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G aphical Models
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Disc e e a iable 3D models in Compu e ex ended Desc ip i e Geome y
(CeDG): Building o polygonal shee -me al elbows and compa ison agains
CAD
Manuel P ado-Velasco ∗, Lau a Ga cía-Ruesgas
Depa men o Enginee ing G aphics, Enginee ing Facul y, Uni e si y o Se ille, ETSI, camino de los descub imien os s/n, Se ille, 41020, Spain
Mul ile el Modelling and Eme ging Technologies in Bioenginee ing, Uni e si y o Se ille, Spain
ARTICLE INFO
Keywo ds:
Compu e ex ended Desc ip i e Geome y
Compu e aided design
3D disc e e modeling
Shee me al
Polygonal elbows
Dynamic geome y so wa e
ABSTRACT
The Compu e ex ended Desc ip i e Geome y (CeDG) is as a no el app oach based on Desc ip i e Geome y
o build 3D models wi hin he amewo k p o ided by Dynamic Geome y So wa e ools. Pa ame ic CeDG
models can be in e ac i ely explo ed when con inuous pa ame e s change, bu his is no he case o disc e e
pa ame e s. This s udy demons a es he capabili y o he GeoGeb a - CeDG app oach o inco po a e algo i hms
ha build disc e e a iable 3D models wi h dynamic pa ame e iza ion. Se e al 3D models and hei la ened
pa e ns (neu al ibe ), based on a new de eloped CeDG algo i hm, we e compa ed o hei LogiTRACE .14
and Solid Edge 2024 (CAD) coun e pa s. The accu acy o he CeDG models su passed ha o CAD models o
nea ly all dimensions de ined as me ics. In addi ion, he CeDG app oach was he unique ha p o ided an
au oma ic solu ion o any alue o he numbe o e ules.
1. In oduc ion
The ascendancy and widesp ead adop ion o Compu e -Aided De-
sign (CAD) in ecen decades ha e p og essi ely ma ginalized Desc ip-
i e Geome y (DG) wi hin bo h academic cu icula and p o essional
p ac ice [1]. This shi has p omp ed e o s o combine adi ional DG
echniques wi h CAD ools o enhance accu acy and achie e goals ha
would be di icul , i no impossible, wi hou digi al suppo [2,3].
A ela i ely new concep , Augmen ed G aphic Thinking (AGT), e e s
o he expansion o DG h ough he use o digi al p ocedu es and
simula ions [4]. Fo ins ance, he d awing o geodesic lines on a
non-de elopable su ace by s ips o ec i ying su aces is ci ed as an
example o a complex ope a ion ha was deduced using AGT [5].
The impac o compu e s on he me hodologies and applica ions
o DG has been he subjec o conside able esea ch, wi h s udies
highligh ing he po en ial o ha nessing he syne gies be ween image
syn hesis and solu ion analysis, as well as ad ancing in o he con-
s uc ion p ocesses o geome ic loci [1]. Howe e , o he bes o ou
knowledge, he e is cu en ly no no el compu e -based me hodology
ha ully accoun s o his syne gy. Ano he a ea ha has expe ienced
signi ican ad ancemen s in ecen decades is dynamic geome y, pa -
icula ly in he ma hema ical domain [6]. Recen ly, he e has been a
g owing in e es in in eg a ing dynamic geome y in o uni e si y-le el
∗Co esponding au ho a : Depa men o Enginee ing G aphics, Enginee ing Facul y, Uni e si y o Se ille, ETSI, camino de los descub imien os
s/n, Se ille, 41020, Spain.
E-mail add esses: [email p o ec ed] (M. P ado-Velasco), [email p o ec ed] (L. Ga cía-Ruesgas).
educa ion, as e idenced by he eme gence o new eaching app oaches
in his ield [7]. I is no ewo hy ha he ini ial explo a ion o he
use o DGS o suppo DG can be aced back o he hesis wo k o
Wo eng [8], in which he in luence o Gaspa d Monge’s de elop-
men s on desc ip i e and di e en ial geome y was analyzed. Wo eng
employed one o he inaugu al dynamic geome y so wa e (DGS)
p og ams, Ske chpad, o acili a e he mo emen o an auxilia y cu -
ing plane du ing he in e sec ion o wo cones, he eby gene a ing a
compu a ional locus. Ske chpad, which was de eloped by Su he land
as pa o his doc o al disse a ion [9], is ega ded as a ounda ional
p ecu so o he exis ing pa adigm o CAD.
The concep o Compu e -ex ended Desc ip i e Geome y (CeDG)
eme ges as a compu e -based me hodology oo ed in DG o 3D model-
ing. I is designed o se e bo h academic and p o essional communi ies
and is aligned wi h he p inciples o AGT. Howe e , CeDG equi es
implemen a ion wi hin a Dynamic Geome y So wa e (DGS) ool.
De eloped in he ea ly 2020s, CeDG esponds o he displacemen
o DG by mode n CAD sys ems. I le e ages he na u al ela ionship
be ween he sequence o ma hema ical en i ies in a DGS model and
he sequence o g aphical p ocedu es in DG [10]. As a esul , CeDG
models in eg a e g aphical and p ojec i e objec s in hei undamen al
ma hema ical o ms, in con as o CAD, which ocuses on con olling
h ps://doi.o g/10.1016/j.gmod.2024.101253
Recei ed 21 Sep embe 2024; Recei ed in e ised o m 20 No embe 2024; Accep ed 24 Decembe 2024
G aphical Models 137 (2025) 101253
1524-0703/© 2025 The Au ho s. Published by Else ie Inc. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ ).
M. P ado-Velasco and L. Ga cía-Ruesgas
3D shapes h ough B-Splines.
Unlike CAD, which uses su ace-cu e app oxima ions [11], CeDG
o e s a undamen ally di e en app oach. A CeDG model consis s
o a sequence o g aphic-ma hema ical en i ies ha a e in e ela ed
h ough speci ic dependencies. These en i ies ep esen he p imi i es
and cu es de i ed om he p ojec ions o a 3D sys em, along wi h
he p ocedu es o Desc ip i e Geome y and o he ma hema ical e-
la ionships. This o ganiza ional s uc u e allows o p ecise and e i-
cien ep esen a ion o complex 3D sys ems by in eg a ing a a ie y o
ma hema ical and geome ic echniques.
CeDG models a e implemen ed using Dynamic Geome y so wa e
(DGS) pla o ms, such as GeoGeb a, which p o ides an in e ac i e
en i onmen o he explo a ion and manipula ion o hese g aphic-
ma hema ical en i ies [6,12,13]. GeoGeb a acili a es he c ea ion o
dynamic pa ame ic models, enabling use s o isualize and adjus p o-
jec ions and cu es in eal- ime. This capabili y enhances he analysis o
he dynamic beha io o geome ic cons uc ions while p ese ing hei
in eg i y. Fu he mo e, i allows designe s and enginee s o obse e
how changes in pa ame e s impac he o e all s uc u e.
Ne e heless, a mo e in ica e challenge a ises when pa ame e s
a e disc e e and lead o quali a i e, non-con inuous changes in he
3D model. To add ess his challenge, a me hodology o cons uc ing
3D CeDG models ha inco po a e quali a i e changes associa ed wi h
disc e e pa ame e s mus be de eloped. One p ac ical example o his
app oach is he c ea ion o polygonal elbows (bo h cylind ical and
conical) made om n- e ules, which can adap o changes in he
disc e e a iable n (whe e n>2).
