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This is an Accep ed Manusc ip o an a icle published by Sp inge Na u e in
Ad ances in Design Enginee ing III Feb ua y 2023, a ailable a :
h ps://doi.o g/10.1007/978-3-031-20325-1_55
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In e sec ion Be ween Su aces Using Compu e Ex ended
Desc ip i e Geome y (CeDG): Applica ion o he Focal
Illumina ion o a Sphe e
Manuel P ado-Velasco and Lau a Ga cía-Ruesgas
Abs ac Compu e Ex ended Desc ip i e Geome y (CeDG) is a new app oach o compu e
modelling o 3D geome ic sys ems ha ies o o e come se e al limi a ions o cu en
CAD sys ems. A p elimina y e sion o CeDG o GeoGeb a has demons a ed ad an ages
in shee me al and mechanisms field. This pape de elops he heo e ical basis o he
Locus-based Su aces’ In e sec ion Me hod (LSIM) o CeDG and compa es i agains he
s anda d Desc ip i e Geome y echnique, h ough he calcula ion o he illumina ion o a
sphe e by a ocal ligh beam. Resul s showed ha , in opposi ion o s anda d Desc ip i e
Geome y echnique, he LSIM CeDG model combines less complexi y (geome ic
p ocedu e equi es only one i e a ion) wi h he capabili y o be ex ended o o he
pa ame e s’ alues and p ojec ions, keeping he compliance wi h geome ical
equi emen s. Accu acy me ics ha e demons a ed ha LSIM can gene a e he exac ( ue)
su ace in e sec ion cu e h ough i s p ojec ions, hanks o he geome ic in eg i y o he
unde lying dynamic geome y so wa e (GeoGeb a).
Keywo ds Desc ip i e geome y · CAD · Compu e pa ame ic g aphic modelling · Dynamic
geome y so wa e · CeDG
1 In oduc ion
Desc ip i e Geome y (DG) and i s de i ed ep esen a ion sys ems, such as dihed al
sys em, ha e defined he undamen als o g aphical ep esen a ion in enginee ing since he
second hal o he 19 h cen u y, es ablishing an impo an scien ific echnical co pus [1].
This one includes many ex s ha con inue o p o ide a solid basis o uni e si y eaching
[2–7], e en hough hei use in he p o essional field has been p ac ically eplaced by
compu e aided design so wa e (CAD) [8]. The geome y o CAD model is mainly
ep esen ed by fi cu es and su aces, such as B-splines, which p o ide high con ol and
accu acy [9, 10].
Compu e Ex ended Desc ip i e Geome y (CeDG) is a new app oach o compu e
modelling o 3D geome ic sys ems [11], which ies o add ess he limi a ions o cu en
CAD sys ems poin ed ou by some au ho s [11–13]. B iefly, CAD ools allow he cons uc ion
o i ual p o o ypes o 3D sys ems ha can be manipula ed in space and easily p ojec ed
acco ding o he chosen ep esen a ion sys em. Howe e , hey do no acili a e he c ea ion
o he model when i depends on some implici pa ame e , in addi ion o p esen ing
sho comings in he calcula ion o fla pa e ns o shee me al su aces.
The CeDG app oach combines he abili y o DG o sol e spa ial geome ic p oblems wi h
he abili y o dynamic geome y in he p ocess o building geome icalgeb aic models [14–
16]. Unlike CAD sys ems, CeDG pa ame ic models p ese e he in eg i y o cu es and
su aces. A p elimina y e sion o he CeDG app oach implemen ed on he dynamic
geome y so wa e GeoGeb a™ [17] has demons a ed i s abili y o o e come some o he
abo e limi a ions [11, 18].
