Jou nal o Geodesy (2024) 98:47
h ps://doi.o g/10.1007/s00190-024-01854-1
ORIGINAL ARTICLE
A no el geome ic me hod based on con o mal geome ic algeb a
applied o he esec ion p oblem in wo and h ee dimensions
Jo ge Ven u a1·Fe nando Ma inez1·F ancisco Manzano-Aguglia o1·Aleš Ná a 2·Ja osla H dina2·
Ahmad H. Eid3·F ancisco G. Mon oya1
Recei ed: 5 Oc obe 2023 / Accep ed: 23 Ap il 2024 / Published online: 27 May 2024
© The Au ho (s) 2024
Abs ac
This pape in oduces a no el me hod o sol ing he esec ion p oblem in wo and h ee dimensions based on con o mal
geome ic algeb a (CGA). Ad an age is aken because o he cha ac e is ics o CGA, which enables he ep esen a ion o
poin s, lines, planes, and olumes in a uni ied ma hema ical amewo k and o e s a mo e in ui i e and geome ic unde s anding
o he p oblem, in con as o exis ing pu ely algeb aic me hods. Se e al nume ical examples a e p esen ed o demons a e he
e icacy o he p oposed me hod and o compa e i s alidi y wi h es ablished echniques in he ield. Nume ical simula ions
indica e ha ou ec o geome ic algeb a implemen a ion is as e han he bes -known algo i hms o da e, sugges ing ha
he p oposed GA-based me hods can p o ide a mo e e icien and comp ehensible solu ion o he wo- and h ee-dimensional
esec ion p oblem, pa ing he way o u he applica ions and ad ances in geodesy esea ch. Fu he mo e, he me hod’s
emphasis on g aphical and geome ic ep esen a ion makes i pa icula ly sui able o educa ional pu poses, allowing he
eade o g asp he concep s and p inciples o esec ion mo e e ec i ely. The p oposed me hod has po en ial applica ions in
a wide ange o o he ields, including su eying, obo ics, compu e ision, o na iga ion.
Keywo ds Resec ion p oblem ·T iangula ion ·Snellius–Po heno ·Con o mal geome ic algeb a
BF ancisco Manzano-Aguglia o
[email p o ec ed]
Jo ge Ven u a
[email p o ec ed]
Fe nando Ma inez
[email p o ec ed]
Aleš Ná a
na[email p o ec ed].cz
Ja osla H dina
[email p o ec ed].cz
Ahmad H. Eid
[email p o ec ed]
F ancisco G. Mon oya
[email p o ec ed]
1Depa men o Enginee ing, Uni e si y o Alme ia, C a.
Sac amen o s/n, 04120 Alme ía, Alme ía, Spain
2Ins i u e o Ma hema ics, Facul y o Mechanical Enginee ing,
B no Uni e si y o Technology, Technická 2896/2, K álo o
Pole, 61669 B no, Czech Republic
3Elec ical Enginee ing Depa men , Po Said Uni e si y,
Po -Said, Egyp
1 In oduc ion
The esec ion p oblem, also known in su eying as he
Snellius–Po heno (SP) o he in e se in e sec ion p oblem,
in ol es calcula ing he posi ion o an unknown poin P(also
called a s a ion) using he posi ions o h ee known poin s A,
B, and C, and ela i e angula measu emen s om P.I isa
ele an p oblem no only in geodesy and su eying, bu also
in o he disciplines such as obo pa h planning (Masselli
and Zell 2014), posi ioning (Pie lo and Van D oogenb oeck
2014), na iga ion (Pe ei a e al. 2018) o compu e g aphics
(Mazahe i and Habib 2015), and can be sol ed bo h geome -
ically and algeb aically. The solu ion o he wo-dimensional
p oblem has been known o se e al cen u ies and has nume -
ous a ian s [almos 500! acco ding o Bock (1956)]. The
h ee-dimensional p oblem is much mo e in ica e and only
complex and sophis ica ed algeb aic solu ions a e known.
The wo- and h ee-dimensional con igu a ions a e illus a ed
in Fig. 1.
123
47 Page 2 o 21 J. Ven u a e al.
Fig. 1 Rep esen a ion o he
esec ion p oblem in 2D (le )
and 3D ( igh )
1.1 Mo i a ion
The de e mina ion o he posi ion o an obse e based on
angula measu emen s om known poin s is o in e es in
se e al disciplines, such as su eying, compu e g aphics,
op ics, and obo ics. T adi ionally, solu ions o his p ob-
lem ha e elied on hea ily algeb aically loaded me hods,
which can be complex and challenging o comp ehend. Fu -
he mo e, hese me hods do no always p o ide an in ui i e
unde s anding o he geome ic ela ionships in ol ed in he
p oblem.
In bo h wo-dimensional (2D) and h ee-dimensional (3D)
p oblems, he e is a need o a mo e geome ic o g aphical
app oach ha simpli ies he s udy o he esec ion p oblem.
Al hough he e a e exis ing g aphical me hods o sol ing
he p oblem in 2D ha ha e been known o some ime,
hei algeb aic implemen a ion can be qui e cumbe some,
hinde ing hei widesp ead adop ion. When he p oblem is
app oached om a geome ic pe spec i e, a be e unde -
s anding o he unde lying s uc u es and ela ionships can
be achie ed, making he p oblem mo e accessible o a wide
ange o esea che s and p ac i ione s. This could po en ially
lead o no el applica ions and ad ancemen s in he ela ed
a eas men ioned abo e.
In ligh o he exposed ideas, he main mo i a ion behind
his pape is o de elop a no el geome ic me hod based on
con o mal geome ic algeb a (CGA) o add ess he esec ion
p oblem in wo- and h ee-dimensional ways. By le e ag-
ing he uni ying p ope ies o CGA o ep esen geome ic
p imi i es wi hin a single ma hema ical amewo k, we aim
o p o ide a mo e in ui i e and geome ic unde s anding o
he p oblem, simpli ying i s s udy, and pa ing he way o
u he applica ions and ad ancemen s in he ield.
