KAMIWAAI INTERACTIVE 3D SKETCHING WITH JAVA BASED ON Cl(4,1) CONFORMAL MODEL OF EUCLIDEAN SPACE

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1 KAMIWAAI ITERACTIVE D SKETCHIG WITH JAVA BASED O Cl(4,) COFORMAL MODEL OF EUCLIDEA SPACE Submitted to (Feb. 8,00): Advaces i Applied Cliffod Algebas, Eckhad M. S. Hitze Dept. of Mech. Egieeig, Fukui Uiv. Bukyo -9-, Fukui, Japa. hitze@mech.fukui-u.ac.jp, homepage: Abstact. This pape itoduces the ew iteactive Java sketchig softwae KamiWaAi, ecetly developed at the Uivesity of Fukui. Its gaphical use iteface eables the use without ay kowledge of both mathematics o compute sciece, to do full thee dimesioal dawigs o the scee. The esultig costuctios ca be eshaped iteactively by daggig its poits ove the scee. The pogammig appoach is ew. KamiWaAi implemets geometic objects like poits, lies, cicles, sphees, etc. diectly as softwae objects (Java classes) of the same ame. These softwae objects ae geometic etities mathematically defied ad maipulated i a cofomal geometic algeba, combiig the five dimesios of oigi, thee space ad ifiity. Simple geometic poducts i this algeba epeset geometic uios, itesectios, abitay otatios ad taslatios, pojectios, distace, etc. To ease the coodiate fee ad matix fee implemetatio of this fudametal geometic poduct, a ew algebaic thee level appoach is peseted. Fially details about the Java classes of the ew GeometicAlgeba softwae package ad thei associated methods ae give. KamiWaAi is available fo fee iteet dowload. Key Wods: Geometic Algeba, Cofomal Geometic Algeba, Geometic Calculus Softwae, GeometicAlgeba Java Package, Iteactive D Softwae, Geometic Objects. Itoductio The ame KamiWaAi of this ew softwae is the Romaized fom of the expessio i vese sixtee of chapte fou, as foud i the Japaese taslatio of the fist Lette of the Apostle Joh, which is pat of the ew Testamet, i.e. the Bible. It simply meas God is love. (Please compae the quotatio at the ed of this itoductio.) The eady-to-use applicatio is witte i the platfom idepedet pogammig laguage Java. Most computes owadays aleady have a Java Rutime Eviomet (JRE) of the Java Platfom, Stadad Editio (JSE) istalled. If ot, JREs ae available fo fee dowloadig fom the Su Micosystem homepage []. KamiWaAi vesio 0.0. beta is available fo fee ocommecial use as a zip compessed biay code set of files fo dowload fom: This site also featues a sceeshot (Fig. ), showig how the KamiWaAi gaphical use iteface ca be used fo iteactive sketchig. Fig. KamiWaAi sceeshot: fot (left) ad ight-side view (cete) paels, fuctio adio buttos (ight).

2 The site futhe gives olie access to the KamiWaAi Readme file with ifomatio about the amig, waaty, cotact fo use commets, istallatio ifomatio, bief ifomatio o how to use it ad efeeces. I the futue ew vesios ad detailed documetatio will be available fom this site. KamiWaAi has bee tested o Widows 98 platfoms with the JREs of JSE vesio.._04 ad.4._0. It should theefoe opeate acoss all opeatig systems (Widows, McItosh, Uix, Liux, ) without poblem ad with all JREs cuetly maitaied by Su Micosystems. A small photo i the lowe ight coe of the KamiWaAi homepage shows models of all egula Platoic polyhedal solids: tetahedo, cube, octahedo, dodecahedo ad icosahedo. Fig. Platoic polyhedos. The eflectio goups of these Platoic solids cotai the otatioal symmety goups of these solids as subgoups, because the combiatio of two eflectios always yields a otatio. [],[] Both the eflectio ad the otatio symmety goups of the Platoic solids ae of eomous impotace fo the study of molecula, cystal ad lattice symmeties i may fields of sciece. The autho has aleady ceated [4] a seies of olie applets [5] fo the study of the two dimesioal symmety goups of egula polygos (=,,6). This two dimesioal poit goup applet seies featues both eflectio ad otatio symmeties. These applets wee desiged with the iteactive (two dimesioal) geomety softwae Cideella. [6] The desig aims to show how the geometic algeba of vectos i two dimesios epesets evey of these two dimesioal goups just usig two physical polygo vectos. It is clea that the two dimesioal poit symmety goups ae othig but plaa subgoups of full thee dimesioal cystal goups. Geometic algeba teats these goups usig thee paticula lattice vectos to geometically epeset all eflectios ad otatios. The challege is to visualize this iteactively, i ode to supply a hads o expeiece of the geat beauty ad simplicity of epesetig the poit goups that leave some thee dimesioal lattice ivaiat ad the full set of 0 space goups, i.e. the complete symmety goups of thee dimesioal cystals. To descibe the space goups with full elegace it poves coveiet to choose a five dimesioal cofomal model of the Euclidea thee space. The otatios ad taslatios both become simple moomial (oe tem) expessios. Simila to pojective geomety, which adds oe dimesio, we add hee two dimesios, epesetig the oigi ad spatial ifiity. Physicists ae by ow vey familia with the fou dimesioal space-time desciptio of special elativity ad the fact that ull-vectos descibe the popagatio of light. Hee we choose a space with two liealy idepedet ull-diectios added to the usual Euclidea space. Fo a physicist this meas simply a sigatue (+4,-) Mikowski space. The ot oly the desciptio of cystal symmeties beefits, but we ideed aive at a ew cosistet algeba whose elemets ae i a obvious oe-to-oe cofomal coespodece with eal geometic objects like poits, lies (ad cicles), plaes (ad sphees), volumes, etc. This obviously demads a mode object oieted compute pogam implemetatio. With such a fudametal set of ecoded geometic objects ad thei geometical methods of uio, itesectio, otatio, taslatio, etc. the pogamme ca wok fee of the estaits of covetioal matix algeba implemetatios, whee coodiates ad matix coefficiets ofte completely hide geometic cocepts ad ivaiats. To show, that this is ot pue fatasy, but ca be implemeted as a matte of piciple is the mai aim of this pape ad of the KamiWaAi softwae. The backboe of KamiWaAi is theefoe a package of epeatedly used GeometicAlgeba objects