The objec i e o his s udy is o p esen and analyze he me hod used in CeDG models o
ob ain he in e sec ion cu e be ween wo su aces. The usual p ocedu e in DG consis s o
defining i by means o i s p ojec ions, which a e calcula ed in u n by in e pola ion on a se
o poin s belonging o hem. The poin s a e ob ained by means o DG echniques, applied
i e a i ely. Acco dingly, he accu acy and complexi y o he esul ing cu e is p opo ional
o he numbe o poin s ob ained [7]. The CeDG app oach uses an ex ension o he DG
p ocedu es, which allows he gene a ion o a specific locus unc ion o each p ojec ion o
he in e sec ion cu e sough . The locus unc ion is au oma ically de e mined om he
sequence o geome ic-algeb aic ins uc ions associa ed wi h he calcula ion o a single
gene ic poin o he in e sec ion [11].
This pape de elops he heo y ha defines he ounda ion o he Locus-based Su aces’
In e sec ion Me hod (LSIM) and compa es his one wi h espec o he s anda d DG
p ocedu es h ough a case s udy.
2 Me hods
The s udy is de eloped acco ding o he ollowing wo kflow:
i. The heo y o LSIM is de eloped using he 2D locus unc ion o he dynamic geome y
so wa e as algeb aic suppo . Se e al examples a e succinc ly p esen ed o desc ibe wo
a ian s o his no el me hod.
ii. A case s udy has been defined o e alua e he goodness o LSIM. This is a sphe e
illumina ed by a ocal (spo ) ligh beam. The p oblem is sol ed using bo h s anda d
desc ip i e geome y p ocedu es and he LSIM-based CeDG app oach.
iii. Solu ions o he illumina ed su ace eached in he p e ious poin a e compa ed o gi e
he me hodological ad an ages and he imp o emen in accu acy o CeDG agains he
s anda d desc ip i e geome y p ocedu es.
The no el LSIM equi es he de elopmen o a heo y ha is succinc ly p esen ed in he
ollowing Sec ion. The examples selec ed o suppo he heo e ical desc ip ion include a
cone-cylinde in e sec ion and he ocal illumina ion o he sphe e ha is subsequen ly
execu ed as case s udy. The case s udy is defined wi h de ail in his Sec ion wi h he aim o
acili a ing he defini ion o me ics o he compa ison s age.
The spa ial sys em ha defines he case s udy appea s in Fig. 1 h ough he p ojec ions. The
ocal spo ha illumina es he sphe e is abo e he ho izon al plane ha con ains he
ci cula di ec ix o he ligh cone (dis ance DV), which in u n has a diame e o 2· BCono.
The diame e o he sphe e is 2· Es and i s cen e o-o’ is abo e he p e ious ho izon al
plane a he dis ance Dsph. These pa ame e s con ol comple ely he ocal illumina ion o
he sphe e h ough a conical ligh beam wi h solid angle equal o Ω.
Those main pa ame e s a e defined in Table 1, oge he wi h he de aul alue ha was used
du ing he model building p ocess. The alue o he solid angle o he conical ligh beam
may be w i en as ollows:
om which he de aul alue o Ω is 0.27 π s .
Fig. 1 Sphe e o be illumina ed om a ocal ligh beam wi h spo in - ’ and cone su ace defined by he spo as
e ex, e ical axis and ci cula di ec ix
Table 1 Main pa ame e s o he sphe e illumina ed om a ocal ligh beam
The illumina ion o he sphe e is compu ed h ough he wa ped cu e in he sphe e ha
encloses he ligh ened zone.
Conce ning he me ics, we use he nex quali a i e p ope ies o compa e he CeDG
me hodology agains ha o he s anda d desc ip i e geome y p ocedu es:
1. Complexi y o he me hod and numbe o i e a ions equi ed o ge he p ope solu ion.
2. Addi ional p ocedu es equi ed o gua an ee ha he solu ion is complian wi h
geome ical and physical equi emen s.
