album de desene realizate cu functii supermatematice selariu – 2012 - supliment

8/22/2019 Album de desene realizate cu functii supermatematice Selariu 2012 - supliment

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Techno-Art of Selariu SuperMathematics Functions

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editor: Florentin Smarandache; translators: Marian Nitu, Florentin Smarandache, preface by Mircea Eugen Selariu;2007, A R Press;

127 pagesISBN: 978-1-59973-037-0;

This is also the case with the complements of ex-centric mathematics, which, reunited with the ordinary mathematics, have

been temporarily named supermathematics. It has been named this way because it generates the multiplication, from one toinfinite, ofall functions, curves, relations, etc., in other words of all actual mathematics entities. The supermathematics has

the same equation for circle as for perfect square or triangles. In supermathematics there is no difference between linear andnonlinear. And, also, as it can be observed from this album, it gets, sometimes, artistic valences. And this is just a smallhuman step in mathematics and a big leap of mathematics for the mankind.

The preparation of this album was made possible only because of the discovery of the mathematics complements. The

mathematical expressions of the new supermathematics functions constitute the base of the colored curves families, as wellas the base of some technical and/or artistic solids.

We hope that some of them will pleasantly impress your eyes. The excitement of the retina, though, is a collateral effect. Thealbum doesnt limit itself at the waves that have the capability of impressing the eye, but intends to extend to the invisible

light: infra-red and ultraviolet through which to impress the thinking, the invisible eye of the brain, the idea. The infra-redwarmly invites you to meditate on the unlimited technical and mathematical possibilities of the new functions. The ultravioletevokes a multiplication chain reaction of the existing mathematical forms/objects. Because, citing again from Grigore C.

Moisil, The most powerful explosive is not the toluene, is not the atomic bomb, but the mans idea. Between circle andsquare, as well as between sphere and cube, there exist an infinity of other supermathematics forms, which pretend the sameright to exist.

The rumor is that After Pythagoras discovered his famous theorem, he sacrificed one hundred oxen. From that time on, after

a new discovery takes place, the big horned animals have great palpitations. This story is credited to Ludwig Bjrne. In factbehind each discovery there is a story. The history records that in December 1989, the so called Romanian polenta exploded.

In 1978 it was published the first article from the domain of the mathematics functions (Ex-centric circular functions) andfrom that time on it is expected an explosion in mathematics. Is it possible that it will start with the arts?

ABSTRACT REDUCED TO 144 WORDS: This album of tehno-art represents the complements of ex-centric mathematics, which,reunited with the ordinary mathematics, have been temporarily named supermathematics. It has been named this way

because it generates the multiplication, from one to infinite, of all functions, curves, relations, etc., in other words of all actualmathematics entities. The supermathematics has the same equation for circle as for perfect square or triangles. Insupermathematics there is no difference between linear and nonlinear. And, also, as it can be observed from this album, it

gets, sometimes, artistic valences. And this is just a small human step in mathematics and a big leap of mathematics for themankind.

The preparation of this album was made possible only because of the discovery of the mathematics complements. Themathematical expressions of the new supermathematics functions constitute the base of the colored curves families, as well

as the base of some technical and/or artistic solids.
http://popupwindow%28%27https//edupublisher.com/EPBookstore/index.php?main_page=popup_image&pID=557&zenid=gunnqcon49ld90n10s45ceifl4%27)http://popupwindow%28%27https//edupublisher.com/EPBookstore/index.php?main_page=popup_image&pID=557&zenid=gunnqcon49ld90n10s45ceifl4%27)http://popupwindow%28%27https//edupublisher.com/EPBookstore/index.php?main_page=popup_image&pID=557&zenid=gunnqcon49ld90n10s45ceifl4%27)http://popupwindow%28%27https//edupublisher.com/EPBookstore/index.php?main_page=popup_image&pID=557&zenid=gunnqcon49ld90n10s45ceifl4%27)http://popupwindow%28%27https//edupublisher.com/EPBookstore/index.php?main_page=popup_image&pID=557&zenid=gunnqcon49ld90n10s45ceifl4%27)


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Traducere automat din limba englez n limba romn

Aceasta este, de asemenea, un alt caz n care matematica ordinara (centric) estecompletat, datorita apariiei unei noi dimensiuni a spaiului, excentricitatea matematic, cumatematica excentric, care, reunite, au fost denumite temporar supermatematic.