3 (Java classes). These classes ae poits, lies (cicles), plaes (sphees) ad moe geeal multivectos. It poved pactical to hieachally implemet multivectos based o simple objects of complex umbes (scalas ad 5D pseudoscalas), ad (complex) quateios. [7],[8] That way the dimesioal multivectos get a atual easy-to-use sub-stuctue. It should be obvious that the GeometicAlgeba package maybe used i may othe cotexts. Wheeve the fa eachig mathematical appaatus of this algeba is wated fo object oieted compute implemetatio. This icludes divese aeas [],[],[],[7],[8] as simulatio, aviatio, stuctual mechaics, avigatio, obotics, compute gaphics, molecula modelig, solid state physics, cosmology, electodyamics, gavitatio, quatum mechaics, Fom a histoical viewpoit five people pioeeed this appoach (alogside may othes): Gottfied W. Leibitz [9], Si Rowa Hamilto [7],[0] of Ielad, Hema Gassma [] of Gemay, William K. Cliffod [0] of Eglad ad the Ameica David Hestees. [0] Cuetly a eseach goup aoud L. Dost (Amstedam) is pusuig a pofessioal Java implemetatio of geometic algeba. The GeometicAlgeba package of KamiWaAi offes woldwide oe of the fist implemetatios of cofomal geometic algeba. Ad to my kowledge KamiWaAi epesets woldwide the fist exclusively iteactive Java softwae based o cofomal geometic algeba. It theefoe takes its place at the fotie of excitig ew developmets i applied mathematics ad compute sciece. Yet I must cautio, that at the momet KamiWaAi is fa fom optimal coceig aspects of speed optimizatio. But I egad this as a techical issue ad ot at all as a picipal poblem. It will be meded i due time. Aothe aspect is, that this is oe of the fist beta vesios. This meas that oly vey, vey few fuctios ae implemeted i the 0.0. beta vesio of KamiWaAi ad may moe fuctios ae scheduled to be implemeted soo: E.g. plaes, the itesectios of sphees, cicles, etc, ext eflectios ad otatios, abitay choice of agula pespective, etc. Cofomal geometic algeba povides staightfowad algebaic expessios fo all these fuctios, they just eed to be added as methods of the coespodig geometical objects. Advice fo uses: If you ae pimaily iteested i kowig how to use KamiWaAi fo D sketchig, just ead sectio. Sectio gives mathematical backgoud ifomatio, some of it oigial. Sectio 4 is of iteest to pogammes, who wat to apply the GeometicAlgeba package fo developig othe applicatios o applets. The ame of KamiWaAi oigiates fom the Japaese vesio [] of: Ad we ouselves kow ad believe the love which God has fo us. God is love, ad those who live i love live i uio with God ad God lives i uio with them. []. What KamiWaAi Ca Do ow Afte statig the applicatio with the commad java cp. KamiWaAi fom the diectoy whee the object KamiWaAi.class ad the package GeometicAlgeba eside, e.g. fom c: KWA, a ew widow (compae Fig. ) will ope o the compute scee. The widow is composed of thee mai paels. The left pael shows the fot view, the cete pael shows he ight side view ad to the ight the available fuctios ae gouped as iteactive adio buttos with thei labels. I the default cofiguatio the fuctio poit is activated. I will ow quickly step you though the pesetly available seve fuctios: poit, lie, cicle, sphee C,, sphee, move ad S it l.. Iteactively Defiig D Poits with poit If the adio butto poit is activated, the use ca defie thee dimesioal poits by fist clickig o the left pael ad ext o the cete pael. Clickig o the left pael immediately ceates a small blue disk at the positio of the click. The left pael stays the deactivated (i.e. ueceptive fo mouse clicks) util a coespodig ight side view poit is defied by clickig o the cete pael. O the cete pael a gee disk will appea at the positio of the click, exactly i the same height as the blue poit just defied o the left pael. The the cete pael becomes deactivated, the left pael becomes activated ad a ew poit ca be defied. Late the udelyig epesetatio of these poits (vectos) i cofomal geometic algeba ad by its softwae implemetatio i the GeometicAlgeba Java package will be explaied late.

4 . Dawig Staight Lies with lies Afte defiig at least two poits with the fuctio poits the use ca activate the adio butto labeled lie fo dawig staight lies though the poits. The choice of poits is pefomed by simply clickig o the small coloed disks epesetig the poits i the fot ad side view paels (left ad cete pael espectively). O the commad lie of the shell o MSDOS pompt widow the messages poit chose! ad poit chose! will appea successively. As soo as the secod message appeas a sky blue staight lie will be daw though both poits, visible both i the fot ad i the side view paels. ow ew poits ca be chose to daw moe lies. It does ot matte fom which pael the poits ae chose, i.e. whethe they ae chose both fom the same pael view o e.g. fist fom the left pael ad the fom the cete pael ad vice vesa.. Dawig Cicles with cicle Dawig cicles afte activatig the adio butto cicle is vey much simila to dawig lies. The oly diffeece is that the use eeds to choose a total of thee poits istead of two. Theefoe befoe beig able to ceate a cicle, the use must at least defie thee poits with the fuctio poit. The messages poit chose!, poit chose! ad poit chose! will appea successively o the shell o MSDOS pompt widow. Afte the thid message appeas, a ed cicle will be daw i both views itesectig all thee chose poits. ow ew poits ca be chose to daw moe cicles. It agai does ot matte how may of the thee poits ae chose fom the fot ad/o the side view paels..4 Defiig a Fou Base Poit Sphee with sphee Fist the adio butto labeled sphee eeds to be activated. The fou poits eed to be chose simila to clickig the poits whe dawig lies o cicles. Theefoe befoe beig able to ceate a sphee by fou base poits, the use must at least defie fou poits with the fuctio poit. The fou messages poit chose! poit 4 chose! will appea successively o the shell o MSDOS pompt widow. Afte the fouth oe appeas, a yellow cicle will be daw, visible both i the fot ad i the side view paels. Agai it does ot matte how may poits ae chose fom the fot ad/o the side view pael, espectively. Sphees ae daw by thei logitudes ad latitudes. The logitudial meidias itesect i two poles. The agula positio of the poles o a sphee is oly fo the coveiece of displayig them ad fo late easig the iteactive selectio of sphees (compae the sectio o the itesectio of sphees ad lies.).5 Defiig a Ceteed Radius Sphee with sphee C, Selectig the adio butto sphee C, povides a moe coveiet way to daw sphees. The use just eeds to iteactively choose the cete poit. Immediately afte that a small ew automatic iteactive dialogue widow will appea. Eteig the desied adius by pessig the umbes o the compute keyboad followed by pessig the ETER key will cause KamiWaAi to immediately daw a yellow sphee of the desied adius ad aoud the selected cete poit. The appeaace of the sphee is othewise simila (composed of logitudes ad latitudes ad with visible poles). As with all othe fuctios, the use ca poceed to defie moe ceteed adius sphees with cetes ad adii at will. O he ca poceed to ay of the othe implemeted fuctios..6 Movig Poits ad Objects that Deped o them with move Up till ow thee is o geat diffeece to dawig thigs o a peace of pape, eve if oe has to admit that the compute ca daw much faste ad moe pecise tha sketchig by had usually allows. But the iteactive move fuctio is so to say the fist fuctio that eally capitalizes o the ability of a compute to memoize gaphical objects ad adapt them to chages of the objects which seved to defie them i the fist place. At the momet the defiig objects ae basically the poits ceated with the poit fuctio. The fist step is to activate the adio butto labeled move. The the use ca click o ay poit i ay of the two paels (fot o ight side view), keep the mouse butto fimly pessed ad slowly move it acoss the scee. The poit will follow this motio. Afte eleasig the poit agai, KamiWaAi will ecostuct all peviously peset objects, adaptig positios, oietatios ad sizes whee it is equied. This is cetaily somethig, which eeds much moe wok, fesh sheets of pape ad a good udestadig of the chagig pojected appeaaces if sketched by had.

