Possible charge ordering and anomalous transport in graphene/graphene quantum dot heterostructure

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 . 123 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, 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 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). 123 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 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 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 123 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. Funding Funding o open access publishing: Uni e sidad de Alme ía/CBUA. Open Access This a icle is licensed unde a C ea i e Commons A ibu ion 4.0 In e na ional License, which pe mi s use, sha ing, adap- a ion, dis ibu ion and ep oduc ion in any medium o o ma , as long as you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons licence, and indi- ca e i changes we e made. 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Mech Mach Theo y 173:104835 123