1.2 Backg ound and li e a u e o e iew
The esec ion p oblem has been ex ensi ely s udied in he
li e a u e, wi h a ious algeb aic and geome ic o g aphical
me hods p oposed o i s solu ion. I appea s ha ancien
G eeks, such as Hippa chus o P olemy, al eady s udied his
p oblem in he con ex o as onomy, al hough he i s pe son
o sol e he p oblem in he con ex o su eying was he
Du ch ma hema ician Willeb o d Snel an Royen (known
Table 1 Classi ica ion o main me hods o sol e he 2D esec ion p ob-
lem (see Pie lo and Van D oogenb oeck 2014)
Me hod G oup Yea
Snellius T igonome ic 1617
Collins Geome ic/G aphical 1671
Kaes ne -Bu kha d T igonome ic 1801
McGillem Geome ic 1988
Cohen and Koss Geome ic 1992
Madsen and Ande sen T igonome ic 1998
Sanchiz e al I e a i e 2004
Es e es e al Geome ic 2006
Eas on and Came on T igonome ic 2006
Tsukiyama Geome ic 2009
Fon -Llagunes Geome ic 2009
Deko I e a i e 2012
Ligas Geome ic 2013
Pie lo e al Geome ic 2014
Wille ding O he 2020
Tiens a O he Unknown
Cassini Geome ic/T igonome ic Unknown
as Snellius) in 1617 (W eede 2007). He achie ed he goal
by using geome ic and algeb aic me hods mainly based on
igonome y. This same p oblem was add essed in 1671 by
John Collins in his wo k Philosophical T ansac ions. Collins
con ibu ed signi ican ly o he opic by p esen ing a new and
elegan geome ical solu ion, which in ol es he use o jus
one ci cle and an auxilia y poin . In 1692, Lau en Po heno ,
who was wo king on he de ini ion o he me idian no h o
Pa is, p esen ed a pape on he subjec . Howe e , acco ding
o McCaw (1918) and o he s, Po heno did no con ibu e
any hing new o he solu ion o he p oblem and all he did
was publish he wo ks o Snellius and Collins unde his own
name. O he au ho s who ha e s udied his issue a e shown
in Table 1.
In he con ex o geodesy and su eying, he esec-
ion p oblem has been a undamen al p oblem o cen-
u ies. The p oblem has been add essed om a ious pe -
spec i es, including algeb aic (Awange e al. 2010), geo-
me ic/g aphical (Masselli and Zell 2014), and nume ical
123
A no el geome ic me hod based on con o mal geome ic algeb a applied o he esec ion p oblem… Page 3 o 21 47
(Deko 2012) me hods. Algeb aic solu ions o he p ob-
lem a e well es ablished and ha e been ex ensi ely s udied.
Howe e , hese me hods o en in ol e complex algeb aic
manipula ions and do no p o ide any in ui i e geome ic
unde s anding o he p oblem.
The wo-dimensional p oblem has been sol ed using
app oaches such as g aphical me hods (Bock 1956), ana-
ly ical geome y (Bil 1992), ma ix me hods (Bäh 1991),
and algeb aic me hods based on Syl es e esul an s (Awange
and G a a end 2002), G oebne bases (Awange 2002), and
o he elimina ion echniques (S u m els 2002). Awange and
G a a end (2005) p o ided a de ailed o e iew o se e al
algeb aic echniques o sol ing 2D and 3D esec ion.
The h ee-dimensional e sion is mo e complex, wi h
mos solu ions elying on in ica e algeb aic me hods. G une
o iginally o mula ed he p oblem in 1841 and de i ed qua -
ic equa ions o de e mine he unknown dis ances (G une
1841). Since hen, nume ous p ocedu es ha e been de eloped
o op imise G une ’s o mula ion and educe he compu a-
ional s eps in ol ed (Awange and G a a end 2005; Fischle
and Bolles 1981). Algeb aic echniques such as G oebne
bases (Awange and G a a end 2003a), polynomial esul-
an s (Awange and G a a end 2003b), and linea homo opy
(Paláncz e al. 2010) ha e been e ec i ely applied o sol e he
3D p oblem. Howe e , geome ic solu ions o he esec ion
p oblem ha e been less explo ed. The geome ic o g aphi-
cal app oach p o ides a mo e in ui i e unde s anding o he
p oblem and can be mo e easily isualised.
In ecen yea s, he e has been a g owing in e es in
he applica ion o geome ic algeb a (GA) o sol e se e al
geome ic p oblems. GA p o ides a uni ied ma hema ical
amewo k o ep esen ing and manipula ing geome ic
objec s, making i a powe ul ool o geome ic compu a ions
(Do s e al. 2010). The applica ion o GA o he esec ion
p oblem was i s p oposed by Smi h (2023b,a).
Howe e , he applica ion o GA o he esec ion p ob-
lem is s ill in i s ea ly s ages, and he e is s ill a long way
o go. In pa icula , he applica ion o con o mal geome -
ic algeb a (CGA) o he esec ion p oblem has no been
ully explo ed. CGA ex ends GA by inco po a ing he con-
cep o con o mal ans o ma ions, which can p o ide a mo e
powe ul and lexible amewo k o geome ic compu a ions
(Hes enes and Sobczyk 2012; Do s e al. 2010). CGA p o-
ides a uni ied amewo k o handling poin s, lines, planes,
ci cles, sphe es, and o he geome ic en i ies. Each en i y
has a unique ep esen a ion, and he ela ionships be ween
en i ies co espond o algeb aic ela ionships be ween hei
CGA ep esen a ions. This elimina es he need o coo di-
na e ans o ma ions when ansi ioning be ween e e ence
ames.
Came on and Lasenby (2008) showed ha CGA subsumes
p ojec i e geome y and is mo e compu a ionally e icien
han ma ix me hods. CGA has ound applica ions in com-
pu e ision (Wa eham e al. 2004), obo ics (Zaplana e al.
2022), and o he geome ic compu ing p oblems (Hi ze
e al. 2022). Howe e , i s po en ial o sol ing esec ion- ype
p oblems in geodesy and su eying emains ela i ely unex-
plo ed. This pape seeks o ill he oid by in oducing a new
geome ic solu ion based on CGA o he 2D and 3D esec-
ion p oblem.
1.3 Con ibu ions
In his pape , a new me hod o sol ing he esec ion p ob-
lem using he ma hema ical amewo k geome ic algeb a in
2D and 3D dimensions is p oposed. The me hod p o ides
a simple solu ion based on pu ely geome ic and g aphical
p inciples. Speci ically, he 3D e sion o he p oblem is ho -
oughly analysed in de ail. Speci ic no el con ibu ions a e he
ollowing:
•A no el con o mal geome ic algeb a (CGA)-based
me hod is p esen ed o sol e he wo- and h ee-
dimensional esec ion p oblem, p o iding a mo e in u-
i i e and uni ied geome ic app oach.
•The p oposed me hod is compa ed wi h es ablished ech-
niques, and i s ad an ages and e icacy a e demons a ed
h ough nume ical examples.
•The po en ial applica ions o he me hod in se e al ields
such as compu e g aphics, op ics, and obo ics a e high-
ligh ed, emphasising i s e sa ili y, and pa ing he way
o u u e ad ancemen s in geome ic esea ch.
•Se e al algo i hms ha e been de eloped ha p o ide be -
e esul s han he bes -known algo i hms o da e om a
compu a ional pe spec i e.
1.4 Ou line
The emainde o he a icle is o ganised as ollows. Sec ion 2
p o ides an in oduc ion o geome ic algeb a and con o mal
geome ic algeb a. Sec ion3 e isi s he esec ion p oblem
and e iews adi ional me hods. Sec ion 4p esen s exis ing
and new GA-based me hods. Sec ion 4p esen s he p o-
posed me hod based on CGA o sol e he esec ion p oblem.