Thus, he goal o his s udy is wo old: (i) o demons a e he
capabili y o he CeDG app oach in building 3D models wi h dynamic
pa ame e iza ion o disc e e a iables, h ough he de elopmen o a
no el algo i hm ha suppo s he au oma ic gene a ion o cylind ical
an conical polygonal elbows and hei la ened pa e ns, and (ii)
o compa e hese models wi h hose c ea ed using indus y-s anda d
so wa e ools - LogiTRACE .14 (specialized in Shee Me al) and Solid
Edge 2024 (a gene al-pu pose so wa e wi h a shee me al module)- in
e ms o scope, cons uc ion me hodology, and p ecision.
2. Me hods and so wa e
We use GeoGeb a (h ps://www.geogeb a.o g), a dynamic geom-
e y so wa e, o implemen he CeDG app oach. The algo i hms o
gene a ing disc e e- a iable polygonal elbows a e implemen ed using
he GeoGeb aSc ip language, which is a ailable in GeoGeb a Classic
5 (desk op-o ien ed) and GeoGeb a Classic 6 (web-o ien ed). GeoGeb a
can be un ei he h ough i s execu able ile (GeoGeb a.exe) o as a web
se ice. Al hough GeoGeb a’s ins alle s a e ee o non-comme cial
use, i s sou ce code is licensed unde he GNU Gene al Public License
( 3). Fo b oade usage, including comme cial applica ions, GeoGeb a
can be execu ed h ough he G adle build ool (h ps://g adle.o g/)
o any o he ool ha u ilizes he GeoGeb a sou ce iles di ec ly.
This app oach o e s mo e lexibili y o comme cial deploymen and
ex ension.
The eliabili y o he no el algo i hms, pa ame e ized wi h espec
o con inuous a iables (dimensions) and disc e e a iables (numbe o
e ules), was analyzed using bo h a cylind ical elbow and a conical
elbow. The e e ence nominal dimensions we e as ollows: diame e
(a unique alue o he cylind ical elbow and he i s alue o he
conical elbow), D1= 150 mm, second diame e o he conical elbow,
D2=70 mm, elbow adius, Relb = 300 mm, and elbow angle, 𝛿= 98◦.
Shee me al models we e de eloped using he neu al su ace, se ing
as a p elimina y s ep in c ea ing he g aphical pa e n o a shee me al
pa . This me hodology mi iga es dependency on ma e ial ype and
emphasizes su ace geome y in bo h 3D and la ened o ms. The
associa ed CeDG models include he p ima y o hog aphic iew and i s
la ened s a e ( la pa e n). Reliabili y was assessed by con i ming ha
hese models adap dynamically o a ia ions in he numbe o e ules
Fig. 1. Canonical p ojec ion o a conical polygonal elbow model (see ex ).
(g ea e han o equal o wo) and con inuous dimensions. Addi ionally,
he models we e cons uc ed using Solid Edge 2024 and LogiTRACE
.14, o compa e hei eliabili y agains he CeDG app oach.
The accu acy o CeDG, Solid Edge 2024 and LogiTRACE .14 models
was e alua ed by compa ing he main dimensions o hei la pa -
e ns. These dimensions a e illus a ed in Fig. 1, using he example
o a conical elbow wi h i s mou hs posi ioned pe pendicula o he
p ojec ion plane and cen e ed wi hin a plane pa allel o his same
p ojec ion plane (canonical p ojec ion). The key dimensions selec ed
o his compa ison included he ollowing: he leng h o he la ened
sec ion cu e be ween e ules i and i+1 , deno ed as Li, he adius
R o he ci cula a ch ep esen ing he cone o e olu ion om which
he e ules a e de i ed, and he angle 𝛼1be ween he gene a ix
h ough X and he angen line, which se es as a local index o he
cu a u e a he i s la ened sec ion cu e ( he in lec ion poin ).
The ue alues o hese dimensions we e calcula ed based on he
sys em’s geome ic p ope ies. The alue o R was de e mined h ough
desc ip i e geome y p ocedu es [14], whe eas he ue alue o Li
co esponds o he pe ime e o he spa ial plane sec ion (illus a ed
in Fig. 1 o he i s sec ion, deno ed as L1 ). The angle 𝛼1 ep esen s
he angle be ween he 3D gene a ix h ough X and he angen o he
plane sec ion a X, le e aging i s in a iance in he la ening p ocess.
Fo cylind ical elbows, he selec ed dimensions included he leng h
o he la ened sec ion cu e be ween e ules, deno ed as L, and he
angle 𝛼1. In cases whe e he model p o ides di e en alues o L, he
mean and s anda d de ia ion o hese alues will be epo ed.
The nume ical compa ison was conduc ed o elbows wi h 3, 5,
and 7 e ules. Accu acy was quan i ied using he Rela i e E o (RE),
de ined as RE = (M-T)/T⋅100%, whe e M ep esen s he model mea-
su emen and 𝑇is he ue alue. Fo cases wi h mul iple measu emen s
o he same ue alue, accu acy was assessed using he Roo Mean
Squa e E o (RMSE), calcula ed as RMSE =√1∕𝑛∑𝑖(M𝑖∕T− 1)2⋅100%,
whe e Miand T ep esen he measu emen alues and he ue alue,
espec i ely. Sensi i i y analyses we e also conduc ed o ensu e he
obus ness o he models.
3. Algo i hms o he gene a ion o polygonal elbows in CeDG
To c ea e a model pa ame e ized by bo h con inuous (dimensions)
and disc e e (numbe o e ules) a iables o modeling o cylind ical
elbows, we begin by analyzing wo pa adigms. In he i s pa adigm, he
model is cons uc ed using i e a i e and con ol low s uc u es, which
mus be upda ed i any pa ame e changes. In con as , he second
pa adigm in ol es building only dynamic objec s a each coding s age.
These objec s main ain in e dependence based on he cons uc ion
sequence and au oma ically espond o pa ame e changes wi hou
equi ing an explici e-execu ion.
G aphical Models 137 (2025) 101253
2
M. P ado-Velasco and L. Ga cía-Ruesgas
Fig. 2. Modeling a disc e e a iable cylind ical polygonal elbow wi h algo i hm-gene a ed auxilia y objec s.
3.1. Fi s pa adigm
We begin wi h he algo i hm o cons uc ing a cylind ical polyg-
onal elbow. Fig. 2p esen s a gene ic cylind ical elbow h ough i s
canonical p ojec ion ( e e o he Me hods sec ion). This posi ion sim-
pli ies he model cons uc ion, allowing i o be subsequen ly mo ed
o any o he p ojec ion iew [10]. An addi ional o hogonal iew is
unnecessa y in his canonical posi ion. The model will also include he
la ened s a e o he elbow (Fig. 2, igh ).
Conside ing ha he e ules cons i u ing he elbow can be de-
i ed om a single e olu ion cylinde h ough 180◦ o a ion be ween
consecu i e pieces [14], he ollowing GeoGeb aSc ip code compu es
he sequence o bo om and uppe con ou poin s as depic ed in he
canonical p ojec ion in Fig. 2.