5 The use ca use this move fuctio visually to iteactively exploe the depedece of his costuctios o the positios of the defiig objects (hee poits) i thee dimesios. All objects will follow this motio. (The oly exceptio is metioed i the ext paagaph, but the ext pogam vesio will elimiate this exceptio.).7 Itesectig Sphees with Lies Usig S it l This fuctio eeds some explaatio. Afte activatig the adio butto S it l the use is expected to click withi thee pixels of the lie, which he wats to use fo the itesectio opeatio. It does ot matte fom which pael the lie is chose. If the use clicks vey close to two lies, KamiWaAi will automatically choose the lie whose pojectio i the pael view is the closest to the clicked positio of the pael. The click positio has i o way to be a poit as defied i sectio.. Ay positio close eough to the pojected view of the staight lie will do! If the clicked positio is ot close eough to ay lie pojectio, the compute will wait fo aothe click i ode to popely select a lie ad will give out the ifomatio message Lie o. selected. i the shell o MSDOS pompt widow. Afte successfully selectig a lie the compute will pit the umbe of the staight lie immediately followed by the message Ad ow select a sphee! To select a sphee the use simply has to click o oe of the two peviously metioed poles of the desied sphee. Remembe that a pole is maked by the itesectio of the logitudial meidias, which ae used to display the sphee. Thee is a small sesitive aea aoud each pole of each sphee i both of the display paels (fot ad ight side). If the clicked positio is ot close eough to ay pole the message Sphee o. selected. will appea. I this case the use ca just cotiue to click i the viciity of ay pole util the message with the selected sphee umbe (a positive itege umbe icludig zeo) will appea i the shell o MSDOS pompt widow. Immediately afte successfully selectig a sphee the compute will do thee actios: Fist it will ifom the use how may poits of itesectio of the chose lie ad sphee exist: Two poits of itesectio! o Oe poit of itesectio! o o itesectio! I the case of o itesectio!, the compute will immediately estat all ove agai with the message: Select a ew lie! Secod i the cases of Two poits of itesectio! o Oe poit of itesectio!, KamiWaAi will daw the poit(s) of itesectio with dak blue disk(s) i the (left) fot view pael ad with dak gee disk(s) i the (cetal) ight side view pael. Thidly the compute will geeate the shell o MSDOS pompt message: Select a ew lie! KamiWaAi theeby idicates that it is eady fo the iteactive selectio of the ext lie fo the ext oud of itesectio of a sphee ad a lie. Hee a emak about the exceptio metioed fo move opeatios at the ed of the last sectio is i ode. The S it l geeated itesectio poits ca be used like all othe poits to defie ew geometic objects (e.g. lies, cicles ad sphees). But i the 0.0. beta elease of KamiWaAi they will ot (yet) follow the move opeatios of poits which cause the defiig sphee o lie of a itesectio poit to chage its positio, oietatio o size. This is a exceptio that will be emedied i the ext elease of KamiWaAi! ow all pesetly implemeted seve fuctios ae explaied i detail ad the iteested eade is heeby ivited to stat exploig thee dimesioal geomety o his ow usig KamiWaAi. Aothe ote of cautio is i ode: The display is ot pemaet. That is, if the KamiWaAi display widow is oveshadowed by ay othe widow, the oveshadowed potio will be lost util aothe move opeatio takes place. A tempoay wokaoud is to just defie a ew poit ad move it with move as descibed i sectio.6. This tempoay loss of display will also be dealt with i the ext elease. As a easo I submit, that KamiWaAi is i fact the fist eve Java pogam which I wite fom scatch. I atually welcome advise o such techical issues. I the ext two sectios fist the backgoud i the theoy of cofomal geometical algeba will be outlied. The the Java package GeometicAlgeba, which implemets this algeba i the fom of ew object oieted Java classes ad thei methods will be documeted. This desciptio ifom about the majo classes, list thei most impotat methods, specifyig thei iput ad output. The easo fo fist dealig with the cofomal geometical algeba backgoud is, that with this kowledge at had, the way the classes ae defied ad the methods they posses will become obvious ad atual.