Sec ion5p o ides se e al applica ion examples o demon-
s a e he e ec i eness o he p oposed me hod, along wi h
benchma ks o compu a ional e iciency. Sec ion 6p o ides
an e o and unce ain y analysis o he p oposed me hods.
Finally, Sec . 7concludes he a icle wi h a summa y and
some sugges ions o u u e wo k.
2 Basics concep s in geome ic algeb a
Geome ic algeb a is a ma hema ical amewo k o ep e-
sen ing geome ic objec s and ans o ma ions in a uni ied
123
47 Page 4 o 21 J. Ven u a e al.
way (Hes enes and Sobczyk 2012). GA ex ends he alge-
b a o ec o s o include o he geome ic objec s such as
poin s, lines, planes, and olumes. GA p o ides a powe ul
ool o sol e geome ic p oblems and has applica ions in a
wide ange o ields, including elec ical enginee ing (Mon-
oya e al. 2019,2021), compu e ision (H dina and Ná a
2017), obo ics (H dina e al. 2017), and o he enginee ing
ields [see Hi ze e al. (2022) and e e ences he e in].
GAs a e used mainly in si ua ions whe e Euclidean ans-
o ma ions play a signi ican ole. A simple way o in oduce
GAs is o unde s and hem h ough mo e amilia ools such
as complex numbe s o qua e nions (indeed, hey a e subalge-
b as o GA). Fo example, implemen ing Euclidean o a ions
using qua e nions is essen ial o imp o e compu a ional capa-
bili ies and adop an objec -o ien ed app oach. We can see
qua e nions Has a na u al ex ension o he complex numbe s
Cin he o m
z=a+bi+cj+dk(1)
whe e a,b,c,d∈Rand he p oduc s o he basis elemen s
i,j,ksa is y he mul iplica ions ules i2=j2=k2=−1
and ij =−ji =k. The qua e nion Im(z)=bi+cj+dk
is called he imagina y pa o z, and Re(z)=ais called he
eal pa o z. No e he use o bold le e s o qua e nions and
egula on o eal numbe s.
The ope a ion o o a ing an objec by an angle θa ound
he axis n=nxi+nyj+nzkcan be ep esen ed by he
qua e nion
R=eθ
2(nxi+nyj+nzk)
=1
2cos θ+nxi+nyj+nzksin θ(2)
ac ing on he ec o q=xi+yj+zk. No e ha ||n|| =
√n¯
n=1, whe e he ba deco a ion s ands o qua e nionic
conjuga ion. The ep esen a ion is done in a simila way as in
he case o complex numbe s. The di e ence is ha qua e -
nions a e no commu a i e and hus hey ac ia he so-called
sandwich p oduc
RqR−1=1
||R||Rq ¯
R(3)
So, we a e wo king in a ou -dimensional linea space. I
we wan o ealise also he ansla ions, we ha e o ex end
he algeb a by ano he dimension using an elemen such
as 2=0 and i=j=k=0. This algeb a is usually
called dual qua e nions. Fo mo e on he use o qua e nions
in gene al enginee ing opics, see Selig (2005).
On he o he hand, he wedge ope a ion on a ec o space
allows us o wo k wi h linea subspaces. A line can be cha -
ac e ised by a ec o ,soxbelongs o a line i and only i
x∧=0. The wedge o wo ec o s hen cha ac e ises he
plane in he same way. A ec o space closed in he wedge
ope a ion is called a G assmannian algeb a. The combina ion
o hese wo concep s leads o he no ion o GA.
2.1 Euclidean ec o GA
Vec o geome ic algeb a (VGA) is pe haps one o he sim-
ples GAs. Fo he wo-dimensional case, he VGA (G2)isa
G assmannian algeb a based on wo o hono mal basis ec-
o s (σ1,σ2) oge he wi h a bilinea ope a ion known as
geome ic p oduc sa is ying he ollowing iden i ies:
σ1σ2=σ1·σ2+σ1∧σ2=σ1∧σ2
σ1·σ2=0
σ2
1=σ2
2=1(4)
The e m σ1∧σ2is known as bi ec o (σ12 o sho ).
Geome ically, a bi ec o ep esen s an o ien ed plane seg-
men spanned by he wo ec o s. We can use bi ec o s o
ep esen o a ions in he ollowing way. The collec ion o
bi ec o s σ1∧σ2 o ms a one-dimensional ec o space ha
is closed unde mul iplica ion. We can hen gene a e o a ions
by applying he exponen ial map o bi ec o s. Fo example,
he exponen ial o he bi ec o
e1
2θσ12 =1
2(cos θ+sin θσ12)(5)
is used as in (3) o pe o m a o a ion by an angle θin
he plane spanned by σ1and σ2. In his way, he g oup
o o a ions gene a ed by he bi ec o s is ma hema ically
equi alen o he g oup o uni a y complex numbe s eiθ,
which also ep esen o a ions in he complex plane (no e ha
(σ1∧σ2)2=−1). Howe e , bi ec o s p o ide a mo e in u-
i i e geome ic ep esen a ion o o a ions di ec ly in ec o
space.
Simila ly, we can in oduce he G3algeb a o he pu -
poses o easoning in he 3D space. The algeb a G3is based
on h ee gene a o s σ1,σ2and σ3 oge he wi h a geome ic
p oduc de ined by he ollowing iden i ies:
σiσj=σi∧σjwhe e i= j
σ2
i=1 whe e i=1,2,3(6)
In Sec .4.1, i will be shown how o use he VGA-based
me hod o sol e a 2D e sion o he esec ion p oblem. This
p ocedu e is, in ac , a use o GA G2.
2.2 Con o mal geome ic algeb a
The goal he e is o c ea e a model o Euclidean geome y.
Speci ically, geome y whose symme y g oup con ains he
123
A no el geome ic me hod based on con o mal geome ic algeb a applied o he esec ion p oblem… Page 5 o 21 47
Euclidean symme ies ( o a ions, ansla ions, e c.). Fo ha
pu pose, a nondegene a e quad a ic o m will be chosen, hus
ob aining he con o mal geome ic algeb a (CGA). In he
case o CGA o a wo-dimensional Euclidean space, a GA
o signa u e (3,1)is ob ained [also known as Compass Rule
Algeb a CRA, see Hildenb and (2018)], wi h basis ec o s
σ1,σ2,σ+and σ−such ha
σ2
i=1,i∈{1,2,+},σ2
−=−1(7)
σiσj=−σjσi,i= j,i,j∈{1,2,+,−} (8)
Fo ma hema ical con enience, i is ad isable o de ine
wo new basis ec o s
σ0=1
2(σ−−σ+)(9)
σ∞=σ−+σ+,(10)
wi h p ope ies
σ2
0=σ2
∞=0 and σ0·σ∞=−1 (11)
Vec o σ0is known o ep esen he Euclidean poin a he
o igin o he coo dina e sys em, and ec o σ∞ ep esen s
he poin a in ini y. Using he ules abo e, i can be p o ed
ha any Euclidean poin Xcan be mapped o he CGA ec o
space as
X−→ x=σ0+(x1σ1+x2σ2)+1
2(x2
1+x2
2)σ∞
(12)
In “Appendix A.1”, explici calcula ions ha jus i y ou
choices can be ound. As in Sec .2.1, he ex ension o highe
dimensions is s aigh o wa d by adding he elemen σ3.This
will make G3appea ins ead o G2, and qua e nions will
appea in he bi ec o s.