1# Fi s de ini ions
2Po A1 = I (Dis ance(ET_1,EC)<Dis ance(ET_2 ,EC),ET_1 ,
ET_2)
3Po A2 = I (Po A1==ET_1, ET_2, ET_1)
4Cen e A cElbow = EC
5Po B1 = Ro a e(Po A1 ,ElbowAngle , Cen e A cElbow)
6Po B2 = Po B1 + Uni Vec o (Vec o (Cen e A cElbow ,
Po B1)) Dis ance(Po A1 ,Po A2)
7# Tangen poin s
8Po ACen e = Midpoin ( Po A1 , Po A2 )
9Po BCen e = Midpoin ( Po B1 , Po B2 )
10 linePo A = Line(Po A1 , Po A2)
11 linePo B = Line(Po B1 , Po B2)
12 Ang2dElbow = Del aElbowAB / (nFe ules - 1)
13 Lis Radius = I e a ionLis (Ro a e(A cElbowPoin ,
Ang2dElbow , Cen e A cElbow), A cElbowPoin , {
Po ACen e }, nFe ules - 1)
14 Lis Planes = Sequence(In e sec (Tangen (Lis Radius(
iFe ules), A cElbow), Tangen (Lis Radius(
iFe ules+1), A cElbow)), iFe ules , 1, nFe ules
- 1)
15 # Con ou poin s
16 Lis Con ou A1 = {}
17 Se Value(Lis Con ou A1, 1, Po A1)
18 Lis Con ou A2 = {}
19 Se Value(Lis Con ou A2, 1, Po A2)
20 iFe ules = 1
21 linePlanes = Line(Cen e A cElbow , Lis Planes(iFe ules
))
22 lineTangen = Tangen (Lis Radius(iFe ules), A cElbow)
23 DO = {" Se Value(Lis Con ou A1 , iFe ules+1, In e sec (
linePlanes , Line(Lis Con ou A1(iFe ules),
lineTangen )))"," Se Value(Lis Con ou A2 ,
iFe ules+1, In e sec (linePlanes , Line(
Lis Con ou A2(iFe ules), lineTangen ))) " ,"
Se Value(iFe ules , iFe ules+1) " }
24 LOOP = {" I (iFe ules <nFe ules ,Execu e(Join(DO, LOOP)
)) " }
25 Execu e(LOOP)
26 Se Value(Lis Con ou A1 , nFe ules+1, Po B1)
27 Se Value(Lis Con ou A2 , nFe ules+1, Po B2)
GGB code 1 cylind ical elbow ( i s pa adigm)
Lines 1 o 6 a ange he mou h ends o ensu e ha po A1 and
Po B1 a e posi ioned on he bo om con ou , while Po A2 and Po B2
a e on he uppe con ou . The ollowing block di ides he elbow a ch -
a ci cula a ch be ween he mou h cen e s - in o n - 1 segmen s, whe e
n ( e e ed o as nFe ules) ep esen s he numbe o e ules. The
Lis Radius poin s a e he in e sec ions be ween hese di ision lines
and he a ch (ma ked by emp y ci cles in Fig. 2). Addi ionally, Lis -
Planes deno es he sequence o poin s c ea ed by he in e sec ion o
angen s o he a ch a each Lis Radius poin .
Planes pe pendicula o he p ojec ion plane de ine sec ions be-
ween adjacen e ules, using lines ha o igina e om he a ch cen e
and in e sec wi h poin s in he Lis Planes sequence. The inal
block cons uc s hese sec ions using he linePlanes and line-
Tangen objec s wi hin a LOOP i e a i e s uc u e, de ined by he
DO s ing lis . As shown line 24, LOOP e e ences i sel , making Exe-
cu e(LOOP) a ecu si e ins uc ion ha e mina es when iFe ules
equals nFe ules.
The model’s canonical p ojec ion is comple ed by ano he ecu si e
LOOP ha gene a es he inal con ou segmen s and sec ion planes.
These can be accessed in he CylElbowP oj1Pa adigm ile in he supple-
men a y ma e ial. As shown in Fig. 2, all in e media e e ules a e wice
he size o he e ules connec ed o he mou hs. The model pa ame e s
- D (diame e o mou hs), 𝛿, and n - a e linked o adjus able slide s;
howe e , he Execu ion block equi es manual ini ia ion whene e
hese pa ame e s change, which limi s he model’s dynamic upda ing.
The algo i hm o la ening he elbow, ollowing he ini ial
pa adigm, is p esen ed below.
1# Ini ializa ion
2ejeCil = Pe pendicula Line(PAC, Pa Re )
3 Eje = Uni Vec o (ejeCil)
4 Re = -Pe pendicula Vec o ( Eje)
5dEje = Dis ance(Lis Planes(1),Lis Planes(2))
6# Rec angle
7PBL = PAC + Dis ance(Po A1 ,Po A2)/2 Re
8PBR = PBL + pi Dis ance(Po A1 , Po A2) Re
9PTL = PBL + (nFe ules -1) dEje Eje
10 PTR = PBR + (nFe ules -1) dEje Eje
11 Segmen (PBL , PBR)
12 Segmen (PTL , PTR)
13 Segmen (PBL , PTL)
14 Segmen (PBR , PTR)
15 # Planes’ sec ions
G aphical Models 137 (2025) 101253
3
M. P ado-Velasco and L. Ga cía-Ruesgas
16 Lis PlanesD = I e a ionLis (PEje + dEje Eje, PEje, {
PAC + dEje/2 Eje}, nFe ules - 2)
17 PhiPlaneAxis = Angle(Po ACen e , Lis Planes(1),
Lis Con ou A2(2))
18 # Fla ened sec ion cu es
19 baseCyl = Ci cle(PAC, PBL)
20 omegaCyl = Poin (baseCyl)
21 gomegaCyl = Line(omegaCyl , ejeCil)
22 A comegaCyl = A c(baseCyl , PBL, omegaCyl)
23 iFe ules = 1
24 DO = {" Locus(In e sec (Pe pendicula Line(In e sec (
Line(Lis PlanesD( " +iFe ules+ " ), Ro a e(PAC , (-1)
^ " + iFe ules+ " *PhiPlaneAxis , Lis PlanesD( " +
iFe ules+ " ))), gomegaCyl),ejeCil), Line(PBL +
Leng h(A comegaCyl) Re , ejeCil)), omegaCyl) " ,"
Se Value(iFe ules , iFe ules+1) " }
25 LOOP = {" I (iFe ules <=nFe ules -1, Execu e(Join(DO,
LOOP))) " }
26 Execu e(LOOP)
GGB code 2 pa e n o cylind ical elbow ( i s pa adigm)
We begin by de ining he cylinde ’s axis, ejeCil, based on he e -
e ence poin PAC and pe pendicula o Pa Re (line 2), as illus a ed
in Fig. 2( igh ). The axis leng h o an in e media e e ule is calcula ed
as dEje (line 5). These elemen s enable he calcula ion o he ec angle
ha ep esen s he ull pa e n, de ined by poin s PBL, PBR, PTL
and PTR (block Rec angle). To compu e he la ened e ules, we
de e mine he poin s whe e sec ion planes in e sec ejeCil, labeled
as Lis PlanesD, as well as he angle be ween he sec ion planes and
he cylinde axis, PhiPlaneAxis (lines 16–17).