3. Capabili y o he me hodology o ex end he solu ion o o he pa ame e s’ alues and
p ojec ions
The sough wa ped cu e ha defines he bounda y o he ligh ened zone in he sphe e is
composed by a piece o he cone-sphe e in e sec ion and an a c o he sepa a ix ci cle
associa ed wi h he ligh spo . We e alua e he accu acy o he cone-sphe e in e sec ion,
aken ad an age ha he p ojec ion o his cu e in he e ical plane o Fig. 1 is an a c o
pa abola. The eason is ha bo h su aces a e e olu ion quad ics wi h axes defining a plane
pa allel o e ical plane [6]. The me ics ha quan i y he compa ison o accu acies a e
ex ac ed as ollows:
1. The conic associa ed o he ue (exac ) cone-sphe e in e sec ion p ojec ion is compu ed
using fi e ue poin s o his conic, in he dynamic geome y so wa e (GeoGeb a).
2. We calcula e he dispe sion o he conics iden ified by GeoGeb a when hei defini ion
poin s a e mo ed along he cone-sphe e in e sec ion p ojec ion, om he ue poin
owa ds i s ex eme poin s, defined as equidis an poin s om hei adjacen ue poin s.
The p ocedu e is cla ified in Fig. 2, which shows a ue poin and hei wo ex eme poin s
M1 and M2. The ue poin s and p ojec ion cu e ha e been ex ac ed and enla ged om
Fig. 9.
3. The ho izon al dis ance be ween any poin in he calcula ed p ojec ion cu e and he ue
p ojec ion cu e (absolu e e o cu e), E (y), is calcula ed as a unc ion o he e ical
dis ance, y.
4. The influence o dimensional pa ame e s o he sys em in E (y) is ob ained and
discussed.
Fig. 2 T ue poin (p’1p’2)in he p ojec ion o cone-sphe e in e sec ion and poin s M1 and M2, which a e
equidis an o he adjacen ue poin s (p’3p’4,p’5p’6)

3 Locus Based Su ace In e sec ion Theo y
The in e sec ion be ween wo su aces p oduces a cu e in space ha in gene al will no be
fla . The calcula ion o hese cu es is a ubiqui ous p oblem in science and enginee ing,
including he 3D defini ion o any indus ial pa o sys em, he spa ial analysis o
biomechanical sys ems, o he s udy o a ocal ligh ing sys em, o name a ew examples.
The gene al echnique o calcula ing he in e sec ion cu e be ween su aces in
desc ip i e geome y is based on he use o a se o auxilia y su aces, defined in such a way
ha he encoun e be ween any one o hem and he wo da a su aces p oduces a pai o
simple cu es ha will be cu by belonging o he same auxilia y su ace, o p o ide poin s
o he in e sec ion cu e sough . This p ocedu e mus be epea ed un il enough poin s is
ob ained o define he cu e wi h he equi ed
accu acy.
The de ails conce ning he ypology o auxilia y su aces o be used depending on he da um
su aces a e pa o he la ge body o exis ing knowledge in desc ip i e geome y. One
ea u e o no e in his gene al echnique is ha he poin s ha mee each auxilia y su ace
a e exac ( ue). This p ope y p o ides some con ol o e he accu acy o he esul ing 3D
cu e.
This p ocedu e on pape is i e a i e and ime-consuming and equi es ob aining no able
poin s o he cu e, such as hose belonging o con ou lines and auxilia y bounda y
su aces, which ensu e ha i mee s a se o minimum quali y c i e ia. As he con ou lines
depend on he di ec ion and ype o p ojec ion, he poin s o con ac wi h he desi ed cu e
also depend on he di ec ion and ype o p ojec ion, which is an addi ional sho coming o
he echnique.