In he no a ion used h oughou his pape o ensu e cla i y,
egula on indica es eal numbe s, bold ace deno es ec o s
in he CGA space, bold ace wi h an o e head a ow (e.g.

x) ep esen s ec o s in he Euclidean space, and uppe case
le e s in bold ace indica e mul i ec o s in CGA.
The use ulness o CGA and CRA is based on he ac ha
he dis ance be ween Euclidean poin s is encoded in he scala
p oduc . The poin s Xand Ya e ep esen ed by he Euclidean
ec o s 
xand 
y, espec i ely. The mapping de ined in (12)
maps hese Euclidean poin s ( ec o s) o ec o elemen s x
and yin he CGA space.
As shown in “Appendix A.2”, he scala p oduc be ween
wo CGA poin s codes he dis ance:
x·y=−1
2
x−
y2(13)
Thus, he Euclidean poin X( ep esen ed by he Euclidean
ec o 
x) lies on he sphe e (ci cle) Swi h cen e in poin C
and adius i and only i i sa is ies he iden i y x·c=−1
2 2
in he con o mal space. Since x·σ∞=−1 o each poin ,
his iden i y may be w i en as
x·c+1
2 2=x·c−1
2 2(x·σ∞)
=x·c−1
2 2σ∞=0(14)
Thus, he ec o elemen c−1
2 2σ∞ ep esen s he sphe e
Sin he CGA space. I should be emphasised ha wi hin
CGA, he e exis wo dis inc i e me hods o e e ence he
iden ical geome ic en i y: IPNS and OPNS (see “Appendix
A.3”). IPNS is supe io o ans o ma ions and in e sec ions,
whe eas OPNS is ad an ageous o blending and mo phing
asks. I is c ucial o be awa e o he ep esen a ion we a e
ope a ing in; howe e , ansi ioning be ween ep esen a ions
can be achie ed seamlessly using he dual ope a o (), as
desc ibed in “Appendix A.3”.
3 Re isi ing he esec ion p oblem
In he con ex o geodesy and su eying, he esec ion p ob-
lem plays an impo an ole in de e mining he posi ion o an
obse e based on angula measu emen s om h ee known
e e ence poin s. O e he yea s, a ious app oaches ha e
been de eloped o add ess his p oblem, anging om g aphi-
cal o geome ical me hods o algeb aic ones. Howe e , many
o hese me hods can be complex o edious, pa icula ly
when add essing he p oblem in h ee dimensions. Wi h he
g owing impo ance o accu a e posi ioning in mode n appli-
ca ions, i is essen ial o e isi he esec ion p oblem and
explo e inno a i e app oaches ha o e mo e in ui i e solu-
ions.
3.1 T adi ional me hods
In his sec ion, adi ional me hods o sol ing he esec-
ion p oblem a e e iewed. They can be classi ied in o ou
basic g oups: igonome ic, geome ic/g aphical, i e a i e
(nume ical) and o he s (see Table 2). A comp ehensi e lis
o me hods o 2D esec ion p oblems is al eady p esen ed in
Table 1. No e ha some imes he on ie be ween igono-
me ic and geome ic me hods is no as clea because bo h
solu ions exis a he same ime (e.g. he Cassini me hod).
T igonome ic me hods a e some o he oldes and mos
amous p ocedu es o sol ing he 2D esec ion p oblem.
They a e based on he use o igonome ic unc ions o com-
pu e he posi ion o he obse e using he angles be ween
he known poin s and he obse e .
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47 Page 6 o 21 J. Ven u a e al.
Table 2 Taxonomy o Resec ion Me hods
G oup Me hods
T igonome ic Use igonome ic unc ions and equa ions o calcula e s a ion/ obo posi ion
Geome ic Compu e in e sec ion o ci cles o lines passing h ough known poin s and s a ion/ obo posi ion
I e a i e S a wi h es ima e o obo posi ion, i e a i ely e ine using e o minimisa ion
O he s Use o Ba ycen ic Coo dina es o Complex Numbe s
The Po heno –Snellius me hod, also known as he Käs ne –
Bu kha d me hod, is one o he mos known and oldes
p ocedu es in his g oup. The Cassini me hod is ano he
example o a igonome ic solu ion (wi h a g aphical solu-
ion, oo), which is simila o he one desc ibed by Es e es e
al. and Cohen and Koss. The Eas on and Came on me hod is
also a igonome ic app oach, simila o he Cassini me hod.
Geome ic o g aphical me hods, on he o he hand, use
geome ical cons uc ions and p ope ies o he known poin s
o calcula e he posi ion o he obse e . These me hods a e
based on he use o geome ic and g aphic p inciples o sol e
he p oblem. Es e es e al. p oposed a geome ic me hod
ha uses he in e sec ion o ci cles o calcula e he posi ion
o he obse e . Cohen and Koss also p oposed a geome ic
me hod ha uses he in e sec ion o ci cles, bu wi h a di -
e en app oach om Es e es e al. Fon -Llagunes p oposed
a me hod ha uses he in e sec ion o lines and simila i y
be ween iangles o calcula e he posi ion o he obse e .
Pie lo e al. and Ligas p oposed a me hod ha uses he in e -
sec ion o powe lines o calcula e he posi ion o he obse e ,
al hough he solu ion is gi en in algeb aic o m.
I e a i e me hods use i e a i e algo i hms o con e ge o
he obse e posi ion. These me hods a e based on he use o
an ini ial es ima e o he obse e posi ion, which is e ined
i e a i ely un il con e gence is achie ed. I e a i e sea ch is an
example o an i e a i e me hod ha uses a sea ch algo i hm
o ind he obo posi ion. Sanchiz e al. p oposed a me hod
ha uses an i e a i e sea ch algo i hm o ind he obse e
posi ion.
Finally, he e a e o he me hods ha do no i he p e ious
g oups. The Tiens a me hod is one such example, which is a
comple ely di e en app oach based on ba ycen ic coo di-
na es. Ano he me hod like Wille ding is based on he use o
complex numbe s o compu e o a ions in he A gand plane
o ind he obse e posi ion.
4 Resec ion using geome ic algeb a
Nowadays, me hods based on geome ic algeb a (GA) ha e
been de eloped, p o iding a new solu ion o he esec ion
p oblem while main aining he ocus on i s geome ical oo s.