The e e ence poin omegaCyl is hen es ablished on he suppo
cu e baseCyl (lines 19–20). This poin acili a es he compu a ion
o he la ened ans o ms o each cu e sec ion, using a ecu si e
LOOP o gene a e hem as pa ame ic locus cu es, in acco dance
wi h Eq. (A.1) (see Appendix A). He e, P𝜏 ep esen s he in e sec ion
o he gomegaCyl gene a ix wi h each sec ion plane, align wi h
co esponding poin s in Lis PlanesD (line 24) and mapped o he
la ened ec angle o each e ule.
As illus a ed in Fig. 2, he pa e n o a ou - e ule cylind ical el-
bow equi es h ee plane cu es o de ine he ou e ules. As expec ed,
he la ened sec ion cu es a e symme ic due o he 180◦ o a ion
be ween consecu i e e ules [14]. Howe e , any change o a pa ame e
alue equi es manual execu ion o he algo i hm o upda es, limi -
ing dynamic pa ame e iza ion. The comple e code is p o ided in he
CylElbowPa 1Pa adigm ile in he supplemen a y ma e ial.
3.2. Second pa adigm
To elimina e he need o manually execu ing he algo i hm, all
Execu e commands mus be emo ed. In he upda ed code, he con ou
poin s a e now calcula ed using he Sequence(command, i, 1,
n) objec . This app oach dynamically gene a es a lis based on com-
mand(i) when he index i anges om 1 o n. This in e nal command
eplica es he logic o GGB code 1(lines 16–27), ensu ing ha he
esul s upda e au oma ically wi hou equi ing explici execu ion.
1# Con ou poin s
2con ou Bo om = Sequence(In e sec (Line(Lis Radius(
iFe ules) +Dis ance(Po ACen e , Po A1)
Uni Vec o (Vec o (Lis Radius(iFe ules),
Cen e A cElbow)), Tangen (Lis Radius(iFe ules),
A cElbow)), Line(Cen e A cElbow , Lis Planes(
iFe ules))), iFe ules , 1, nFe ules - 1)
3con ou Uppe = Sequence(In e sec (Line(Lis Radius(
iFe ules) -Dis ance(Po ACen e , Po A2)
Uni Vec o (Vec o (Lis Radius(iFe ules),
Cen e A cElbow)), Tangen (Lis Radius(iFe ules),
A cElbow)), Line(Cen e A cElbow , Lis Planes(
iFe ules))), iFe ules , 1, nFe ules - 1)
GGB code 3 cylind ical elbow (second pa adigm)
A simila app oach allows us o eplace he ecu si e LOOP in GGB
code 2 o cons uc he pa ame ic locus cu es, as demons a ed below.
Fig. 3. Fla pa e ns o cylind ical elbows wi h 3- e ule (le ), 5- e ule (cen e ), and
7- e ule ( igh ) con igu a ions, compu ed using he CeDG app oach.
1# Fla ened sec ion cu es
2Lis PlanesFla = Sequence(Locus(In e sec (
Pe pendicula Line(In e sec (Line(Lis PlanesD(
iFe ules), Ro a e(PAC, (-1)^iFe ules*(-
PhiPlaneAxis), Lis PlanesD(iFe ules))),
gomegaCyl), ejeCil), Line(PBL + Leng h(
A comegaCyl) Re , ejeCil)), omegaCyl), iFe ules
, 1, nFe ules)
GGB code 4 pa e n o cylind ical elbow (second pa adigm)
Models based on his ype o algo i hm can dynamically espond
o changes in bo h con inuous and disc e e a iables, making hem an
ideal choice. The comple e sou ce code o he algo i hm applied o
he cylind ical elbow is a ailable in he CylElbowAlg ile wi hin he
supplemen a y ma e ial. A disc e e a iable model o a gene al polyg-
onal conical elbow can also be cons uc ed using he same algo i hmic
app oach, as demons a ed in Appendix B.
4. CeDG models o cylind ical and conical elbows
The algo i hms in he p e ious sec ion we e execu ed on GeoGeb a
Classic 5 o de elop CeDG models o he cylind ical and conical
polygonal elbows de ined in he Me hods sec ion. Fig. 3shows he la
pa e ns calcula ed o he h ee cylind ical elbows using he speci ied
nominal dimensions. The alues eached o 𝛼1and L (see Table 1)
show ha he models yield exac alues o he de ined me ics, consis-
en wi h he capabili y o he CeDG app oach o main ain geome ic
in eg i y du ing he la ening ans o ma ion.
The conical elbow models pa ame e ized using he nominal dimen-
sions alues, p oduced he la pa e ns shown in Fig. 4. To acili a e
inspec ion and compa ison o he la ened geome y in he pape , he
pa e ns a e shown a di e en scales. Full-sized pa e ns a e a ailable
in PDF iles in he supplemen a y ma e ial.
Tables 2–4 p esen selec ed dimension alues o he conical elbow
pa e ns. As seen, excep o R, he alues o 𝛼1and Lishow mino
di e ences om he ue pa ame e s. Al hough he ela i e e o s a e
negligible, hese di e ences a e discussed below, as CeDG main ains
he geome ical in eg i y o hese cu es.
5. CAD models o cylind ical and conical elbows
The cons uc ion o cylind ical and conical elbow models, as p e-
sen ed in he p e ious sec ion, is de ailed below using CAD echnology.
The so wa e u ilized includes Solid Edge 2024, a pa ame ic CAD
p og am o h ee-dimensional pa s, and Logi ace .14, specialized
enginee ing so wa e designed o de eloping la layou s o duc s,
componen s, and ools used in en ila ion and hea ing sys ems.
G aphical Models 137 (2025) 101253
4
M. P ado-Velasco and L. Ga cía-Ruesgas
Fig. 4. Fla pa e ns o conical elbows wi h 3- e ule (le ), 5- e ule (cen e ), and 7- e ule ( igh ) con igu a ions, compu ed using he CeDG app oach.
Table 1
Leng hs o sec ions be ween e ules in la pa e ns, L (mm)a, oge he wi h 𝛼1(◦) (see Fig. 1) and hei ela i e e o sb(%)cin cylind ical elbows.
3- e ules 5- e ules 7 e ules
App oach 𝛼1L𝛼1L𝛼1L
T ue 114.5 230.92 102.25 222.48 98.17 221.04
CeDG 114.5 (0) 230.92 (0) 102.25,(0) 222.48 (0) 98.17 (0) 221.04 (0)
LogiTRACE 114.5 (0) 230.67 ±0.29 (0.15) 102.25 (0) 220.28 ±0.16 (0.11) 98.24 (0.07) 220.93 ±0.08 (0.06)
Solid Edge 114.32 (0.16) 230.93 (0) 102.17 (0.08) 222.49 (0) 98.17 (0) 221.05 (0)
aIn he case o se e al alues o L, mean ±SD is w i en.
bRoo mean squa e e o , RMSE (%), i he e exis se e al measu es o L.
cValues <0.01% a e ounded o 0.
Table 2
Leng hs o sec ions be ween e ules in la pa e ns, Li(mm), oge he wi h 𝛼1(◦) and R (mm) (see Fig. 1) and hei ela i e e o s (%)ain
3- e ules conical elbows.