6 . The Cofomal Geometic Algeba of Oigi, Euclidea Thee Space, ad Ifiity I this sectio I will give a staightfowad desciptio of essetial elemets ad elatios of the five dimesioal cofomal geometic algeba coceed. Fo futhe details ad some poves, I efe the iteested eadeship to publicatios. []-[5] Howeve it appeas to me that the essetial employmet of a thee level sub-algeba stuctue is a ew ad oigial appoach to the subject.. Multivectos ad Geeal Multivecto Poduct Fie itoductoy desciptios of the geometic algeba of the Euclidea (two ad) thee space ca be exist. [5],[6] This algeba cosistig of gade 0 scalas, thee liealy idepedet othoomal gade vectos { e, e e }, thee liealy idepedet gade bivectos, {i, i, i e, e, e } () e e e (equivalet to the i,j,k of Hamilto s quateios ad udestood as the oieted side aeas of a uit cube) ad oe squae mius oe pseudoscala thee-volume i with squae mius oe i = -. This = 8 dimesioal geometic algeba has theefoe the algebaic basis: {, e, e, e, i, i, i, i e, e e }. () This geometic algeba of the Euclidea thee space i itself is aleady of geat iteest fo mechaics, obotics, etc. [0],[6]-[8] But without pause, I will ow add the two liealy idepedet ull-vectos fo the oigi { } ad fo ifiity { } to give the full set of five basic vectos: {, e, e, e, }. () Equipped with the geometic poduct iveted by H. Gassma [9] ad W.K. Cliffod [0] ad augmeted by poduct ules fo highe gade objects we get the full 5 = dimesioal cofomal geometic algeba with its geeal multivecto poduct. The biliea ad associative but ot commutative geometic poduct was fist itoduced by Gassma to itegate Hamilto s quateios [0] ito his ow extesive algeba. Idepedet of the dimesios of the space coceed, it uifies the covetioal scala poduct of vectos ad Gassma s ealie atisymmetic oute bivecto poduct of vectos. ab a b a b. (4) Befoe tabulatig the full multiplicatio table of all basic elemets it poves to be vey istuctive ad immediately useful to fist look at the poduct table of the oigi ad ifiity {ull-vectos ad }. A physicist ca easily deive these elatioships by addig two vectos of opposite sigatue ( e, e ) to the basis of the Euclidea thee dimesioal space ad defiig 0 4 the two ull-vectos i tems of these two exta vectos: e 0 e 4, e e ). (5) ( 0 4 We the have the followig fudametal ull-vecto poducts: 0, 0, Defiig we futhe obtai,, (7). (8),, (9) (6)

7 . (0) All these poducts ca be coveietly summaized i the followig sub-algeba {,,, }multiplicatio table: Table Sub-algeba {,,, } multiplicatio table. 0 0 This table will ideed povide a vey coveiet top level algebaic sub-stuctue fo the complete multiplicatio table of the cofomal geometic algeba. ext the quaduple {, i, i, i } has the followig quateioic multiplicatio table: Table Quateioic sub-algeba multiplicatio table. i i i i i i i i - i -i i i -i - i i i i -i - I futhe give a explicit expessio fo the five dimesioal cofomal algeba pseudoscala I as: I e e e i, I II. () ow eveythig is eady to give the complete list of all liealy idepedet basis elemets of the five dimesioal cofomal algeba i Table. Table All liealy idepedet basis elemets of the 5D cofomal algeba. Gades i fist lie e Ii i Ii i e e Ii i Ii i e e Ii i Ii i e e Ii i I e Ii i I ee e e Ii i e Ii i e Ii i I e Ii i ee e i I e e e

8 Table featues gade by gade oe scala, five vectos, 0 bivectos ad the dual 0 tivectos, 5 quadivectos ad the 5D pseudoscala I. The gades ae give i the top lie of Table. The cetal vetical dividig lie betwee the gades ad idicates that all elemets o the ight ae dual to coespodig elemets o the left ad vice vesa. Duality hee simply meas multiplicatio with the pseudoscala I. The geometic poduct is associative, but i geeal ot commutative. Yet scalas ad pseudoscalas I commute with all elemets ad fom because of Eq. () a sub-algeba isomophic to complex umbes (Table 4.) Table 4 {, I } subalgeba isomophic to complex umbes. I I I I - Please ote that ou pseudoscala I has a defiite geometic meaig as the poduct () of the thee dimesioal Euclidea pseudoscala i, as defied i Eq. () times the two dimesioal oigi-ifiity ull-subspace pseudoscala. With the aim of edeig the full multiplicatio table of all elemets listed i Table, I gouped the tems ito fou complex quateios with the help of the subalgebas of Tables ad. By complex quateio I mea the followig expessio: The eight coefficiets { q q q s q v, with qs qs Iqsi, ad q v q q ( q Iq ) i ( q Iq ) i ( q Iq ) i. () q s, qsi, q, q i, q, qi, q, qi geometic multiplicatio of two complex quateios p ad q yields: i i i } ae all eal scalas. Usig the multiplicatio Table the cofomal pq p q p q p q p ) ( p q p q ) i ( p q p q ) i ( p q p q i s s ( q ) which is a ew quateio with its complex scala pat i the fist ight had side lie of Eq. () ad its complex bivecto pati the secod lie. The fist tem p q s s of the complex scala pat e.g. simply is accodig to Table 4: p q s s p s q s p si q si I ( psqsi psiqs () ). (4) A geeal (up to elemet) multivecto elemet of the cofomal geometic algeba, defied by addig scala multiples of all elemets of gades oe to five listed i Table, ca ow be ewitte i a vey elegat way as: m q q q q. (5) m compises ideed degees of feedom, because each of the fou complex quateios { q, q, q, q } has itself eight eal scala degees of feedom as show i Eq. (). The multiplicatio of two multivectos m, m is ow simply achieved usig Table ad the multiplicatio of the fou complex quateio coefficiets accodig to Eq. (): mm qq q q q q q q ( qq q q q q q q ) ( qq q q q q q q ) ( qq q q q q q q. (6) ) It is obvious that this 6 tem expessio is much easie to hadle tha a full by = 04 tem expessio ivolvig all eal 64 degees of feedom of the two multivectos m, m. It would ideed make o sese to povide such a extesive listig of tems, because we would ecessaily be lost i udestadig thei geometic sigificace ad it would be had fo a pogamme to coectly pogam the 04 elemet matix without eo. Opposed to that ou thee level hieachy of the full multivecto algeba ad its subalgebas of complex quateios ad complex umbes also povides a staightfowad ad absolutely eo fee way of implemetig the full poduct of Eq. (6), by fist implemetig the poduct of complex umbes as i Eq. (4), the implemetig