GA o e s a e sa ile amewo k ha can be adap ed o di -
e en con ex s based on he selec ion o speci ic me ics
and he numbe o dimensions. Fo example, when all ele-
men s o he base σisqua e o +1, he esul ing algeb a is
known as ec o geome ic algeb a (VGA) (see Sec .2.1).
By ex ending he abili y o he basis elemen s o squa e o
−1 o 0 o by accommoda ing mo e dimensions, i becomes
possible o explo e al e na i e o ms o GA. One such exam-
ple is CGA, which inco po a es wo addi ional dimensions:
one dimension squa ing o +1 and ano he squa ing o −1.
This lexibili y enables GA o sol e a wide a ay o applica-
ions and p oblem domains, seamlessly scaling he numbe
o dimensions in a s aigh o wa d way.
4.1 Vec o GA me hod
The 2D VGA-based me hod has been ecen ly p oposed by
Smi h (2023a) and published as dissemina i e ma e ial. The
p ocess is mainly geome ic and esul s in ob aining a ec o
p ha desc ibes he posi ion o he poin Pwhen choosing
he middle poin (B) as he o igin. In his case, we s a wi h
a ec o basis consis ing o wo elemen s σ={σ1,σ2}.
Figu e 2illus a es a ep esen a ion o he p oblem, as well
as a de ailed sequence o s eps ca ied ou .
Fi s , using he known da a (A,B,C,α, and β), ci cles
c1and c2a e d awn using poin s A,B,Pand B,C,P,
espec i ely (see Fig. 2a). These ci cles se e as an auxilia y
elemen o be e unde s anding he solu ion, bu a e no
equi ed as such. Wi h he help o he cen al angle heo em,
he ec o s d1and d2a e ob ained
d1= 1+ 1
an α
σ12 = 1
sin αe(90−α)σ12
d2= 2− 2
an β
σ12 = 2
sin βe(β−90)σ12 (15)
whe e 1=A−Band 2=C−B. No e ha Bwas chosen
as he o igin, bu any o he poin can also be selec ed unde he
condi ion ha αo βis no null. This si ua ion occu s when
Pis collinea wi h wo o he h ee known poin s. In such a
case, o he poin s can be selec ed as he o igin. Equa ion (15)
indica es ha ec o s dia e he esul o o a ing and scaling
ec o s i, as shown in Fig. 2b. No e ha he o a ion angle
is gi en by (90 −α) and (90 −β), espec i ely. The nex
s ep in ol es de e mining he ec o das d2−d1. Finally,
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A no el geome ic me hod based on con o mal geome ic algeb a applied o he esec ion p oblem… Page 7 o 21 47
Fig. 2 Vec o GA-based me hod
s eps o sol e he 2D esec ion
p oblem
(a) Ini ial se up, wi h (unknown) ci cles c1(A,
Band P)andc2(B,Cand P).
(b) Using known ec o s 1and 2, ge ec o s
d1and d2by o a ion and scaling.
(c) Ge ec o das d2−d1. Rejec ei he d1
o d2on d o compu e p.
he desi ed ec o pis he ejec ion o d1o d2on d(see
Fig. 2c). In VGA, he abo e s eps a e summa ised in he
ollowing equa ion
p=(d1∧d)d−1=−(d2∧d)d−1=(d1∧d2)d−1(16)
I should be no ed ha he p oposed solu ion is ema kably
simple and does no in ol e he use o any ype o coo di-
na es. The esul is ob ained by simple geome ic ope a ions,
such as o a ion, scaling, and ejec ion, applied o he inhe en
p imi i es o VGA, such as ec o s in his speci ic case.
The p oposed me hod has ad an ages o e he exis ing
ones. I a oids some limi a ions as in Tiens a’s me hod
whe e no solu ion can be ound i he poin s A,B, and C
a e collinea . Fu he mo e, i is easible o ob ain an indi-
ca o o how close Pis o he o bidden ci cle (de ined by
poin s A,B, and C) by means o he leng h o ec o d.I
he poin Pis on his ci cle, hen i can be easily checked
ha d=d=0. Consequen ly, small alues o dsugges
ha we should eloca e he s a ion o ano he si e o ensu e a
educed sou ce o e o (see Sec . 6 o a de ailed e o anal-
ysis).
4.2 Con o mal GA me hod
The me hodology employing VGA, as ou lined in Sec . 4.1,
p ima ily u ilised he GA G2. Howe e , gi en ha he
esec ion p oblem p ima ily deals wi h ci cles, u ilising i s
con o mal ex ension, speci ically he compass ule algeb a
(CRA), appea s o be mo e sui able (see Hildenb and 2018).
When dealing wi h geome ic p oblems, i is highly ad an-
ageous o ha e a ool ha can exp ess g aphical me hods
algeb aically.
On he basis o he pos ula es p esen ed in Sec . 2.2, wo
adi ional and well-known g aphical me hods a e p oposed
o sol e he esec ion p oblem using CRA: Cassini and
Collins. By le e aging CRA, bo h me hods can ecei e clea
algeb aic in e p e a ions, as explained in he ollowing sec-
ions.
Fo a mo e in-dep h example o CRA using he Cli o d
lib a y in Py hon, see he Gi Hub eposi o y.
4.2.1 Cassini cons uc ion
The Cassini me hod p o ides a solu ion o he esec ion p ob-
lem by le e aging he insc ibed angle heo em. The solu ion
is ob ained by de e mining he in e sec ion o wo ci cles: one
passing h ough poin s A,B, and P, and he o he h ough
poin s B,C, and Pas shown in Fig. 3. To de e mine he
cen es o he ci cles, wo lines mus be in e sec ed. The s ep-
by-s ep g aphical app oach unde lying he Cassini me hod
can be elucida ed, along wi h he equi alen s eps, using he
CRA algeb a (see Fig. 4).
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47 Page 8 o 21 J. Ven u a e al.
Fig. 3 2D g aphical esec ion p ocedu e using Cassini me hod
1. CRA Mapping: The p oblem s a s by mapping h ee
known Euclidean poin s, A,B, and C, o he CRA
domain. Fo example, he poin Awi h coo dina es
(a1,a2)is mapped as
A=(a1,a2)−→ 
a=a1σ1+a2σ2
−→ a=σ0+
a+1
2
a2σ∞(17)
2. Auxilia y Lines: To ob ain he posi ion o he cen e o
he ci cle de ined by A,B, and P, he line AB joining A
and Bmus i s be cons uc ed along wi h he line mAB
as he pe pendicula bisec o o AB. The line AO is hen
cons uc ed by o a ing AB by (π/2−α) ela i e o A
clockwise. This p ocess is epea ed o BC and mBC bu
wi h CO being o a ed some (π/2−β)coun e clockwise
ela i e o C.