Au fost numit n acest fel, (supermatematic) deoarece genereaz o multiplicare, de launul la infinit, a tuturor funciilor, curbelor, relaiilor, etc, cu alte cuvinte a tututror entitilormatematice. Entiti supermatematice diferite au aceleai ecuaii, ca de exemplu, cercul iptratul perfect, sau triunghiul perfect.

n supermatematic nu exist nici o diferen ntre liniari neliniar. i, de asemenea, aacum se poate observa din acest album, ea prezint, uneori, valene "artistice".

i acesta este doar un nceput, un prim pas, un pas mic pentru om i un salt mare nmatematic pentru omenire.

Realizarea acestui album a fost posibil numai datorit descoperirii matematiciiexcentrice. Expresiile matematice, ale noilorfuncii supermatematice, constituie baza familiilorde curbe colorate prezentate n album, secondate de solide cunotine tehnice de programare i /sau artistice.

Sperm c unele dintre ele v vor impresiona plcut ochii. Dei, excitarea retinei este unefect colateral. Albumul nu se limiteaz la oscilaiile care au capacitatea de a impresiona ochiul.El intenioneaz s se extind i la "lumina invizibil: n infrarou i n ultraviolet", prin care simpresioneze gndire, "ochiul invizibil" a creierului, ideea.

Infrarosul clduros v invit s meditai asupra posibilitilor nelimitate tehnice i

matematice ale noilorfuncii.Ultravioletul evoc o reacie n lan de multiplicare a formelor matematice existente i aobiectelor. Deoarece, citnd din nou de la Grigore C. Moisil, "Explozivul cel mai puternic nu estetrinitrotoluenul, nu este bomba atomic, ci ideile umane". ntre cerc i ptrat, precum i ntresfer i cub, exist o infinitate de alte forme supermatematice, care pretind acelai drept de aexista.

Se zvonete c "Dup ce a descoperitcelebra sa teorem, Pitagora a sacrificat o sut deboi. Din acel moment, la fiecare nou descoperire, animalele coarnute mari au maripalpitaii. Aceasta poveste este atribuit la Ludwig Bjrne.

n fapt, n spatele fiecare descoperire este o poveste. Evenimentele istorice arat c n

decembrie 1989 aa-numita " mamalig romneasc " a facut explozie. n 1978 a fost publicatprimul articol din domeniul funciilorsupermatematice (Funcii circulare excentrice) i, din acelmoment, este de ateptat o explozie i n matematic.

Este posibil ca aceast explozie s ncep cu arta?


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REZUMAT REDUS LA 144 cuvinte:

Acest album, de art-tehnic, completeaz matematica ordinar (centric) cu matematica

excentric, care, reunite, au fost temporar numite supermatematic. Acest numire este justificatde multiplicare, de la unul la infinit, a tuturor funciilor, curbelor, relaiilor, etc, cu alte cuvinte, atuturorentitilormatematice cunoscute n matematica centric.

Graie supermatematicii cercul i ptrat perfect, sfera i cubul poerfect ca i cercul itriunghiul perfect au aceleai ecuaii parametrice.

n supermatematic nu exist nici o diferen ntre liniar i neliniar. i, de asemenea, aacum se poate observa i din acest album, ea prezint, uneori, i valene "artistice". i acesta estedoar un pas mic fcut de om n matematic i un salt mare al matematicii fcut pentru omenire.

Realizarea acestui album a fost posibil numai datorit descoperirii excentricitii i,astfel, a completrii matematicii (centrice).