The abo e limi a ions a e inhe en o he manual echnique o pe o ming he p ocedu es,
bu no o he desc ip i e geome y i sel . Thus, he gene al echnique o calcula ing he
in e sec ion cu e can be o mula ed as ollows. Calling σ he plane cu e esul ing om
he p ojec ion o he in e sec ion cu e C on o a plane o in e es , σ will be defined by he
se o all i s poin s pσ. On he o he hand, i he auxilia y su aces ha gi e he poin s pσ o
σ a e exp essed as S(ω), whe e he pa ame e ω in a se Ω iden ifies each o he possible
su aces, hen he cu e σ can be exp essed as he locus o he poin s pσ (ω) o all ω. Tha
is:
whe e L (ω) is a unc ion ha ep esen s he cu e pa ame ically, p o iding poin s in fla
space (plane on which C is p ojec ed) and aking ω ∈ Ω as a pa ame e . The unc ion L exis s
and will be con inuous unde ce ain usual con inui y assump ions on S(ω) and on da a
su aces. Unde hese condi ions, CeDG allows o cons uc he unc ions L i associa ed
wi h he p ojec ions o he in e sec ion cu e C on o he planes o in e es deno ed by he
index i (σi), using he Geogeb a locus command.
This command is mainly employed in wo o ms: wi h a e e ence poin , locus (pσ, p e ) and
wi h a pa ame e , locus (pσ, ). In he fi s o m, he se Ω is defined by a e e ence locus
wi h poin s p e , while he second o m se s Ω as an in e al in R defining he alues o . To
cla i y he cons uc i e p ocedu e in each case, wo examples a e discussed below.
3.1 Ω Defined as a Re e ence Locus
The fi s example uses he cylinde -cone bi e o Fig. 3. The in e sec ion be ween hese
su aces is a wa ped cu e ha can be ob ained on pape by he gene al echnique o
in e sec ion be ween su aces o desc ip i e geome y, bu he p ocess is e y sensi i e o
d awing e o s due o he na u e o he cu e. Figu e 3 shows h ee p ojec ions o such a
cu e, ob ained using he LSIM-based CeDG model.
Auxilia y su aces used o his sys em a e planes defined by he e ex o he cone (V) and
one o he gene a ixes o he cylinde . Since he cylinde is pe pendicula o he p ofile
plane, i s p ojec ion in his plane coincides wi h i s ci cula di ec ix. Figu e 3 shows a
gene ic auxilia y plane pe pendicula o he p ofile plane and defined by he line passing
h ough he poin s " and 3"4" in he p ofile. This auxilia y plane allows ob aining wo poin s
o he in e sec ion cu e sough , defined by 3 and 4 in he ho izon al p ojec ion, and 3’ and
4’ in he e ical one.
To build he LSIM-based CeDG model o his sys em, we s a by calcula ing he poin s 3
and 4 in 3D s a ing om he auxilia y plane men ioned abo e, defined in e ms o a ee
poin on he e e ence locus. The ee poin is 3"4" (p e ) and he e e ence locus is he a c
o ci cle inside he cone be ween he bounda y poin s 1" and 2" in he p ofile, as shown in
Fig. 3. This poin can be c ea ed g aphically by clicking on he a c a e selec ing he poin
ool (Poin (A c) command). Once c ea ed, i can be d agged wi h he mouse along he a c.
Once he chosen plane is placed in a com o able si ua ion o wo k (p e e ably an
in e media e one), he gene al in e sec ion echnique is applied, which will p o ide he
poin s 3 and 4 belonging o he in e sec ion cu e sough , symme ically placed on he le
and igh side o he plan and ele a ion iews.
Acco ding o he p e ious analysis, he a c o ci cle is heΩlocus ha defines he
p ojec ions o he in e sec ing cu e acco ding o Eq. (2). Thus, he ho izon al p ojec ion o
he cu e, σ1, will be:
Fig. 3 LSIM based CeDG model o cylinde -cone bi e
whe e 3 and 4 deno e he ho izon al p ojec ions o he poin s ob ained h ough he auxilia y
plane. In he same way he e ical p ojec ion will be:
whe e 3’ and 4’ a e he e ical p ojec ions o he poin s ob ained h ough he auxilia y
plane.