LAB =a∧b∧σ∞,LCB =c∧b∧σ∞
MAB =(a−b)MBC =(b−c)(18)
Fo he o a ed lines AO1and CO
2, he ollowing o o s
and ansla o s mus be i s de ined
Rα=e−1
2(α−π/2)σ12 ,Rβ=e−1
2(π/2−β)σ12
TA=1−
a
2
σ∞,TC=1−
c
2
σ∞
DA=TARα
TA,DC=TCRβ
TC(19)
, and hus,
LAO1=DALAB
DA,LCO
2=DCLBC 
DC(20)
No e ha o a ions a e always ela i e o he o igin. To
o a e a ound an a bi a y poin , i mus i s be ans-
la ed o he o igin. The o a ion is hen applied, ollowed
by ano he ansla ion ha e u ns he poin back o i s
o iginal posi ion.
3. Ci cle building: The cen es o he ci cles can now be
ound as he in e sec ion o he lines AB and AO1and
BC and CO
2
O1=o1∧σ∞=LAB ∨LAO =(L
AB ∧L
AO)
O2=o2∧σ∞=LBC ∨LCO =(L
BC ∧L
CO)(21)
Fig. 4 G aphical solu ion o
Cassini me hod s ep by s ep.
The poin Pis ound by
in e sec ing he wo ci cles c1
and c2whe e all he poin s see
AB wi h angle αand BC wi h
angle β, espec i ely
(a) Lines de ined by AB and BC (b) Bisec o o AB and BC
(c) Cen es O1and O2(d) The sough poin P
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A no el geome ic me hod based on con o mal geome ic algeb a applied o he esec ion p oblem… Page 9 o 21 47
Fig. 5 CRA e sion o Cassini’s me hod s ep by s ep
The esul is a la poin , ep esen ing he wedge o he
sough poin and he poin a in ini y.1The ex ac ion o
he poin o in e es is s aigh o wa d by ac o ing ou
he poin a in ini y
o1=(σ0·O1−σ0)
o2=(σ0·O2−σ0)(22)
The adius o he ci cles can be compu ed as
1=−2(o1·a), 2=−2(o2·c)(23)
, and he ci cles hemsel es a e de e mined as
c1=o1−1
2 2
1σ∞c2=o2−1
2 2
2σ∞(24)
4. In e sec ion o ci cles: The desi ed esul can be ob ained
om he in e sec ion o he wo ci cles c1and c2as
P=c1∧c2(25)
Two in e sec ing ci cles in CGA yield a couple o poin s
(1D sphe e), also known as pai -poin P. Finally, he
1A line can be conside ed as a ci cle wi h in ini e adius, so he in e -
sec ion o wo lines esul s in wo poin s, one a in ini y.
Euclidean poin Pis eco e ed by he classic o mula
(see Hildenb and 2018)
P±=±√P·P+P
σ∞·P(26)
One o he poin s P±is exac ly he poin B, and he o he
is he sough poin P(see Fig. 3).
Figu e 5shows a concise and condensed summa y o he
key s eps in ol ed and discussed abo e. I p o ides alu-
able isual depic ions ha enhance he geome ic in ui ion
unde lying he me hod, o e ing an algeb aic in e p e a ion
o he g aphical app oach. I also helps o ein o ce he s ong
connec ion be ween he g aphical Cassini me hod and i s
algeb aic ansla ion using CRA.
The Gi Hub eposi o y shows se e al examples de i ed
om he code de eloped by he au ho s. The compu a ional
p ocedu es in CGA a e ound o be s aigh o wa d. The
me hodology in ol es he manipula ion and combina ion o
geome ic objec s, hus jus i ying he occasional e e ence o
GA as an objec -o ien ed app oach.
4.2.2 Collins cons uc ion
The g aphical me hod o Collins p o ides a solu ion o he
esec ion p oblem using he in e sec ion o he line passing
h ough he poin Band he so-called Collins auxilia y poin
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47 Page 16 o 21 J. Ven u a e al.
Table 5 Pe o mance compa ison o esec ion algo i hms using geo-
me ic algeb a and s a e-o - he-a algo i hms (see Pie lo and Van
D oogenb oeck 2014)
Me hod Mean (μs) Rank #
VGA 18997.8 1
To al #1 21919.1 2
To al #2 27340.0 3
CollinsCGA 196122.2 4
CassiniCGA 220135.8 5
Ou indings e eal ha ou VGA-based algo i hm ou -
pe o ms s a e-o - he-a me hods (see Table 5), execu ing
app oxima ely 13.3% and 30% as e han he p e iously
bes -known algo i hms by Pie lo wi h To al #1 and To al
#2.
The p ima y ad an age o ou me hodology and imple-
men a ion is he u ilisa ion o GA-FuL’s comp ehensi e code
gene a ion capabili ies. These capabili ies ange om gene -
a ing code o indi idual mul i ec o ope a ions o c ea ing
ull so wa e lib a ies wi h p ope so wa e a chi ec u e and
nes ed olde / ile s uc u e, enabling e icien and op imised
geome ic algeb a compu a ions.
6 Unce ain y analysis
This sec ion in es iga es he impac o measu emen unce -
ain ies on he e icacy o he p oposed GA me hods. Gi en
he in insic p esence o noise in p ac ical measu emen s, i is
c ucial o assess he sensi i i y and esilience o he me hod o
such pe u ba ions (Fon -Llagunes and Ba lle 2009a;Pie lo
and Van D oogenb oeck 2014).
P e ious discussions ha e highligh ed ha he esec ion
p oblem aces an in ac able challenge wi hin he o bidden
ci cle de ined by poin s A,B, and C. This limi a ion is inhe -
en in he na u e o he p oblem a he han a e lec ion o he
me hods used o sol e i . No el me hodological app oaches
should be conside ed wi h cau ion as hey may inad e en ly
in oduce undesi able scena ios and singula i ies no co e ed
by he o iginal o mula ion o he p oblem. In his ega d, he
p oposed me hods handle degene a e o limi ing cases in a
con enien and elegan g aphical way, as shown in Fig. 11.
This s udy p esen s h ee app oaches o add ess he p ob-
lem o 2D and 3D esec ion. In o de o imp o e he abili y
o e alua e he accu acy o he algo i hms and de e mine he
s a ion posi ion e o , a new me ic has been de eloped. Con-
sequen ly, we p opose h ee o mula ions ha a e essen ially
di e en a ia ions o he same unde lying app oach, each
de ining he me ic Dbased on he squa e o a dis ance o
deno e he p oximi y o he s a ion P o he o bidden egion.
•VGA me hod: The ec o dis used as he di e ence
be ween he ec o s d1and d2( e e o Eq. (15)). When
he s a ion Pis loca ed on he o bidden ci cle, he ec-
o s d1and d2con e ge, esul ing in a null ec o d.
