube 1063) in e shown cells and

en which e Eclipse 3D solid iangle is rtices. In a vertex

ces, each ring the

M model ed by its rwise an

incremeregulariexist hathese cotheir impoint nu(1997) irobustnebased oadjacenedges ar

Fig

9. Con

The proprocessechain, completnowadapresent The app. The prtreats vwork in

At the coriginalfaces anensure knowledconformtriangultime.

Furthermwill probetweenframew

ental flippinity of desigaving more omputations

mplementatiumbers. In introduced ess of his al

on this worknt columns. re removed

gure 9 – Expo

nclusion an

oblem desces of buildiit has to teness. Co

ays geometra potential

proach combrimal geome

voids and thn progress.

current statl polyhedralnd hence, bothat bounddge, there

mance as colation mean

more, thereoduce a CDn persistencork is a test

ng algorithmgn in buildin

than four vs are accuraon, these cspecial, trathe conceplgorithm. Tk. Figure (9The triang(i.e. wall ed

orted polyhed

nd Future W

cription inding informabe treated

oncerning crical buildin

method whbines existietrical reprehe exterior

te the triangl boundariesoundary edgary edges ’s no incnstraint crit

nwhile valid

is no knowDM. Since ece and in-mted candidat

m is starteng componvertices on ate calculat

calculations ansitivity dt of adaptiv

The distincti9) presents gulation is dges close c

dra (left) and

Work

dicates an ation model

as a comcompletenesng models. hich furtherng work onesentation has compac

gulation in s. The finalges are requof a cell c

cremental Dteria. Recendation has to

wledge on peach tetrahememory stote to achiev

9

ed based onnents (i.e. ra

the circumtions of det

suffer fromdoesn’t holdve precisionion of casesan exampleDelaunay wcolumn vert

d Delaunay tboundaries

upcoming ling. In ordmputational ss, a more In this pap

rmore can bn dual data shas been enhct closed 2-

CD has nol tetrahedraluired to exisomplex exiDelaunay

nt work assuo deal with

performanceedron has 4ore is needeve suitable p

n the workaster axes)

msphere of aterminants (m inaccuracd. To overcn floating p of the incre of a wall which caustices).

riangulation

issue on er to delega

model. Tgeneric m

per we utilibe an input structures ahanced to a-manifolds.

on-conformalisation musst. Additionist (Si 200algorithm

umes a bounadding and

e and the up48 edges, aned. The Eclperformance

k of Ledoumainly deg

a single tetr(Shewchuk cy of the mcome this point arithmemental Dewith a doo

es that som

(right) with

model valate a modelhis includeethod is neized the Poto semanti

nd Delaunan embeddedFinally, th

ant boundarst be constra

nal points ha8). Accordiwith incr

ndary as a sd removing

pcoming amn efficient clipse Connee.

ux (2006). generated prahedron. B

1997). Conmapping to problem, Sh

metic. He proelaunay algoor opening me initial b

non-conform

lidation wil into an eves correctneeded to d

oincaré – Duic model vaay tetrahedrd 3-manifol

his paper pr

aries concerraint since bave to be ining to the remental bstatic prereq

of boundar

mount of datcaching meected Data

Due to oint sets

Basically, ncerning floating

hewchuk oved the orithm is and two

boundary

mant

thin the valuation ness and eal with uality to

alidation. ralisation ld which resents a

rning the boundary serted to author’s

boundary quisite of ries over

ta which echanism

Objects

10

Two major assumptions are that the geometrical orientation of a face is specified correctly and that there are no self-intersections of an object. Tests with existing modeling software indicate that sometimes a model does not fulfill these requirements. Further research is needed to deal with such geometrical inconsistent models.

References

Asghari, M., M.T. Sadat, and P. Boguslawski (2013). „An Overview on Boundary Representation Data Structures for 3D Models Representation“, International Symposium & Exhibition on Geoinformation (ISG).

Bilchuk, I., and P.J. Pahl (2012). „A Half-Face Data Structure for Topological Models of Buildings“. Moscau, ICCCBE 2012.

Boguslawski, P., and C. Gold (2007). „AUGMENTED QUAD-EDGE – 3D DATA STRUCTURE FOR MODELLING OF BUILDING INTERIORS“. In Computer Information Systems and Industrial Management Applications, 2007. CISIM’07. 6th International Conference on, 125–28, 2007.