9 the poduct of quateios of Eq. () ad oly fially the multivecto poduct of Eq. (6) at the top of this thee level hieachy. Eve the eade void of ay kowledge of Java will theefoe aticipate that we will have thee fudametal objects of complex umbe (Complexumbe.class), complex quateio (ComplexQuat.class) ad multivecto (MultiVecto.class). The complex umbes cosist of two dimesioal aays of eal scalas double[], the complex quateios of fou dimesioal aays of complex umbes Complexumbe[4] ad the multivectos of fou dimesioal aays of quateios ComplexQuat[4]. double idicates the decimal poit eal umbe type vaiables of Java. atually each object has its ow method fo additio [add()], subtactio [sub()], scala multiplicatio [multsc()] ad multiplicatio [mult()], give i Eqs. (4), () ad (6). I the Java souce code we will theefoe be able to multiply multivectos m ad mp by simply witig: m.mult(mp); automatically usig the coect multiplicatio method mult() of the istace m of the MultiVecto.class object. The hadest pat of the cofomal geometic algeba foudatio is ow established. ext we eed to ask ouselves how to assig meaigful geometic etities (like spatial D poits, lies, cicles, sphees, etc.) to cetai multivectos ad how to let these geometic etity objects iteact with each othe. With iteactios I mea geometical uios of (poit, lie ad plae) subspaces, itesectios (of lies, cicles, plaes, sphees, etc.), pojectios, etc., i.e. geometic set theoetic opeatios keepig tack of the dimesios ivolved. Othe opeatios of iteest will be taslatios, otatios, dilatios ad eflectios, etc. It is staightfowad to aswe the fist questio fo meaigful idetificatio of geometic etities (objects) i the cofomal geometic algeba. But i ode to establish the geometical object iteactios ad opeatios it poves vey coveiet to fist itoduce some gade depedet opeatios i the multivecto algeba ad to deive a hadful of fomidable poducts fom the geeal multivecto poduct of Eq. (6). The stategy pesued i the followig fou sectios will theefoe fist itoduce basic idetificatios of geometic etities used by KamiWaAi i its peset vesio. Secod I will give a oveview of impotat gade depedet chages of multivectos ad thid I will show how these gade depedet opeatios o multivectos ca be used to modify ad combie multivectos i deived poducts. Ad fouth this will eable us to wite dow vey elegat moomial (oe tem) expessios fo the desied geometic object iteactios (uios, ) ad opeatios (taslatios, ).. Basic Geometic Etities Idetified i the Cofomal Geometic Algeba.. Poits The fist kid of geometic etities we ae coceed with ae poits. We ae familia with the epesetatio of poits by positio vectos i a thee dimesioal Euclidea vecto space. A thee dimesioal vecto space i ou cofomal algeba is give by the subspace chaacteized by the thee vectos e, e e o by thei poduct i e e. Ay liea vecto combiatio, x x e xe xe e. (7) of these thee vectos coespods oe to oe with a poit i a thee dimesioal Euclidea vecto space. The thee eal scalas { x, x, x } ae called the coodiates of the poit. I ode to eap some beefit fom ou five dimesioal cofomal oigi-space-ifiity algeba, we ow modify the defiitio (7) to epeset poits as cofomal vectos X with additioal oigi ad ifiity compoets: X x x. (8) Two emaks ae i ode: Fist, if x 0 we get X. This shows why the vecto is said to epeset the oigi. If x becomes vey log, the squae x xx x x will dwaf the two othe tems i (8). This is why the vecto is associated with spatial ifiity. Secod, it is vey easy to switch betwee (7) ad (8). To get fom (7) to (8) we simply add the two last tems o the ight had side of equatio (8). To get back to (7) we simply stip away the two oigi ad ifiity vecto pats of (8) to obtai (7) [pojectio ito the Euclidea thee space, also called cofomal split i special elativity.]

10 The easo fo the paticula fom of (8) ca be easily see by pefomig the poduct of two cofomal poits A, B AB A B A B. (9) Usig Table we get fo the scala ie poduct pat A B ( a b). (0) This is of geat pactical iteest, because we see that the ie poduct of cofomal vectos (8) simply gives us the distace betwee the coespodig Euclidea thee space vectos a b,. The oute poduct pat of (9) gives A B a b ( b a a b) ( a b) ( a b ) A B of () allows us to ecove the oigial cofomal vectos A, B : B v u. () u v, u v, a ( ) / u, b ( ) / u, a ( a u v) /, b ( b u v) /, () ad by isetig a, a, b, b ito (8). Equatio () ca be poved by isetio, yet the actual deivatio is a diect geometic algeba calculatio. We lea theefoe that the oute poduct pat A B of the geometic poduct of geometic vectos completely peseves the ifomatio about whee to fid each of the two poits i Euclidea thee space. Extactig the ifomatio of the costitutig poits out of a bivecto like A B will have futhe applicatios whe itesectig e.g. staight lies ad sphees, etc. Let me fially emak that accodig to Eq. (0) the geometic poduct of a cofomal vectos with itself becomes zeo (o ull): AA A A ( a a) 0. ) This shows that the special defiitio (8) ot oly leads to the elegat poduct fom fo calculatig Euclidea distaces of Eq. (0) but also implies that the cofomal vectos of fom (8) ae themselves zeo squae vectos, i.e. ull-vectos. To epeset the Euclidea thee space by cofomal vectos with squae zeo may at fist seem athe atificial. But aleady the scala poduct esult of Eq. (0) ad the possibility of Eq. () to fully extact the oigial vectos show that we ea a umbe of impotat advatages. Thee ae moe to come!.. Cicles ad Lies A volume elemet chaacteizes the space of which it is pat. A lie vecto u of testig whethe ay othe vecto x is paallel to the lie o ot: i Euclidea thee dimesioal space gives a way u, x paallel u x 0. (4) The geometic itepetatio of the oute poduct a b of two vectos i thee dimesios is the aea swept out by displacig oe the fist vecto a alog the secod b ad vice vesa (up to a sigificat sig chage). If we take a thid vecto x ad sweep the aea of the oute poduct of the fist two alog this thid, we get a volume, if the thid vecto is ot i the plae of the fist two.

11 (Compae the aimated ad iteactive illustatios of the oute poduct. [5] ) If x happes to be i the plae of the fist two, we must have fo the oute poduct volume a b x 0. (5) This ca be geealized. The oute poduct of d =,, o 5 liealy idepedet vectos a will idetify a d dimesioal sub-space of the cofomal vecto space spaed by the basis (). The test, whethe ay othe vecto x is pat of this d dimesioal subspace o ot is, whethe the oute poduct with x vaishes: x d-subspace a ad x 0. (6) Afte cosideig the oute poduct A B of two cofomal vectos i the last subsectio, it is pefectly atual to ask what a cofomal thee-volume A V A A costucted fom thee cofomal poits A, A, A coespods to i the thee dimesioal Euclidea subspace. I give the aswe without pove: It coespods to a (geealized) cicle passig though a ad a. The cofomal tivecto V ecodes all ifomatio about this cicle: its cete poit C, its adius ad its plae,a bivecto B. The the adius is VV. (7) ( V ) The plae B is chaacteized by simply takig the bivecto coefficiet of i V B ad the cofomal cete poit C is give by C F F /( ), (8) with F BV. The D Euclidea spatial compoets of C give accodig to Eq. (8) the vecto c. But what happes if the adius of Eq. (7) becomes ifiite, i.e. if V 0? (9) Well a cicle with ifiite adius has zeo cuvatue ad is theefoe othig but a staight lie. Compaig the subspace citeio Eq. (6) with Eq. (9) we fid that i this case the ifiity vecto will be pat of the geealized cicle. This is absolutely atual, sice we expect that a staight lie passes though ifiity. This gives us a easy ecipe fo ceatig lies: Simply take the oute poduct of two fiite cofomal poits o the lie A A ad take the oute poduct with the ifiity vecto : A A. (0) V lie It also costitutes a easy test to see if thee poits ae colliea (i.e. all o oe staight lie) o ot: V A A A 0. () a d.. Sphees ad Plaes The tasitio fom cicles to sphees happes exactly as ituitio may lead to expect. A sphee is chaacteized by the oute poduct of fou cofomal poits:

12 The adius is ow give by (mak the sig!) V A A A A4. () VV () ( V ) ad becomes ifiite fo V 0, i.e. agai if the ifiity vecto is o the sphee. A sphee with ifiite adius (zeo cuvatue) passig though ifiity is othig but a plae. The V of Eq. () theefoe chaacteizes both sphees ad plaes. The test is agai pefomed as i Eq. (9), by pefomig the oute poduct with. Fo a geuie sphee with fiite adius we ca immediately wite dow the cofomal cete C IV /( ). (4) At the momet I just state this as a cojectue, albeit oe that has show to wok well, whe implemeted as Java object method. Agai Eq. (8) tells us how to covet the cofomal cete of Eq. (4) to the Euclidea D space cete c of the sphee. The fomulas of Eqs. (), () ad (4) ae vey simple ad vey easy to implemet with a compute pogam, which is able to multiply cofomal multivectos. I thik this aleady eveals some of the tue geometic computatio powe of the cofomal model ad agai, moe is to come, especially i sectio.5.. Gade Sesitive Modificatios of Multivectos I this subsectio I will biefly itoduce basic gade sesitive modificatios of multivectos, []-[4] which ae highly useful fo deivig multivecto poducts of special sigificace ad ae fequetly applied i the maipulatio of multivectos: gade selectio, the two cojugatios of (the atiautomophic ivolutio) evesio ad (the automophic) gade ivolutio. Ay cofomal multivecto ca be witte i tems of its six gade pats whee the gade idexes of the gade selectos m m, (5) m m m m m m idicate the gades g as tabulated i Table. As a example the scala g poduct of Eq. (0) has gade 0, all vectos i Eqs. (7), (8), have gade, the oute poduct of two vectos of Eq. () has gade, oute poducts of thee vectos of Eq. (0) have gade, the oute poducts of fou vectos i Eq. () have gade 4 ad the five dimesioal pseudoscala of Eq. () has gade 5. Gade selectio applied to the geometic poduct [compae Eq. (4)] of vectos esults i ab a b a b ab ab, (6) which is aothe way of otig that the scala poduct pat is scala (of gade 0) ad that the oute poduct pat is a bivecto (of gade ). I kow immediately apply gade selectio i ode to defie the evesio (idicated by the tilde) as m ~ m m m m m m. (7) The ame evesio efes to the fact that the evese ode of a oute poduct of g liea idepedet vectos is its evesio as defied i Eq. (7). E.g. accodig to Table we have fo i i ~ i e e e e e e e e e i i. (8) Fially the gade ivolutio (o mai ivolutio) meas to chage the sig of all odd gade pats, idicated by a small hat mˆ m m m m m m. (9) 0 4 5

13 The emak is i ode that the symbols fo gade selectio, evese ad gade ivolutio sometimes diffe depedig o the authos. Kowig about gade selectio, evesio ad gade ivolutio we ae eady to deive some futhe useful poducts fom the geeal multivecto poduct (6)..4 Useful Poducts Deived fom the Geeal Multivecto Poduct We aleady kow besides the geeal multivecto poduct about the scala poduct of vectos as i Eq. (0) ad about oute poducts of two o moe vectos i Eqs. (), (0), (), etc. But it is impotat to fid out whethe the geeal multivecto poduct has some ivaiat pats, which ae of geometic sigificace ad which may fo example geealize the otios of scala poduct ad oute poduct to multivectos. The aswe to this questio is a affimative yes! Reseaches i this field have developed a small zoo of poducts deived fom the geeal multivecto poduct. [],[4],[5] I attempt hee oly to ame but a few, which ae aleady implemeted i the Java package GeometicAlgeba as methods of the object MultiVecto.class. These ae geeal scala ad oute poducts, the socalled left ad ight cotactios ad the geeal scala magitude of multivectos. ow the gade sesitive multivecto modificatios of the pecedig sectio become impotat. The geeal multivecto poduct i Eq. (6) is ot idicated by ay poduct symbol. It is always assumed if two multivectos ae witte oe afte the othe (juxtapositio). All deived poducts have thei special symbols, which sometimes diffe, depedig o the authos. The scala poduct of two multivectos m, m is defied as the scala (gade zeo) pat of the geeal multivecto poduct i Eq. (6): m m mm. (40) The Java method ScPod() implemetatio is theefoe uttely simple: The method fist applies the geeal mult() method ad tha applies the getgade(0) method to the esult. The oute poduct of two gade selected pats m, m is give by s 0 m m m m. (4) s s s This meas to fist select the gade pat m of m, the to select the gade s pat m of m, to compute the full s multivecto poduct of m ad m s accodig to Eq. (6) ad fially take the gade (+s) pat of the esult. This is pecisely the way the compute does it, usig the methods getgadei o m, ad getgade(s) o m, the the method mult() o the two gade pats ad fially the method getgade(+s) o the esult of the multiplicatio. The geeal oute poduct of two multivectos is the easily defied as the sum ove the oute poducts of all gade pats of the two factos (bilieaity of the oute poduct): m m s0 m m s. (4) I the past some eseaches i this field used a symmetic ie poduct of multivectos, which is ot to be cofused with the scala poduct of Eq. (40). But this itoduced some complicatios. [5] It theefoe seems to be moe cosistet to cotiue to use istead two poducts which ae called left ad ight cotactios. The left cotactio is to be udestood as cotactig a multivecto of lowe gades fom the left oto a multivecto of highe gades o the ight. 5 m m m m. (4),s0 The calculatio meas to fist extact the gade pat of m ad the gade s pat of m, ext to use the geeal multivecto poduct of Eq. (6) to pai wise compute all poducts of the gade pats ad the to select the (s-) pats of the poducts fo the fial summatio. Tyig to left cotact a highe gade pat fom the left side with a lowe gade pat o the ight side of this poduct (i.e. fo s<) will simply poduce zeo, sice i geometic algeba o egative gaded pats exist. Fo the case of two vectos the left cotactio is s s