App oach 𝛼1R L1L2
T ue 114.6 1028.12 426.46 295.24
CeDG 116.13 (1.3) 1028.12 (0) 426.6 (0.03) 295.34 (0.03)
LogiTRACE 113.81 (0.69) 1028.12 (0) 426.42 (0) 295.2 (0.01)
Solid Edge 114.56 (0.03) 1026.37 (0.17) 425.16 (0.3) 294.33 (0.3)
aValues <0.01% a e ounded o 0.
Fig. 5. Fla pa e ns o cylind ical elbows wi h 3- e ule (le ), 5- e ule (cen e ), and
7- e ule ( igh ) con igu a ions, compu ed using he LogiTRACE app oach.
5.1. LogiTRACE models
The cons uc ion o cylind ical elbows wi h 3, 5, and 7 e ules was
ca ied ou in Logi ace using model numbe 066. Fig. 5p esen s he
la pa e ns de i ed o hese cylind ical elbows, based on he speci ied
nominal dimensions.
Rega ding he me hodology, he p og am p omp s he use o inpu
essen ial design pa ame e s, which in his case included he diame e
o he elbow opening, he adius and angle o he elbow, he numbe
o e ules, hickness, and he numbe o gene a ices. Fo his model,
a o al o 120 gene a ices and ze o hickness we e speci ied. No
cylind ical supplemen s we e added o adjus he pipe openings.
The model is displayed p omp ly in one o se e al 3D isualiza ion
o ma s, allowing o inspec ion o he in e sec ions wi hin he igu es.
This 3D isualiza ion ensu es ha he shape con o ms o he equi ed
speci ica ions. Addi ionally, he shape is ende ed in 2D, and a d awing
wi h dimensions is gene a ed and sa ed in he s anda d DXF o ma .
The shee ’s pe ime e , weigh , and a ea a e calcula ed.
As illus a ed in Table 1, he 𝛼1 alue o he 3- and 5- e ule elbows
is p ecise; howe e , o he 7- e ule elbow, his alue shows a sligh
de ia ion om he ue alue. Fo he L alue, i was necessa y o
calcula e he mean alue in all cases, along wi h i s oo mean squa e
e o , as he e ules did no all p esen he same alue.
The cons uc ion o conical elbows wi h 3, 5, and 7 e ules was
pe o med using model numbe 120. Fig. 6displays he esul ing la
pa e ns. The da a equi ed by he applica ion a e iden ical o hose o
cylind ical elbows, wi h he only di e ence being ha he diame e s o
he duc openings a y. A o al o 120 gene a ices and ze o hickness
we e used. I should be no ed ha he la pa e ns in Fig. 5we e
compu ed wi hou al e na ing he posi ion o he joins, in con as o
hose in Fig. 6.
As shown in Tables 2–4, he esul s ob ained o Liclosely app oxi-
ma e he ue alues, while R alues a e exac . Scaled la pa e ns a e
p o ided in DFT iles wi hin he supplemen a y ma e ial.
5.2. Solid edge models
The modeling o cylind ical elbows in Solid Edge equi es using
he‘‘Sweep’’ ool wi hin he ‘‘Su aces’’ ab o he shee me al module.
This ool equi es inpu da a comp ising a pa h and a c oss-sec ional
G aphical Models 137 (2025) 101253
5
M. P ado-Velasco and L. Ga cía-Ruesgas
Fig. 6. Fla pa e ns o conical elbows wi h 3- e ule (le ), 5- e ule (cen e ), and 7- e ule ( igh ) con igu a ions, compu ed using he LogiTRACE app oach.
Fig. 7. Fla pa e ns o cylind ical elbows wi h 3- e ule (le ), 5- e ule (cen e ), and
7- e ule ( igh ) con igu a ions, compu ed using he Solid Edge app oach.
p o ile o he la e o ollow. In his wo k, he c oss-sec ional p o ile
is a ci cle wi h a pa ame e ized diame e posi ioned in a plane pe pen-
dicula o he pa h. A nominal diame e o 70 mm was selec ed o he
elbow opening, as speci ied in Sec ion 2, wi h a small 0.01 mm opening
added o allow o each cylind ical e ule’s la layou pa e n.
The pa h, d awn on he canonical p ojec ion (Fig. 1), is calcula ed
based on he heo e ical amewo k ha de e mines he equi ed num-
be o e ules om a single cylinde [15]. Since disc e e a iable
pa ame e iza ion is no possible in he p ojec ion, each elbow model
was cons uc ed manually.
To add hickness, each e ule is indi idually c ea ed by ollowing
hese s eps:
1. Add a new shee elemen named ‘‘ e ule1’’ o he body.
2. T ans e he shee elemen o he solid module o apply hick-
ness, using he ‘‘Swi ch o’’ command in he ‘‘Tools’’ ab.
3. Wi hin he ‘‘Sweep’’ command in he ‘‘S a ’’ ab, add a minimum
hickness o 0.01 mm, con e ing he i s e ule in o a solid.
4. Repea he p ocess o each e ule o he cylind ical elbow.
Once all segmen s a e c ea ed, hey a e assembled using he ‘‘Mul i-
Body Publish’’ command. Each e ule’s ile and he assembly ile mus
be sa ed sepa a ely. To gene a e he la pa e n o he elbow e ules:
1. Edi each e ule o swi ch o he shee module.
2. Selec he ‘‘Thin Pa o Shee Me al’’ icon, con e ing i in o a
shee pa .
3. Choose he ‘‘Un old’’ op ion, speci ying he e ule, edge, and
o igin o ob ain he la pa e n.
4. Finally, gene a e a d awing o ep esen he la pa e n o he
e ule.
Using he de ined nominal dimensions, la pa e ns we e calcula ed
o cylind ical elbows wi h 3, 5, and 7 e ules, as shown in Fig. 7.
Table 1p esen s he esul ing alues o each case. The alues o 𝛼1
closely app oxima e he ue alues, wi h ela i e e o s below 0.16%,
while he L alues ma ch exac ly.
In con as , modeling he comple e conical elbow wi hin he Solid
Edge shee me al module p o ed un easible, as only a single e ule
can be gene a ed using he ‘‘Lo ’’ command. The e o e, each e ule
was indi idually gene a e in he solid module using he ‘‘P o usion
by Sec ions’’ command, which equi es closed p o iles. P o iles wi h
a 0.01 mm hickness and a one-deg ee la e al opening we e c ea ed
and posi ioned on planes o hogonal o he canonical p ojec ion plane,
de ined by h ee poin s.
A e gene a ing each e ule in he solids module, we p oceed
simila ly o he cylind ical bends. By selec ing he ‘‘Thin Pa o Shee
Me al’’ icon, he hin shee is con e ed in o a shee me al componen .
Using he ‘‘un old’’ op ion, he la pa e n is ob ained by selec ing he
e ule, he edge, and o igin o de elopmen . Following he modeling
o conical elbows based on nominal dimension alues, he la pa e ns
shown in Fig. 8we e de i ed. Scaled la pa e ns a e a ailable as DFT
iles in he supplemen a y ma e ial. The alues o 𝛼1, R, and L ob ained
om he la pa e ns o he conical elbows a e p esen ed in Tables 2–4,
showing sligh de ia ions om he exac alues.