Expresiile matematice ale noilor funcii supermatematice constituie baza realizrii

familiilor de curbele colorate, secondat de solide cunotinte tehnice de programare i / sauartistice.https://edupublisher.com/EPBookstore/index.php?main_page=products_new&disp_order=2&p

age=8
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TEHNOART ofELARIU SUPERMATHEMATICS FUNCTIONS

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2.1 AERONAUTICS CAPSULE / CAPSULA AERONAUTIC

www.supermathematica.com www.eng.upt.ro/~mselariu www.supermatematica.ro


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2.1,b AERONAUTICS CAPSULES /CAPSULE AERONAUTICE

,

, [ 0, 2]www.supermathematica.com www.eng.upt.ro/~mselariu www.supermatematica.ro


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2.2 D O U B L E C L E P S I D R A / C L E P S I D R A D U B L

, S(s = 1; = 0), [0, 3]; u [0, 2]


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TECHNO-ART ofELARIU SUPERMATHEMATICS FUNCTIONS

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2.2,b D O U B L E C L E P S Y D R A / C L E P S I D R A D U B L

, S(s = 1; = 0), [0, 3]; u [0, 2]


-0.5

0

0.5

-0.5

0

0.5

-1

-0.5

0

0.5

1

-0.5

0

0.5


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MIRCEA EUGEN ELARIU

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2.3 THE GATE OF THE KISS / POARTA SRUTULUI

X = (3+cos u).cos t, u [ 0, 2]Y = (3+cos u).siq t, S(s = 1 ; = 0)

Z = 2.siq t, t [ 0, 2]www.supermathematica.com www.eng.upt.ro/~mselariu www.supermatematica.ro

-4

-2

0

2

4

-4

-2

0

2

4

-10

-5

0

5

10

-4

-2

0

2

4


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MIRCEA EUGEN ELARIU

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2.3, b THE GATE OF THE KISS / POARTA SRUTULUI



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2.4 V A S E / V A Z



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2.5 SUPERMATHEMATICX HELIX / ELICE SUPERMATEMATIC



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2.6 SUPERMATHEMATICS HELIX / ELICE SUPERMATEMATICE



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2.5, b SUPERMATHEMATICS HELIX / ELICEA SUPERMATEMATIC



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2.6 CONE and ECCENTRICS PYRAMID / CON i PIRAMIDE EXCENTRICE

www.supermathematica.com www.eng.upt.ro/~mselariu

www.supermatematica.ro


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2.7 CONOPIRAMYDS / CONOPIRAMIDE

,


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2.7,b CONOPIRAMYDS ENSEMBLE / ANSAMBLU DE CONOPIRAMIDE



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TECHN0-ART ofELARIU SUPERMATHEMATICS FUNCTIONS

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K N I T T I N G S / M B R L I G R I

"Six Interwoven Loops" from The Wolfram DemonstrationsProject_http://demonstrations.wolfram.com/SixInterwovenLoops/

www.supermathematica.com www.eng.upt.ro/~mselariu

http://www.eng.upt.ro/http://www.eng.upt.ro/


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www.supermathematica.com www.eng.upt.ro/

~mselariu

http://www.eng.upt.ro/http://www.eng.upt.ro/


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S W I N D L E / P A N G L I C R I I

www.Supermathematica.com www.Supermatematica.ro


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S Q U A R E T O R U S / T O R P T R A T



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C O L O A N E L E I M B R A I R I I C U L O R I L O R



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TECHNO-ART OF ELARIU SUPERMATHEMATICS FUNCTIONS

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Motto:The book of Nature ist writtten in mathematical caractersGalileo Galilei

Chapter 1

INTRODUCTION( AT SUPERMATHEMATICS FUNCTIONS )

The Functions, which are the base to generate the most often, technical objects, so less artistical,neogeometrical, included in this album, are namedSupermathematics functions (SMF)

These functions are are the fruit of 38 research years, begun in 1969 at University of Stuttgart.Meanwhile, 42 related works were published, written by over 19 authors, as can be seen in the

bibliography.