14 idetical to the scala poduct of Eq. (40). The ight cotactio is defied i aalogy to the left cotactio, just that ow the cotactios has to be see as cotactig lowe gade elemets fom the ight side oto highe gades elemets o the left side of the poduct. 5 m m m m. (44),s0 s s As expected tyig to ight cotact a highe gade elemet fom the ight oto a low gade elemet to the left (i.e. s>) yields zeo. Fo the case of two vectos the ight cotactio is agai equal to the scala poduct of Eq. (40). Fo = s, the ight cotactio ad the left cotactio give the same scala valued esults. Thee is a atual geometic meaig to left ad ight cotactios. If we thik i the tems of pojectio, it is impossible to poject e.g. a aea oto a mee lie, that is a highe dimesioal etity caot be cotaied i a lowe dimesioal etity..5 Maipulatios of Basic Geometic Etities A commo obsevatio is that we ca assig cetai magitudes i geomety: leth, aea, volume, Ad as we expect theefoe, cofomal geometic algeba povides a staightfowad method of calculatio fo this. The squae of the eal scala magitude of a multivecto m is m m~ m, (45) i.e. the scala poduct of Eq. (40) of the evese (7) of a multivecto with m itself. The poduct i Eq. (45) will be positive fo ay o-zeo m fom ay positive defiite sub-algeba of the cofomal geometic algeba. I paticula the thee dimesioal Euclidea subalgeba () cotais oly positive defiite elemets. atually this coditio is obviously ot fulfilled fo the ull-vectos of oigi, ifiity, fo the cofomal poits (8) o fo algebaic elemets that cotai oe (but ot both) of the oigi ad ifiity ull-vectos as a vecto facto. Though this may ot be ou daily expeiece, special elativists ae used to such facts, which ae costituet fo the vey stuctue of ou space-time. Give that we have a positive defiite sub-algeba multivecto, we ca poceed to omalize it by dividig it by its magitude m m. (46) m This omalized multivecto ow has accodig to Eqs. (45) ad (46) the magitude (oe). Apat fom allowig us to defie the otio of agle betwee multivectos, aothe use fo the magitude is the defiitio of a ivese elemet fo positive defiite sigle gade multivectos multivecto poduct of Eq. (6): m (sometimes called blades) with espect to the geeal m m m~. (47) I emphasize that i paticula each elemet listed i the geometical algeba of Euclidea thee space () has a multiplicative ivese. We ca theefoe ow divide ot oly by scalas, but also by vectos (!), bivectos ad tivectos. It is also possible to divide by I ad of Table. The oly aspect we eed to take cae of is that divisio fom the ight ad divisio fom the left ae o loge the same thig, because geometic poducts ae geeally ot commutative. This divisio popety is of immese value fo the solutio of algebaic equatios ad it is somethig that taditioal vecto algeba completely misses out o. This last fact is quite egettable, egadig the histoical developmet of the teachig of mathematics ad should defiitely be emedied as soo as possible. I geeal it is possible to calculate itege powes of multivectos by just multiplyig them k times: k m mm m. (47) k Utilizig this we ca defie the expoetial of a multivecto m usig the familia powe seies expasio ( ) k k m exp( ) m. (48) k 0 k! As stated ealie, Hamilto s quateios ae isomophic to the set () addig the scala. This is demostated i detail by Table. It is theefoe absolutely o supise to lea that i the geometic algeba of Euclidea thee space pecisely the same two sided

15 desciptio of otatios [8] is used which made quateios so pecious to Hamilto: ~ ~ m RmR, with RR,(49) whee ow R exp( b), (50) ad b is the uit bivecto chaacteizig the plae i which the otatio takes place. I emphasize that Eq. (49) ot oly applies to ~ vectos, but to all elemets of (), because of the popety RR. But eve moe, the otos R commute by costuctio [compae Eqs. (-), (48) ad (50)] with the oigi ad ifiity ull vectos,, hece otatios about the oigi of cofomal poits (8) o highe gade elemets epesetig accodig to sectio. cicles, lies, sphees, plaes ad volumes have exactly the same fom as the otatio i Eq. (49). This makes otatios of ay geometic etity icedibly easy ad also i compute implemetatios the same oto R applied to all multivecto objects epesetig geometic etities achieves thei otatios, which geatly simplifies the pogammig. A vey special aspect of cofomal geometic algeba is, that it implemets taslatios i the vey same way as otatios i Eq. (49) which may i some degee be eve aticipated cosideig that ow multiplicatio is elated to a measue of distace as i Eq. (0). The taslatio opeato, the taslato T fo taslatig a multivecto by the Euclidea thee space vecto a is All highe ode tems of the expasio i Eq. (48) of T i powes of T exp( a) a. (5) a vaish, because of Eq. (6), i.e. the ull popety of. So fo taslatig e.g. the poit X we simply wite i full aalogy to (49) ~ X TXT, (5) agai with ~. (5) TT The ext step is to combie otos ad taslatos to achieve otatios about abitay cetes. Fo that we fist taslate fom a desied cete of otatio a back to the oigi, usig the ivese of T, i.e. T ~, the otate ad fially taslate with T back to the cete positio a : with the oto R fo otatios about as i Eq. (49). a ~ ~ ~ ~ X TRTXTRT RXR, (54) ~ ~ R TRT ad RR, (55) Fially we ca feely combie agle otatios R about ay cete a with taslatios T to ay othe positio b i space to give us combied motio-otatio opeatos also called motos D ~ D T ( b) R(, a), with DD. (56) Applyig a moto D, e.g. to a cofomal poit X would just agai be ~ X DXD. (57)