The e o e, minimal alues o Dindica e close p oximi y
o he o bidden ci cle
D=d2(41)
•Cassini me hod: The de e mina ion o Dusing he CGA
Cassini me hod equi es he calcula ion o he squa ed
dis ance be ween he cen es o he ci cles c1and c2(as
shown in Fig. 4), ma ked by O1and O2, espec i ely. In
si ua ions whe e he s a ion is posi ioned on he o bidden
ci cle, he wo ci cles con e ge, causing hei cen es o
o e lap and he in e cen e dis ance o become 0. Thus,
acco ding o Eq. (A3), Dis de ined as
D=−2(o1·o2)(42)
•Collins me hod: Simila ly o he p e ious me hod, Dis
de e mined as he squa ed dis ance be ween wo speci ic
poin s. The auxilia y Collins poin Eand he poin Ba e
used o his pu pose. They coincide when he s a ion P
is si ua ed on he o bidden ci cle. The o mula used o
compu e his dis ance is as ollows
D=−2(e·b)(43)
To alida e he sensi i i y o he p oposed algo i hms, a
se ies o simula ions ha e been p oposed. The simula ion
amewo k is designed wi hin a squa e a ea measu ing 4 by
4 squa e me es, inco po a ing wo unique con igu a ions o
h ee known poin s. The i s con igu a ion is an equila e al
iangle, wi h he poin s loca ed a he posi ions A=(0,1),
B=(−0.866,−0.5), and C=(0.866,−0.5). The poin s
in he second con igu a ion a e linea ly a anged a A=
(−0.866,0),B=(0,0), and C=(0.866,0). A spacing o
2cm is used in each di ec ion ac oss he g id. A each poin
on he g id, he angles αand βseen om Pa e calcula ed.
Gaussian noise is in oduced in o hese angles, cha ac e ised
by a ze o mean and wo dis inc s anda d de ia ions (σ=
0.01 deg ees and σ= 0.1 deg ees). The algo i hms use hese
modi ied angles as inpu o de e mine he es ima ed posi ion
o he unknown poin . The disc epancy in posi ion (d)is
quan i ied by he Euclidean dis ance be ween he exac and
es ima ed loca ion o he poin P.
The s udy pe o ms 1000 i e a ions o each posi ion o
de e mine he s anda d de ia ion o he posi ion e o . The
esul ing s anda d de ia ions a e shown in Fig.14. The esul s
o he equila e al iangle con igu a ion, wi h s anda d de i-
a ions o σ= 0.01 deg ees and σ= 0.1 deg ees, a e p esen ed
in he i s and second columns, espec i ely. Simila ly, he
123

A no el geome ic me hod based on con o mal geome ic algeb a applied o he esec ion p oblem… Page 17 o 21 47
Fig. 14 E o analysis o wo poin con igu a ions using a squa e win-
dow o 2 ×2 m.. Con ig #1 is an equila e al iangle, while in con ig #2
he poin s a e collinea . Bo h con igu a ions ha e been es ed wi h s an-
da d de ia ion σ=0.01 and σ=0.1. The h ee GA me hods (VGA,
CollinsCGA, and CassiniCGA) a e compa ed based on hei posi ion
e o ( i s ow) and me ic 1/D(second ow), espec i ely
123
47 Page 18 o 21 J. Ven u a e al.
esul s o he second con igu a ion (collinea poin s) a e
de ailed in he hi d and ou h columns, espec i ely, using
he same s anda d de ia ions. The igu e shows he s anda d
de ia ion o he posi ion e o and he mean e o measu e
1/Din he i s and second ows, in ha o de . I is impo an
o no e ha he scales used in he g aphic ep esen a ion a e
no linea . To enhance he isual cla i y o hese images and
emphasise he co ela ion be ween he posi ion e o s and
ou new e o me ic, we applied his og am equalisa ion o
he images.
6.1 Discussion o he esul s
The simula ions pe o med a e in ag eemen wi h hose
epo ed in he li e a u e and suppo he case s udies o
each o he h ee me hods p esen ed. Figu e 14 shows he
di e en poin con igu a ions, whe e he o bidden ci cle is
clea ly iden i iable due o he inc ease in s anda d de ia ion
o he posi ion e o as i is app oached. Minimal e o s a e
obse ed inside he ci cle, while e o s inc ease wi h dis ance
ou side he ci cle. I is ema kable ha he h ee me hods p o-
ide almos iden ical esul s o he e o posi ion.
The 1/Dme ic plo s main ain a shape simila o he posi-
ion e o plo s nea he o bidden ci cle. High ed alues a e
obse ed as Dbecomes smalle , while minimum alues a e
obse ed in he lines AB and BC in Cassini and AC in
Collins due o an inc ease o D(see Figs. 9and 10).
Con igu a ion 2 ans o ms he o bidden ci cle in o a
s aigh line. In his egion, he e o dis ibu ion is high and
inc eases p og essi ely wi h dis ance om he line. Because
his con igu a ion ep esen s a degene a e si ua ion, he case
whe e Dbecomes ze o does no occu . The di e en me hods
used o de ine Dlead o o e lapping poin s a in ini y, caus-
ing D o ake high alues nea he c i ical line and esul ing
in a minimal alue o 1/D. The me ic Dwas chosen o
con igu a ion 2 due o his p ope y.
All me hods p oduce iden ical posi ion e o plo s, indi-
ca ing consis ency and con o mi y wi h he esul s ob ained
by mos algo i hms o sol e he esec ion p oblem. This
con i ms ha he sensi i i y o calcula e he posi ion o he
poin P, e en wi h noisy measu ed angles, is independen
o he me hod used and unique, as discussed in Pie lo and
Van D oogenb oeck (2014) and Fon -Llagunes and Ba lle
(2009b). The me ic 1/Dcan be used as an indica o o
p oximi y o he o bidden ci cle. When dealing wi h aligned
beacons, he alue o Dshould be used di ec ly. In o he
cases, he simila i y be ween his me ic and he posi ion
e o sugges s ha he o me can app oxima e he la e i a
unc ion o he o he p oblem pa ame e s is applied.
7 Conclusions
This a icle p esen s a no el app oach o sol ing he esec-
ion p oblem in wo and h ee dimensions using con o mal
geome ic algeb a (CGA). The CGA amewo k allowed
ep esen ing poin s, lines, planes, ci cles, and sphe es in a
uni ied ma hema ical s uc u e and o e ed a mo e in ui i e
unde s anding and e icien solu ion o he esec ion p oblem
compa ed o exis ing algeb aic echniques.
The p oposed me hod le e aged he abili y o CGA o
ansi ion be ween di e en e e ence ames wi hou equi -
ing coo dina e ans o ma ions. This elimina ed he need o
mul iple calcula ion s eps and complex algeb aic manipula-
ions ha a e cha ac e is ic o adi ional algeb aic solu ions.