The name belongs to the regretted mathematician Prof. em. dr. doc.ing. Gheorghe Silas which, atthe presentation of the very first work in this domain, at the First National Conference in Vibrations in

Machine Constructions, Timisoara, Romania, 1978, named CIRCULAR ECCENTRIC FUNCTIONS ,declared : Young man, you just descovered not only some functions , but a new mathematics, asupermathematics ! I was glad, at my 40 years old, like a teenager. And I proudly found that he might

be right !The prefixsuper is justified today, to point out the birth of new complements in mathematics,

joined together under the name ofEccentrical Mathematics ( EM)with much more important entitiesand infinitely more numerous than the existing entities in the actual mathematics , which weare obliged

to name it Centric Mathematics (CM.)To each entity from CM is corresponding an infinity of similar entities in EM, so

Supermathematics (SM) is the reunion of the two domains, it means SM = MC ME and MC is aparticular case, of null eccentricity ofME. Namely,MC = SM( e = 0 ). To each known function in MCis corresponding an infinity family of functions in ME , and in addition, a series of new functionsappears, with a wide range of applications in mathematics and technology.

In this way, to x = cosis corressponding the functions family x = cex = cex (, s, )wheres =e /R andare the polar coordinates of theeccenter S(s, ), corresponding to the unity/trigonometricalcircle orE(e, )corresponding to certain circle of R radius, consedered aspole of a straight line d which isrotating aroundE orSwith position angle, generating in this way the eccentrictrigonometric functions,or eccentric circular supermathematical functions (EC-SMF), by the cross ofd with unity circle (v.Fig.1).Among them the eccentric cosine of, notedcex = x, where x is the projection ofWpoint, the cross ofthe straight line with the trigonometric circle C(1,O), or the carthesian coordinate of W point. Because a

straight line, taken by S, interior to the circle (s 1 e < R), is crossing the circle in two points, W1 siW2, briefly named W1,2,

It results that two determinations of the Eccentric circular supermathematics functions (EC-SMF) will exist, one principal of indice1- cex1and one secodary cex2, of indice 2,noted brieflycex1,2.E and Swere named ex-centrebecause they were droped out of the centerO(0,0). This expulsion leadto the birth of EM and implicitly, of SM. Trough this, all the mathematical objects were multiplied fromone to infinity : To the unique function from CM, lets say cos , is corresponding an infinity offunctionscex, thanks to the possibilities to place theeccenter S and/orE in the plane.


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Mircea Eugen elariuS(e, )can take an infinity of positions in the plane where is the unity or trigonometric circle . For

each position of S andEa functioncex is obtained. IfS is a fixed point, then eccentric circular SMfunctions are obtained (EC-SMF) , with fixed eccenter, or with constantsand. ButSorEcan move, in

the plane, by various rules or laws, while the straight line which generates the functions by crossing thecircle, is rotating by the angle aroundSandE.

Fig.1 Defining ofEccentric CircularSupermathematics Functions (EC-SMF)

In this last case, we have aEC-SMFby variable point S/E eccenter, it meanss = s () and/or = ().If the variableposition ofS/Eis represented still byEC-SMFof same eccenterS(s, )or by anothereccenterS1[s1 = s1(), 1 = 1 ()], then double eccentricity functions are obtained. Trough extrapolation,triple and multiple eccentricity functions are obtained. Therefore, SMF-CE are functions of so many

variables as we wish or we need.If the distances from O to W1,2 points from C(1,O) circle are constants and equals with the

radius R = 1 of the trigonometric circle C, distances we will name eccentric radiuses pe care le vomdenumi raze centrice, the distances from Sto W1,2 noted byr1,2are variables andare namedeccentricradiuses of the unity circle C( 1,O) and represent, at the same time, new eccentric circular

supermathematics functions (EC-SMF), which were named eccentric radial functions and are noted withrex1,2,if is expressed as function of thevariable namedeccentric and motor, which is the angle fromthe eccenterE. Or, noted Rex1,2 , if it is expressed as function of the angle or centric variable, theangle ofO(0,0). The points W1,2 are seen under the angles 1,2from O(0,0) and under the angles and +fromS(e, ) and E. The straight line d is divided by S din twosemi-straight lines,one positive d +and the other negative d . Therefore, we can consider r1 = rex1a positively oriented segment on d (