Equa ions (3) and (4) define he cylinde -cone in e sec ion cu e by means o algeb aic
en i ies c ea ed in he p ocess o de e mining he pai o poin s associa ed wi h he gene ic
auxilia y plane, hus defining an exac ( ue) and comple e solu ion. The in e sec ion cu e
compu a ion p ocess in CeDG only equi es he use o a gene ic auxilia y su ace and does
no need o calcula e poin s ela ed o bounda ies o o he s o ob ain he comple e solu ion
o he sys em. The L unc ion ha defines each p ojec ion is specific o he modeled sys em
and can be composed o se e al lea es, as in his example. Each lea has been ep esen ed
in a diLe en colo in Fig. 3.
I is ema kable o no e ha , in ag eemen wi h he small dis ance be ween poin s 1” and
2”, almos he en i e cylinde lies inside he cone. A sligh shi o he cylin-d ical di ec ix o
he le is enough o con e he bi e in a pene a ion cha ac e ized by wo unconnec ed
wa ped cu es. Each cu e will be defined in such a case o a single lea . The LSIM-based
CeDG model may au oma ically add esses his change in he na u e o he in e sec ion,
al hough he analysis exceeds he scope o his pape .
3.2 Ω Defined as In e al in R
The second example shows he in e sec ion o a conical su ace wi h a sphe e, a si ua ion
ha can appea in diLe en si ua ions o echnical in e es . One o hem is he illumina ion
o a sphe e om a ocal ligh beam, which was defined as case s udy in Me hods sec ion,
and i is shown in Fig. 1.
When sol ing his sys em, i is necessa y o calcula e he in e sec ion o he ligh cone wi h
he illumina ing sphe e, o ob ain a po ion o he cu e ha limi s he illumina ed a ea. The
cu e is comple ed by a po ion o he sepa a ix ci cle defined om he ligh sou ce, which
dis inguishes he sphe ical a ea ha can be illumina ed om ha which canno . The
sphe ical illumina ed a ea is he e o e defined as he a ea ha can be illumina ed (in e nal
o he sepa a ix ci cum e ence) and ha which is eLec i ely illumina ed by he eLec i e
ligh cone (in e sec ion o ligh cone sphe e).
The CeDG solu ion o his sys em is p esen ed in Fig. 4. The bounda y cu e o he
illumina ed su ace, gi en by i s p ojec ions, is gi en as he sum o a po ion o he sepa a ix
cu e (fla ) connec ed by poin s P1 and P2 wi h he cone-sphe e in e sec ion cu e
(wa ped).
F om he comple e cons uc ion p ocess, we a e in e es ed in he LSIM-based solu ion o
he cone-sphe e in e sec ion cu e. Choosing ho izon al planes as auxil-ia y su aces,
S(ω), whe e ω is some eal alue pa ame e ha allows o iden i y hem, each o hem
gene a es a ci cle in he cone and ano he in he sphe e.
These ci cles may encoun e each o he o p oduce one ( angen ci cles) o wo (secan
ci cles) poin s o he in e sec ion cu e sough . Se ing as ω pa ame e he e ical dis ance
o he ho izon al plane in ela ion o he poin I1 (see Fig. 4), he in e sec ion cu e will be
gi en by i s p ojec ions σ1 (ho izon al) and σ2 ( e ical), de i ed om Eq. (2) o = ω ∈Ω,
beingΩan in e al in R gi en by [0, Lim2] wi h Lim2 equal o he e ical dis ance o poin I2
wi h espec o I1.
The ho izon al p ojec ion consis s o wo lea es:
Fig. 9 Conic dispe sion in cone-sphe e in e sec ion om in e pola ion spline-based solu ion
Fig. 10 Conic associa ed o he cone-sphe e in e sec ion ob ained by he LSIM
As expec ed, he e o s o he ue fi e poin s pe aining o he e ical p ojec ion o he
cone-sphe e in e sec ion a e null. Thei e ical dis ances a e ma ked in he abscissa axis
o Fig. 11. The maximum absolu e e o is 0.35 mm (0.71 % wi h espec o he ho izon al
dis ance be ween I’1 and I’2).