6. Compa ison o CAD and CeDG models
The i s ow o Table 1p esen s he ue alues o he cylind ical
elbow pa e n dimensions, allowing o a compa ison be ween he
CeDG app oach and adi ional CAD. These alues we e ob ained based
on he geome ic p ope ies o he 3D model, as desc ibed in he
Me hods sec ion.
CAD app oaches yielded highly accu a e alues, wi h ela i e e o s
less han 0.16% in all cases. Howe e , he CeDG app oach p o ed
e en mo e accu a e, wi h ela i e e o s below 0.01% ac oss all cases.
In LogiTRACE, he leng h o he ans o med sec ion cu es be ween
adjacen e ules (L) a ied sligh ly depending on he e ule. Table 1
displays he mean ±s anda d de ia ion o L alues ob ained using
LogiTRACE o quan i y his a ia ion, wi h dispe sions o 0.15, 0.11,
and 0.06 mm o elbows wi h 3, 5 and 7 e ules, espec i ely. In
con as , Solid Edge p o ided a single alue o L, consis en ac oss all
e ules o he same elbow and close o he ue alue.
The i s ows o Tables 2–4 display he ue alues o selec ed
dimensions o compa ing conical elbow pa e ns, acco ding o hei
geome ic p ope ies.
LogiTRACE p o ided he mos accu a e pa e ns o he 3- and
5- e ule elbows, wi h sligh ly lowe ela i e e o s han o he ap-
p oaches. CeDG’s accu acy was nea ly equi alen o LogiTRACE o
non-angula dimensions, wi h ela i e e o s below 0.03%. Solid Edge
p o ided he bes accu acy o he angula alues, wi h e o s o 0.03%
and 0% o he 3- and 5- e ule elbows.
Fo he 7- e ule elbow, CeDG was he mos accu a e app oach, as
shown in Table 4, hough LogiTRACE accu acy was simila ly high. Non-
angula dimensions we e calcula ed wi h g ea e p ecision han angula
dimension.
G aphical Models 137 (2025) 101253
6
M. P ado-Velasco and L. Ga cía-Ruesgas
Fig. 8. Fla pa e ns o conical elbows wi h 3- e ule (le ), 5- e ule (cen e ), and 7- e ule ( igh ) con igu a ions, compu ed using he Solid Edge app oach.
Table 3
Leng hs o sec ions be ween e ules in la pa e ns, Li(mm), oge he wi h 𝛼1(◦) and R (mm) (see Fig. 1) and hei ela i e e o s (%)ain 5- e ules conical elbows.
App oach 𝛼1R L1L2L3L4
T ue 102.3 979.92 442.32 379.14 315.95 252.76
CeDG 103.16 (0.8) 979.92 (0) 442.37 (0) 379.17 (0) 315.98 (0) 252.78 (0)
LogiTRACE 102.24 (0.06) 979.92 (0) 442.28 (0) 379.09 (0.01) 316.05 (0.03) 252.72 (0.02)
Solid Edge 102.29 (0) 979.59 (0.03) 441.04 (0.3) 381.11 (0.5) 315.02 (0.3) 252.01 (0.3)
aValues <0.01% a e ounded o 0.
Table 4
Leng hs o sec ions be ween e ules in la pa e ns, Li(mm), oge he wi h 𝛼1(◦) and R (mm) (see Fig. 1) and hei ela i e e o s (%)ain 7- e ules conical elbows.
App oach 𝛼1R L1L2L3L4L5L6
T ue 98.2 971.58 449.89 408.04 366.19 324.34 282.49 240.64
CeDG 98.75 (0.5) 971.58 (0) 449.91 (0) 408.06 (0) 366.21 (0) 324.36 (0) 282.5 (0) 240.65 (0)
LogiTRACE 98.82 (0.6) 971.58 (0) 449.84 (0.01) 408 (0) 366.14 (0.01) 324.3(0.01) 282.46 (0.01) 240.61 (0.01)
Solid Edge 98.9 (0.7) 972.25 (0.07) 448.62 (0.28) 406.88 (0.24) 365.16 (0.28) 323.42 (0.28) 281.65 (0.29) 239.92 (0.3)
aValues <0.01% a e ounded o 0.
Due o he equi emen o gene a e and un old each e ule indi-
idually in Solid Edge, he alue o R was compu ed using he la ges
e ule, as R is leas sensi i e o dimensional pe u ba ions in his
e ule.
In summa y, his s udy highligh s h ee no able di e ences be ween
CeDG and adi ional CAD app oaches:
1. Geome ic Gene a ion: CeDG algo i hms o polygonal elbow
gene a ion a e based on gene al desc ip i e geome y p ocedu es
and au oma ically gene a e he desi ed elbow and la ened
pa e n. In con as , Solid Edge has limi a ions wi hin i s shee
me al module, equi ing he manual d awing o he ske ch o
he main p ojec ion o each elbow model. LogiTRACE add esses
his limi a ion by p o iding a se o shee me al pa s ha can
be combined in o mo e complex sys ems, hough i es ic s
polygonal elbows o a maximum o 12 e ules.
2. Fla ened Cu e In eg i y: CeDG main ains he in eg i y o he
la ened cu e pa e ns h ough he 𝐿(𝜔) unc ions om
Eq. (A.1), unlike he B-spline-based cu e app oxima ions used
in CAD. While he p ecision o B-splines is adequa e o mos
enginee ing applica ions, main aining geome ic in eg i y allows
o heo e ical deduc ions ha B-splines canno suppo . The
simila accu acy be ween CeDG and CAD, despi e his di e ence,
is discussed in he ollowing sec ion.
3. Dynamic Pa ame e iza ion: CeDG suppo s dynamic pa ame e i-
za ion o bo h con inuous and disc e e a iables. Solid Edge only
p o ides dynamic pa ame e iza ion o con inuous a iables, as
s anda d me hods could no achie ed a uni e sal elbow. Log-
iTRACE does no suppo dynamic pa ame e iza ion. A ideo
demons a ing he dynamic pa ame e iza ion o CeDG models o
polygonal elbows is a ailable in he supplemen a y ma e ial.
7. Discussion and conclusion
Nume ous s udies ha e shown ha DG-based echnical d awing o -
e s a deepe unde s anding o spa ial geome y compa ed o CAD [16,
17] and se es as a o mal ool o sol ing spa ial p oblems [4,18].
While he i s asse ion is well-es ablished and suppo s he inclusion
o DG in uni e si y p og ams, he second s a emen is mo e con o-
e sial. The p o iciency o mode n CAD ools in modeling 3D sys ems
has con ibu ed o he pe cep ion ha DG is no longe ele an [19].
Howe e , g aphical p ocedu es in DG align closely wi h a body o
ma hema ical esul s ela ed o he spa ial p ope ies o 3D sys ems.
This alignmen enables solu ions o 3D geome y p ope ies h ough
deduc i e me hods, in con as o he ial-and-e o app oaches o en
equi ed in adi ional CAD ools.