Boguslawski, P., C. M Gold, and H. Ledoux (2010). „Modelling and analysing 3D buildings with a primal/dual data structure“. ISPRS Journal of Photogrammetry and Remote Sensing.

Boguslawski, P. (2011). „Modeling And Analysing 3D Building Interiors With The Dual Half-Edge Data Structure“. The University of Glamorgan, 2011.

Borrmann, A. (2007). „Computerunterstützung verteilt-kooperativer Bauplanung durch Integration interaktiver Simulationen und räumlicher Datenbanken“. Technische Universität München, 2007.

Borrmann, A., and J. Beetz (2010). „Towards spatial reasoning on building information models“. In Proc. of the 8th European Conference on Product and Process Modeling (ECPPM).

Dimyadi, J., and R. Amor (2013). „Automated Building Code Compliance Checking – Where is it at?“ In 13th International Conference on Construction Application of Virtual Reality. London.

Eastman, C., J. Lee, Y. Jeong and J. Lee (2009). „Automatic rule-based checking of building designs“. Automation in Construction 18, Nr. 8: 1011–1033

Guibas, L., and J. Stolfi (1985). „Primitives for the manipulation of general subdivisions and the computation of Voronoi“. ACM Transactions on Graphics (TOG) 4, Nr. 2 : 74–123.

Kraft, B. and Huhnt, W. (2013). „Identifying Incorrectly Specified Objects for Specific Information Derivations in Digital Building Models”. Proceedings of the 20th International Workshop: Intelligent Computing in Engineering 2013. Springer, 62-64.

Ledoux, H. „Modelling three-dimensional fields in geoscience with the Voronoi diagram and its dual“. Delft University of Technology, Netherlands, 2006.

Lienhardt, P. (1994). N-dimensional generalized combinatorial maps and cellular quasi-manifolds. Internat. J. Comput. Geom. Appl., 4(3):275–324

Lopes, H., and G. Tavares (1997). „Structural operators for modeling 3-manifolds“. In Proceedings of the fourth ACM symposium on Solid modeling and applications, pp. 10–18.

Mäntylä, M. (1988). An Introduction to Solid Modeling. Computer Science Press, 1988.

Ohori, K.A., H. Ledoux, and J. Stoter (2013). „Modelling higher dimensional data for GIS using generalised maps“. In Computational Science and Its Applications–ICCSA 2013, 526–39. Springer, 2013. http://link.springer.com/chapter/10.1007/978-3-642-39637-3_41.

Pion, S., and M. Teillaud. (2014). „CGAL 4.4 - 3D Triangulation Data Structure: User Manual“. Last access on April, 15th, 2014. http://doc.cgal.org/latest/TDS_3/index.html#chapterTDS3.

Sallee, J.F. (1982). „A triangulation of the n-cube“, Discrete Mathematics 40(1): pp. 81–86.

Shewchuk, J. R. (1997). „Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates“. Discrete & Computational Geometry 18, Nr. 3 (October 1997): 305–63.

Si, H. (2008). „Three Dimensional Boundary Conforming Delaunay Mesh Generation“. Technische Universität Berlin.

Solibri Model Checker, Solibri.com (2014), Feature portfolio, Last access on April, 23rd , http://www.solibri.com/products/solibri-model-checker/,

Verbree, E. (2010). „Delaunay Tetrahedralisation s - Honor Degenerated Cases“. In 5th 3D geoinfo conference, vol international archives of the photogrammetry, Remote Sensing and Spatial Information Sciences XXXVIII-4, part W, 15:69–72. Berlin.

11

Verbree, E., and H. Si. (2008). „Validation and storage of polyhedra through constrained Delaunay tetrahedralisation “. In Geographic Information Science, 5266:354–69. Lecture Notes in Computer Science. Springer, 2008. http://link.springer.com/chapter/10.1007/978-3-540-87473-7_23.

Weiler, K., (1988). „The Radial Edge Data Structure: A Topological Representation For Non-Manifold Geometric Boundary Modeling”. In Geometric Modeling for CAD Applications, Amsterdam, pp. 3-36