Al hough he e o s a e no la ge, in ag eemen wi h p ojec ions p esen ed in Fig. 8 ( igh ),
hey could inc ease in o he p ojec ions. P e ious sho comings in he geome ical and
physical compliance, oge he wi h his dimensional e o , educe he capabili y o he
s anda d desc ip i e geome y echnique o ex end he solu ion o o he pa ame e s’ alues
and p ojec ions, in opposi ion o LSIM-based CeDG.
Fig. 11 Ho izon al absolu e e o (cm)o he e ical p ojec ion o he in e pola ion spline-based cone-sphe e
solu ion, as a unc ion o he e ical dimension
We ha e finally e alua ed he influence o he model’s pa ame e s on he E (y) dimensional
e o o he in e pola ion spline-based cone-sphe e solu ion o he same p ojec ion. The
Fig. 12 shows how he ligh beam ocus adius, defined by BCono pa ame e , aLec s o
E (y) when i goes down om 70 mm o 60 mm ( ed ajec o- ies), and when i goes up om
70 mm o 80 mm (blue ajec o ies). The maximum absolu e e o s we e 0.32 mm (0.72 %)
o BCono = 60 mm and 0.38 mm (0.74 %) o BCono = 80 mm. The alues o he e ical
dis ances ma ked in he abscissa axis e e o BCono = 70 mm.
The dimensional e o beha io is simila o he o he pa ame e s. The e o e, al hough he
in e pola ion spline-based cone-sphe e in e sec ion solu ion has limi a- ions ha diLicul
he ex ension o o he pa ame e s’ alues and p ojec ions when he compliance wi h
geome ical and physical equi emen s is kep , he model is obus agains pe u ba ions in
hese pa ame e s.
The diLe ence be ween he conic eccen ici y (c/a) o he e e ence pa abola (1.00028) and
he ue pa abola (1) is associa ed o he accu acy o GeoGeb a o sol e he 5-equa ions
sys em defined by he 5 poin s ha define he pa abola coeLicien s. The ela i e e o is
0.028 %. Al hough i is a small e o , i aLec s o he capabili y o CeDG o con ol he
in eg i y o geome y (c/a > 1 is a hype bola) and he accu acy and hus mo e esea ch is
equi ed o imp o e his issue.
Fig. 12 Influence o he ligh beam ocus adius on he ho izon al absolu e e o o he in e pola ion spline-
based cone-sphe e in e sec ion (60–70 mm ed; 70–80 mm blue, inc emen s o 1mm)
5 Conclusions
CeDG is a no el app oach o compu e modelling o 3D geome ic sys ems based on desc ip i e geome y,
which o e comes se e al limi a ions o cu en CAD sys ems. This s udy has p esen ed fi s he heo y
unde lying he Locus-based Su aces’ In e sec ion Me hod (LSIM), and second a case s udy abou he ocal
illumina ion o a sphe e ha allowed compa ing LSIM agains s anda d desc ip i e geome y p ocedu es.
The ou comes demons a e ha he complexi y o he LSIM CeDG solu ion, in e ms o he numbe o
i e a ions equi ed, is lesse han ha o he s anda d desc ip- i e geome y p ocedu es. In addi ion, he
accu acy o LSIM CeDG was much highe han ha o s anda d p ocedu es. In ac , he accu acy was only
limi ed by he nume -ical p ecision o GeoGeb a, since he compu ed in e sec ion cu e is o mula ed h ough
algeb aic objec s ha keep he geome ic in eg i y o he cu e p ojec ions. We may conclude ha CeDG
p o ides he ue cone-sphe e in e sec ion cu e.
Finally, he LSIM CeDG solu ion kep he compliance wi h geome ical equi e-men s when he p ojec ion and
pa ame e s’ alues o he model we e changed, in opposi ion wi h he s anda d echnique ha equi es
addi ional geome ical es ic ions.
CeDG is cu en ly in a s a ing e sion ha will e ol e o include he au oma- ion and imp o emen o many
geome ic p ocedu es o acili a e i s diLusion o p o essional and academic domains.
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