P e ious s udies ha e demons a ed he eliabili y o CeDG in mod-
eling 3D sys ems, showcasing new DG-based echniques o calcula ing
he la s a es o su aces and hei in e sec ions, as well as sol ing
physical p oblems ela ed o g aphical p ope ies [10,20,21]. One ma-
jo s eng h o CeDG agains adi ional CAD is i s capabili y o use
he co pus o DG knowledge as a o mal language o 3D compu e
modeling, he eby o e coming he es ic ion o old manual echnical
d awing. CeDG le e ages he ma hema ical ounda ion o geome ical
DG p ocedu es h ough he algeb aic-g aphical pe spec i e o he DGS,
enabling he o mal deduc ion o geome ical solu ions. A pa icula ly
ele an ca ego y o h ee-dimensional geome ies a e hose exp essed
in e ms o wo-dimensional p ojec ions o ans o ms based on wo-
dimensional loci. This is exempli ied by he la ened s a e o a cu e, as
gi en by Eq. (A.1). The ield o loci compu a ion is a signi ican opic in
dynamic geome y, and ecen heo e ical de elopmen s a e ueling he
e olu ion o new gene a ions o DGS [22]. One limi a ion o CeDG in
compa ison o CAD is he lack o solid-based ope a ions like ex usion,
sweep, o lo , which a e in eg al o he p ima y wo k low in adi ional
CAD. Addi ionally, he cu en s a e o CeDG is s ill e ol ing, sugges ing
ha i may se e as a complemen a y ool o CAD.
G aphical Models 137 (2025) 101253
7
M. P ado-Velasco and L. Ga cía-Ruesgas
Fig. 9. Main p ojec ion (see Fig. 1) o a 15- e ules elbow wi h 𝛿= 190◦compu ed
h ough CeDG.
In his s udy, we de eloped and implemen ed a DG-based echnique
in CeDG ha gene a es disc e e a iable 3D models o cylind ical and
conical polygonal shee -me al elbows along wi h and hei la pa e ns.
The e iciency o hese algo i hms was e alua ed h ough he c ea ion
o six elbows, which we e also cons uc ed using Solid Edge 2024 and
LogiTRACE .14 o p o ide a compa a i e analysis o he esul s.
Sec ion 3ou lined he s a egies ha GeoGeb aSc ip algo i hms
mus ul ill o deli e au oma ic models ha dynamically espond o
changes in con inuous and disc e e a iables. The algo i hms we e used
o gene a e cylind ical and conical elbows, and we compa ed hei la
pa e ns o hose ob ained using CAD app oaches.
Ou esul s demons a e ha bo h CeDG and CAD p oduce highly
accu a e pa e ns; howe e , CeDG uniquely a oids disc epancies in
alues o he same dimension (as seen wi h LogiTRACE o cylind ical
elbows in Table 1and Solid Edge o conical elbows whe e he la ges
e ule dimension was selec ed o (R)).
Rega ding au oma ion, models in LogiTRACE and CeDG we e con-
s uc ed au oma ically based on he disc e e and con inuous pa ame e
alues. In con as , Solid Edge aced challenges in building conical
elbows unde he shee me al module, equi ing a cus om manual
app oach using he gene al pa design module. Ne e heless, we com-
pu ed he canonical p ojec ion o each elbow (Fig. 1- le ) manually
acco ding o speci ic pa ame e alues. This limi a ion is common in
gene al CAD ools; howe e , au oma ing model cons uc ion could be
achie ed by de eloping algo i hms wi hin he ool’s suppo ed p o-
g amming languages and APIs. Fo ins ance, Rojas-Sola e al. [23]
c ea ed a Visual Basic-based algo i hm o CATIA ha allows selec ion
om p e-p og ammed polygonal elbows ( anging om 2 o 8 e ules).
The supe io pe o mance o LogiTRACE o e Solid Edge o shee
me al pa s is expec ed, gi en LogiTRACE’s specializa ion in ha do-
main. Howe e , LogiTRACE es ic s he numbe o e ules o 12
o designs exceeding 360 gene a ices. While his limi a ion may be
negligible o s anda d dimensions and ole ances, CeDG p o ides e -
sa ili y by accommoda ing any numbe o e ules. Fo example, Fig. 9
illus a es he main p ojec ion o a conical elbow wi h 15 e ules o
he speci ied angles and dimensions, wi h i s la pa e n shown in
Fig. 10.
The limi a ion o Solid Edge o au oma ically gene a e he elbow
when he numbe o e ules changes, he limi a ion o LogiTRACE o
12 e ules and he ac ha enginee ing designs a ely equi e mo e
han 7 e ules jus i y limi ing he accu acy compa ison o 7 e ules.
Howe e , he beha io o LogiTRACE and CeDG up o 12 e ules was
simila o ha p esen ed he e.
The dynamic esponse o changes in he numbe o e ules and
dimensions o any alid pa ame e alue has been demons a ed and
can be con i med in he a ached ideo. This ea u e allows o explo e
Fig. 10. CeDG la pa e n o he 15- e ules elbow wi h 𝛿= 190◦shown in Fig. 9.
he model sys em design space, he iden i ica ion o a alid ange
o model pa ame e s, and he de ini ion o ini ial alues o model
a iables associa ed wi h op imiza ion asks. Fo an illus a i e s udy
conce ning he explo a ion o he design space, see [10]. The analysis o
alid model pa ame e s and op imiza ion p oblems is desc ibed in [24,
Chap e s 2 and 8].
Despi e he high deg ee o accu acy achie ed by CeDG, he calcu-
la ions o exac dimensions s uggle o mee expec a ions, pa icula ly
o 𝐿(𝜔)(Eq. (A.1)). While ela i e e o s o 𝛼1in conical elbows we e
app oxima ely 1% ( e e o Tables 2,3, and 4), he ela i e e o s o
leng hs (Li) (leng h o he segmen s) we e a ound 0.01%. The disc ep-
ancy can be a ibu ed o he ac ha 𝐿(𝜔)is de i ed om he Locus
command, which gene a es a se o poin s whose dis ibu ion does no
encompass he solu ion in a manne ha is con olled by he use in
DGS (GeoGeb a). As 𝛼1is a local pa ame e , i s accu acy is limi ed
by he dis ances among hese poin s. Howe e , he alues o Lia e
in eg al, and hei accu acy can be main ained as long as he e a e no
signi ican changes in cu a u e and he numbe o poin s is su icien ly
la ge. Since GeoGeb a lacks a buil -in command o compu e he leng h
o a Locus-based cu e, a cus om command was c ea ed using a ecen ly
de eloped GeoGeb aSc ip -based ool [25]. This command p ocesses
he locus in wo s ages o imp o e leng h compu a ion; howe e , he
inal accu acy emains cons ained by he limi a ions o he Locus
compu a ion, explaining he non-ze o ela i e e o s obse ed in he
3- e ules conical elbow.
We conclude ha he new disc e e a iable algo i hm accu acy
models complies wi h he dynamic pa ame ic beha io o CeDG mod-
els o cons uc ing polygonal shee -me al elbows and hei pa e ns.
These disc e e a iable condi ional models ep esen a signi ican ad-
ancemen in in eg a ing disc e e pa ame e s wi hin compu e -ex ended
desc ip i e geome y [26], expanding po en ial applica ions and in-
c easing he u ili y o CeDG ac oss a ious ields.