M1

W1

x

y

O

SE

cex1

sex1

cex2

sex

OS = s OE = eOW1 = OW2 = 1

OM1 = OM2 = R

SW1 = r1 = rex1SW2 = r2 = rex2 EM1 = R.r1 = R.rex1EM2 = R.r2 = R.rex2

W1OA = 1W2OA = 2SOA=

S(s,)E(e,)M1,2 (R, 1,2)W1,2 (1, 1,2)

A

aex1,2 = 1,2 () = 1,2()= bex1,2 == arcsin[s.sin(-)]cex1,2 = cos 1,2sex1,2 = sin 1,2

W2

dex1,2 =

d

d 2,1=

=1 -

)(sin1

)cos(.

22

s

s

Dex 1,2 =21

d

d


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r1 > 0) and r2= rex2a negatively orientedsegmenton d (r2 < 0 ) and in the sense of the negativesemi-straight line d .

Trough simple trigonometric relations, in certain triangles OEW1,2, or, more precisely, writing

sine theoreme ( as function of) and Pitagoras generalized theoreme (for1,2 variables) in these triangles,immediately we find the invariant expressions of the eccentric radial functions :

r1,2() = rex1,2= s.cos( ) )(sin122

s and

r1,2(1,2) = Rex1,2 = )cos(..212

ss .

All EC-SMF has invariant expressions, and because this they dont need to be tabulated ;tabulated being the centric functions, from MC, which help to express them. In all their expressions, wewill find constantly, one of the square roots of previous expressions, of eccentric radial functions.

Finding these two determinations is simple : for + (plus) in the front of square roots we alwaysobtain the first determination (r1 > 0) and for the (minus) sign we obtain the second determination (r2 0), iarr2= rex2un segment orientat n sens negativ pe d (r2 < 0 ) i n sensulsemidreptei negative d

.

1 2 3 4 5 6

2

1

1

2

1 2 3 4 5 6

2

1

1

2


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Mircea Eugen elariuPrin relaii trigonometrice simple, n triunghiurile oarecare OSW1,2, sau, mai precis, scriind

teorema sinusului (n funcie de) i teorema lui Pitagora generalizat (pentru variabilele 1,2) n acestetriunghiuri, rezult imediat expresiile invariante ale funciilor radial excentrice, i anume:

r1,2() = rex1,2= s.cos( ) )(sin1 22 s i

r1,2(1,2) = Rex1,2 = )cos(..21 2,12

ss .

Cteva observaii, legate de aceste funcii REX(rege), se impun : Funciile radial excentrice exprim distana, n plan, n coordonate polare, dintre dou puncte :S(s, )i W1,2(R=1, 1,2), pe direcia dreptei excentrice d,nclinat cu unghiul fa de axa Ox;Ele au

fost normate, adic au devenit adimensionale, la sugestia Prof. Dr. Ing. Dan Perju. Ca urmare, cu ajutorul lor, i numai al lor, pot fi exprimate ecuaiile tuturor curbelor planecunoscute, ct i a altora noi, care au aprut odat cu apariiaME. Aceast constatarea, ca i denumirea derege, aparine Prof. Dr. Math. Octavian Emilian Gheorghiu, eful, de atunci, al Catedrei deMatematica 1 a Universitii POLITEHNICA din Timioara, anterior, n tineree, asistent al Acad.Grigore C. Moisil. Un exemplu il reprezintlemniscatele lui Booth (v. Fig.6), exprimate prin relaiile, ncoordonate polare, de ecuaia

() = R(rex1 + rex 2) = 2 s.R cos( - ) pentruR = 1, = 0 is [0, 3]i care constituie o transformare continu a unui cerc n dou cercuri tangente exterior (v. Fig.6, n 2D),dar care, d.p.d.v. tehnic,poate constitui un amestectorde fluide, cu dou conducte de aduciune la ntrarei una sau dou la ieire, mai dificil deproiectat, asistat de calculator, n mod obinuit.