Th ough ex ensi e nume ical simula ions, we ha e demon-
s a ed he alidi y and e icacy o ou GA-based app oach,
achie ing accu acy compa able o ha o es ablished alge-
b aic echniques, while signi ican ly imp o ing compu a-
ional e iciency and p o iding aluable geome ic insigh s.
Ou indings sugges ha he geome ic algeb a ame-
wo k has s ong po en ial o sol ing esec ion- ype p oblems
no only in su eying and geodesy bu also in compu e
g aphics, obo ics, compu e ision, and na iga ion. By
exploi ing geome ic ela ionships be ween en i ies, CGA
pa es he way o mo e in ui i e solu ions ha uni y com-
pu a ions in ol ing di e en geome ic p imi i es.
Fu u e esea ch can build upon he ideas p esen ed he e o
add ess mo e complex a ian s o he esec ion p oblem ha
in ol e addi ional cons ain s. The CGA me hod can also be
ex ended o add ess in e sec ion p oblems and o he spa ial
geome ic compu a ions ac oss di e se disciplines. By ha -
nessing he powe o geome ic algeb a and he e sa ili y o
con o mal geome ic me hods, his wo k opens up new possi-
bili ies o ad ancing geome ic esea ch and compu a ional
echniques.
Appendix A Some calcula ions o e CGA
CGA has been b ie ly in oduced in Sec .2.2.In his
Appendix, some undamen al calcula ions a e p esen ed o
illus a e he compu a ional e iciency inhe en in his alge-
b a. Fo u he in o ma ion and a de ailed unde s anding, he
eade is e e ed o H dina e al. (2021), Do s e al. (2010),
Hes enes and Sobczyk (2012), Hildenb and (2018).
A.1 Con o mal inclusion
We’ll show how he ansla ions and o a ions a e connec ed
and ela ed o he inse ion o he Euclidean space. I we
choose ec o σ0as he o igin o he coo dina e sys em and
use he elemen
123
A no el geome ic me hod based on con o mal geome ic algeb a applied o he esec ion p oblem… Page 19 o 21 47
T=e1
2(xσ1+yσ2+zσ3)∧σ∞=1
+1
2(xσ1+yσ2+zσ3)∧σ∞(A1)
, hen i can ac as a ansla ion, allowing us o de ine an
inclusion as a ansla ion o he o igin σ0in he di ec ion o
he ec o 
x=(xσ1+yσ2+zσ3)by he sandwich p oduc
as
ι(
x)=
Tσ0T=1−1
2
x∧σ∞σ01+1
2
x∧σ∞
=σ0−1
2
xσ∞σ01+1
2
xσ∞
=σ0+σ0
1
2
xσ∞−1
2
xσ∞σ01+1
2
xσ∞
=σ0+σ0
1
2
xσ∞−1
2
xσ∞σ0−1
2
xσ∞σ0
1
2
xσ∞
=σ0−1
2
x(σ0σ∞+σ∞σ0)
−
x21
4
σ∞σ0σ∞=σ0+
x+1
2
x2σ∞
So, we see ha we can iden i y poin s in R3wi h ec o s in
CGA h ough he inclusion
ι:
x=(x,y,z)→ x=σ0+
x+1
2
x2σ∞(A2)
whe e he elemen Tac s as a ansla ion. Using a simila
easoning, he elemen R=en1σ2σ3+n2σ1σ3+n3σ1σ2ac s as
a o a ion due o iden i ica ion Im H=σ2σ3,σ1σ3,σ1σ2
and he ollowing compu a ion
Rι(
x)
R=Rx
R=Rσ0+
x+1
2
x2σ∞
R
=Rσ0
R+R
x
R+1
2
x2Rσ∞
R
=σ0+R
x
R+1
2
x2σ∞
A.2 Dis ance be ween poin s
One o he key dis inc ions be ween CGA and VGA lies in
he in e p e a ion o he inne p oduc . In CGA, he inne
p oduc ep esen s he dis ance be ween poin s, which is un-
damen ally a quad a ic concep in VGA. By inco po a ing
wo addi ional dimensions, CGA linea ises his quad a ic
unc ion. The equa ion below elucida es his:
a·b=σ0+
a+1
2
a2σ∞·σ0+
b+1
2
b2σ∞
=
a·
b−1
2
a2−1
2
b2=−1
2(
a−
b)2(A3)
This calcula ion explici ly demons a es how CGA lin-
ea ises he quad a ic na u e o he dis ance ep esen a ion
ound in VGA, o e ing nuanced insigh s in o geome ic ela-
ions and in e ac ions be ween poin s.
Fu he mo e, his cha ac e is ic is u ilised o depic he
midline Mas illus a ed in exp ession (18). A midline is
es ablished by wo con ol poin s, aand b, and encompasses
poin s ha main ain equal dis ances o hese con ol poin s.
This means ha (x−a)2=(x−b)2. In he con ex o CGA,
his condi ion can be a icula ed as ollows:
x·a=x·b⇒0=x·a−x·b=x·(a−b),
i.e. x∈MAB ↔x·(a−b)=0, so (a−b)is he IPNS
ep esen a ion o he midline de ined by con ol poin s aand
b.
A.3 IPNS s OPNS ep esen a ion o objec s
In GA, we ha e se e al ypes o p oduc , so i is possible o
ep esen objec s in di e en ways. Wi h he help o he inne
p oduc , we can de ine he objec Cas ollows.
x∈C⇔x·C=0(A4)
In CGA, o example, he objec C=n1σ1+n2σ2+n3σ3+
dσ∞can be es ed and e i y ha i ep esen s a plane
(σ0+
x+1
2
x2σ∞)·(
n+dσ∞)=
x·
n−d=0(A5)
whe e 
n=n1σ1+n2σ2+n3σ3. Equa ion (A5) desc ibes
a plane wi h no mal ec o 
nand dis ance om he o igin d.
On he o he hand, he wedge p oduc de ines an objec Das
ollows
x∈D⇔x∧D=0.
Again, in CGA, o example, he objec
D=a∧b∧σ∞=σ0+
a+1
2
a2σ∞
∧σ0+
b+1
2
b2σ∞∧σ∞
=(σ0+
a)∧(σ0+
b)∧σ∞
=σ0∧
b∧σ∞+
a∧σ0∧σ∞+
a∧
b∧σ∞
de ines he line goes h ough he poin s aand b:

x∧D=σ0+
x+1
2
x2σ∞
∧(σ0∧
b∧σ∞+
a∧σ0∧σ∞+
a∧
b∧σ∞)
123
47 Page 20 o 21 J. Ven u a e al.
=σ0∧
a∧
b∧σ∞+
x
∧(σ0∧
b∧σ∞+
a∧σ0∧σ∞+
a∧
b∧σ∞)
=(σ0∧(
a∧
b+
x∧(
a−
b)) +
x∧
a∧
b)∧σ∞
=0
So

a∧
b+
x∧(
a−
b)=0 and 
x∧
a∧
b=0,(A6)
which ep esen ed he line based on he poin s aand b.
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