The selec ion o conical and cylind ical polygonal elbows as disc e e
a iable 3D sys ems o model in CeDG was mo i a ed by he ac ha
hese sys ems a e based on uled single-cu a u e su aces wi h nume -
ous heo e ical and p ac ical applica ions in enginee ing [27]. Among
hese applica ions, he e olu ion polygonal elbows a e pa icula ly
no ewo hy. Mo eo e , he me hodology unde lying he algo i hms
de eloped in his s udy can be applied o o he ypes o su aces. Fu u e
esea ch will ocus on de eloping a specialized b anch o Geogeb a o
CeDG ha includes his ype o algo i hm and enhanced con ol o e
he accu acy o locus calcula ions and o he dynamic objec s.
CRediT au ho ship con ibu ion s a emen
Manuel P ado-Velasco: W i ing – e iew & edi ing, W i ing –
o iginal d a , Valida ion, So wa e, Me hodology, Concep ualiza ion.
Lau a Ga cía-Ruesgas: W i ing – e iew & edi ing, W i ing – o iginal
d a , So wa e, Concep ualiza ion.
G aphical Models 137 (2025) 101253
8
M. P ado-Velasco and L. Ga cía-Ruesgas
Fig. A.11. O hog aphic iews o a 3- e ules conical elbow (le ), e olu ion cone wi h plane sec ions om which he e ules can be ob ained (cen e ), la ened ans o m o
he e olu ion cone and sec ion cu es ha sepa a e he e ules ( igh ).
Decla a ion o compe ing in e es
The au ho s decla e he ollowing inancial in e es s/pe sonal ela-
ionships which may be conside ed as po en ial compe ing in e es s:
Manuel P ado-Velasco epo s inancial suppo was p o ided by Uni-
e si y o Se ille. I he e a e o he au ho s, hey decla e ha hey ha e
no known compe ing inancial in e es s o pe sonal ela ionships ha
could ha e appea ed o in luence he wo k epo ed in his pape .
Acknowledgmen s
This wo k was suppo ed in pa by he Uni e sidad de Se illa,
Spain unde G an s 2024/00000615 and 2024/00000544.
Appendix A. Cedg algo i hm o su aces’ la ening
In CeDG, he la pa e n o any de elopable su ace can be com-
pu ed using he locus me hod. The equa ion o he la ened s a e,
𝜏, o a cu e ela ed o he su ace is gi en by:
𝜏≡𝐿(𝜔) = locus(P𝜏( 𝐷(𝜔)), 𝜔∈𝛺),(A.1)
He e, 𝐿(𝜔)is a unc ion ha ans o ms poin s, 𝜔, in he su ace’s
plane sec ion, 𝛺, o poin s, 𝑃𝜏, o he la ened cu e, while 𝑓𝐷(𝜔)is a
mapping unc ion ma ching poin s om 𝛺 o he la ened ans o med
s a e, 𝐷, o he suppo cu e . The cu e on he su ace acili a es
he anspo o gene a ices o i s la ened s a e. Fo cylinde s, is
de ined as he in e sec ion o he su ace wi h a plane pe pendicula
o he axis, esul ing in 𝐷being a s aigh line. A simila choice o
is used o e olu ion cones, whe e co esponds o a ci cula a c o
𝐷. Fo any o he conical su ace, can be de ined as i s in e sec ion
wi h a sphe e cen e ed a he cone’s e ex, which esul s in 𝐷being a
ci cula a ch.
The unc ion 𝐿(𝜔)is cons uc ed using desc ip i e geome y p oce-
du es and, as such, i does no equi e o be algeb aic. The la ened
ans o m 𝜏keeps he dynamic dependency on he pa ame e s o ,
acco ding o he CeDG app oach. A deepe analysis o Eq. (A.1) can
be ound in [21].
The applica ion o Eq. (A.1) o la en he e olu ion cone used in
gene a ing he 3- e ules conical elbow shown in Fig. A.11 is p esen ed
below o cla i y his echnique.
Once a i s gene a ix o he de elopable su ace is de ined in he
la ened domain (segmen (V)-(C) in Fig. A.11 - igh ), he gene a ix
co esponding o 𝜔is anspo ed in o he la ened domain. Fo e o-
lu ion cones, is de ined as a plane sec ion pe pendicula o he axis,
making i equal o 𝛺, as illus a ed.
Since he dis ance be ween c and 𝜔along co esponds o he
dis ance be ween (C) and (𝜔)along D, he posi ion o (𝜔)can be
calcula ed, and he associa ed gene a ix is hen anspo ed. Dis ances
V-2 and V-1 (in 3D space) a e p ese ed in he la ened s a e, enabling
(2) and (1) o be posi ioned along he gene a ix de ined by (𝜔). The
sec ion cu es ha di ide he cone in o h ee e ules can hen be
la ened as locus((2), 𝜔) and locus((1), 𝜔) (Fig. A.11 - igh ), acco ding
o Eq. (A.1).
Appendix B. Algo i hm o a disc e e a iable conical polygonal
elbow model
We now conside he cons uc ion o a gene ic conical elbow wi h
n e ules and i s la ened pa e n, wi h pa ame e s (see Sec ion 2)
linked o slide s o dynamically explo e he beha io o he model. The
diame e D2is con olled by he pa ame e D ac by means o he equa-
ion D2= D ac ⋅D1. All e ules comp ising a conical polygonal elbow
can also be ob ained om a single cone by o a ing adjacen pieces
180◦[14]. The GGB code snippe below shows he main ins uc ions
o he algo i hm ha gene a es he o hogonal iew in he canonical
posi ion.
1# Poin s in elbow a c
2Ang2dElbow = Del aElbowAB / (nFe ules - 1)
3Lis Radius = I e a ionLis (Ro a e(A cElbowPoin ,
Ang2dElbow , Cen e A cElbow), A cElbowPoin , {
Po ACen e }, nFe ules - 1)
4Lis Sphe eCen e = Sequence(In e sec (Tangen (
Lis Radius(iFe ules), A cElbow), Tangen (
Lis Radius(iFe ules+1), A cElbow)), iFe ules ,
1, nFe ules - 1)
5# Sphe es insc ibed in cone
6ejeCone = Pe pendicula Line(PAC, Pa Re )
7 Eje = Uni Vec o (ejeCone)
8 Base = Pe pendicula Vec o ( Eje)
9hal Fe uleL = Dis ance(Po ACen e ,Lis Sphe eCen e
(1))
10 End1Gen = PAC + Dis ance(Po A1 ,Po A2)/2 Base
11 End2Gen = PAC + 2 (nFe ules -1)hal Fe uleL Eje +
Dis ance(Po A1 ,Po A2)D ac/2 Base
12 Gen = Line(End1Gen ,End2Gen)
13 Lis Sphe eCen e Gen = I e a ionLis (Ps a +2
hal Fe uleL Eje , Ps a , {PAC + hal Fe uleL
Eje}, nFe ules -2)
14 Lis Sphe eTanGen = Sequence(In e sec (
Pe pendicula Line(Lis Sphe eCen e Gen(iFe ules),
Gen), Gen), iFe ules , 1, nFe ules -1)
15 adSphe es = Sequence(Dis ance(Lis Sphe eCen e Gen(
iFe ules), Lis Sphe eTanGen(iFe ules)),
iFe ules , 1, nFe ules -1)
G aphical Models 137 (2025) 101253
9