16 Because motos D commute by costuctio with the ifiity vecto ~ ad because of DD, the desciptio (57) of combied taslatios ad otatios by motos D also applies to the othe geometic etities of sectio. (cicles, lies, sphees, plaes ad volumes, etc.) Afte kowig about magitudes, the ivese ad ways to move aoud geometid objects, the questio is fo how to combie, (e.g. two lies to fom a plae) lowe dimesioal objects like two lies, o a lie ad a plae, etc. to fom highe dimesioal objects, like as plaes o the whole space, espectively. I Eqs. (), (0), () we aleady used the oute poduct of cofomal poits to ceate highe dimesioal geometic etities, like pais of poits, lies, sphees, etc. Ad ideed the oute poduct of Eq. (4) is togethe with the cotactios of Eqs. (4) ad (44) exactly what we eed i ode to descibe the set theoetic uio of poits o joi of cofomal subspaces, which ae epeseted by sigle gade multivectos (blades) as explaied i Eq. (6). Equatio (6) applies ot oly to the thee dimesioal Euclidea sub-algeba, but i fact to the whole cofomal algeba. Let us assume, that we have two cofomal subspaces epeseted by the blades W ad V, with the possibility of a commo subspace blade facto M: W W M, ad V M V. (58) The the joi J of these two subspaces will simply be J W ( M V ). (59) This is without poblem, as log as the commo blade M does ot iclude the vecto factos o by themselves. (If it cotais both, i.e., the we have o poblem, because accodig to Eq. (0), is ivese to itself.) So fo example i the case of a M of the fom of Eq. (0) M A A M, (60) (with M 0 ) we simply have to eplace i Eq. (59) M V ( ( M V )). (6) I the evet of we have to eplace i (59) M M, ad M 0, (6) M V ( ( M V )). (6) atually the easiest case is W ad V to be disjoit. The we simply have J W V. (64) Examples of the last case ae Eqs. (), (0), (), etc. The commo subspace blade facto M of (58) is the set theoetic meet o itesectio of the two subspaces W ad V. Oce we kow the joi of two subspaces, we ca i tu calculate the meet as M ( V J ) W. (65) I case that V icludes oe of the vecto factos o, i.e. V V o V V, aalogous eplacemets apply to the poduct ( VJ ) of Eq. (65), which wee made fo M V i Eq. (6) o i Eq. (6), espectively. Aothe impotat set theoetic opeatio is the pojectio of a subspace B oto aothe (sigle gade) subspace, which we deote hee as A, epeseted accodig to Eq. (6) by coespodig blades B ad A, espectively. I the case of the whole space we would simply have A = I. This is achieved [5] by ( B) ( A B) B. (66) P A By the bilieaity of the left cotactio, which is based o the bilieaity of the multivecto poduct ad of the gade selectio

17 opeatios ivolved, this ca be exteded to pojectig whole multivectos m [udestood accodig to Eq. (5) as collectios of subspaces] oto B: ( m) ( m B) B. (67) P A To coclude this selective vista of ways to maipulate geometic etities, which ae epeseted by multivectos, let us look close at the popeties of the cofomal epesetatio of a lie as i Eq. (0) utilizig Eq. (). Eq. (0) poduces V lie a a a a ). (68) ( a a is the socalled momet bivecto of the lie ad is its vecto of dietio. Fom Eq. (68) we ca calculate the a a distace vecto of ay poit x [6] fom the lie d x ( a a) ( a a). (69) ( a a) This is a diect way of calculatig the distace of ay poit x fom the lie, oly usig the two compoets of V cofomally epesetig the lie. ow we have leat eough about five dimesioal cofomal geomety to get a glimpse of its elegat multivecto epesetatios of geometic etities ad its algebaically simple but wide agig computatioal tools to feely maipulate these geometic etities. The ext sectio will theefoe be completely devoted to the way the ew Java package GeometicAlgeba implemets the geometic multivecto etities as objects (classes) ad how it allows to pefom the geometic maipulatios by meas of the methods of these objects. 4. The ew Java Package GeometicAlgeba lie This sectio gives bief ifomatio o the Java object classes cotaied i the package GeometicAlgeba, listig the costuctu of each class togethe with the available methods. Dowloadig KamiWaAi 0.0. beta fom the KamiWaAi idex page metioed i the itoductio automatically dowloads the biay code of the package GeometicAlgeba as well. Simply ispect the sub-folde GeometicAlgeba to fid the biay code! 4. Class Complexumbe This is the public class Complexumbe with the costucto: Complexumbe(double ealpat, double imagiaypat). It cotais the methods listed i Table 5. Table 5 Methods of class Complexumbe. double RealPat() double ImagiayPat() double Magitude() void setrepat(double p) void setimpat(double im) Complexumbe add(complexumbe c) Complexumbe sub(complexumbe c) Complexumbe mult(complexumbe c) 4. Class ComplexQuat This is the public class ComplexQuat with the costucto: ComplexQuat(Complexumbe[] cq). It cotais the methods listed i Table 6. Table 6 Methods of class ComplexQuat. Complexumbe getscpat() Complexumbe getscpat()

18 void setscpat(complexumbe sp) void setbvpat(complexumbe[] bp) ComplexQuat add(complexquat cq) ComplexQuat sub(complexquat cq) ComplexQuat mult(complexquat cq) 4. Class MultiVecto This is the public class MultiVecto with the costucto: MultiVecto(ComplexQuat[] mv). Table 7 shows its methods. Table 7 Methods of class MultiVecto. ComplexQuat getbapat() ComplexQuat gethbpat() ComplexQuat getscpat() ComplexQuat getpat() MultiVecto getdmvecto() MultiVecto getgade(it g) double magitude() MultiVecto omalize() MultiVecto evese() void setscpat(complexquat sp) void setpat(complexquat p) void setbapat(complexquat bap) void sethbpat(complexquat hbp) void setzeo() MultiVecto add(multivecto mv) MultiVecto sub(multivecto mv) MultiVecto mult(multivecto mv) MultiVecto Poweof(it powe) double ScPod(MultiVecto mv) MultiVecto OutPod(MultiVecto mv) MultiVecto Lcotact(MultiVecto mv) MultiVecto Rcotact(MultiVecto mv) MultiVecto multsc(double facto) void show() 4.4 Class fmv This is the public class fmv, a collectio of fequetly used multivectos. The costucto is fmv().its methods ae listed i Table 8. Table 8 Methods of class fmv. MultiVecto () MultiVecto I() MultiVecto ba() MultiVecto e() MultiVecto () MultiVecto e() MultiVecto Oe() MultiVecto e() 4.5 Class Lie This is the public class Lie with the costucto: Lie(MultiVecto L). Its methods ae listed i Table 9. Table 9 Methods of class Lie. MultiVecto Ltivecto() void setcolo( Colo c ) MultiVecto getmomet() MultiVecto getlievecto() MultiVecto LiePoit(double pa) double[] liepoitd(double pa) void dawl(gaphics g) void dawlz(gaphics g)