Fig. 7,b FSM-CE radial excentrice,de variabil centric, modificate

Graie acestui obiect 3D, autorul a fost invitat de Prof. Dr. Horvat, eful Departamentului deTehnologie al Universitii din Budapesta, unde, la 3 decembrie 1998, a inut o Conferin despreSUPERMATEMATIC, la care a fost invitat i Catedra de Matematic a Universitii din Budapesta. Caurmare, au fost parafate dou colaborri n acest domeniu. O alt consecin, consist n generalizarea definiiei cercului: Cerculeste curba plana, ale crei puncte Mse gsesc la distaneler() = R.rex [, E(e, )] = R.Rex[,E(e, )],fa de un punctoarecaredin planul cerculuiE(e, ).

DacS O(0,0), atunci s = 0i rex = 1 constant i r() = R constant, obinndu-se


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definiia clasic a cercului: puncte situate la aceeai distanRde centrul cercului O. Funciilerexi Rex exprim funciile de transmitere de ordinul zero, sau de transfer al

poziiei, din teoria mecanismelor i este raportul dintre parametrul R(1,2), ce poziioneaz elementul

condus OM1,2 i parametrul r1,2() = R rex1,2 ce poziioneaz elementul conductorEM1,2.

Fig.8 FSM-CE beta excentricede variabil excentric

3 2 1 1 2 3

1

1

2

3

4


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Mircea Eugen elariuntre aceti doi parametri, exist urmatoarele relaii, care se deduc la fel de simplu din figura /

schia de definire aFSMCE (Fig. 1 ).ntre unghiurile de poziie ale celor dou elemente, condus i conductor, exist relaiile

i = 1,2 1,2(1,2) = 1,2 arcsin[

)cos(..21

)sin(.

2,1

2

2,1

ss

s] = Aex(1,2), n care sunt

unghiurile din punctele W1,2 sub care se vd centrul O i excentrul S, privind pe direciile dreptelorcentrice OW1,2i excentrice W1,2S n sensul lor pozitiv i rotind privirea, n sens trigonometric pozitiv,adic sinistrorum sau levogin. Se va putea constata c 1+ 2= .

Toate FSMCE au expresii invariante, din care cauz ele nu trebuie tabelate; tabelate fiindfunciile centrice, din MC, cu ajutorul crora se exprim. n toate expresiile lor, se va gsi, invariabil, unuldintre radicalii funciilor radial excentrice de variabil excentric

del1,2 =

Depistarea celor dou determinari este simpl: pentru + (plus) n faa radicalilor se obine,ntotdeauna, prima determinare (r1> 0), principal1i pentru semnul (minus) se obine cea de a douadeterminare (r2< 0), secundar 2. Regula ramne valabil pentru toateFSMCE.

Prin convenie, prima determinare, principal, de indice 1, se poate utiliza / scrie i fr indice,cnd confuziile sunt excluse.

Funciile aex1,2 iAex1,2 suntFSM-CEdenumiteamplitudine excentric deoarece elese pot utiliza la definirea FSM-CE cosinus i sinus excentrice tot aa cum funcia amplitudine sauamplitudinus am(k,u) a lui Jacobise foloseste la definirea funciilor eliptice Jacobi:

sn(k,u) = sin[am(k,u)] i cn(k,u) = cos[am(k,u)] .Adic:cex1,2 = cos[aex1,2(, S)] i Cex1,2 = cos[Aex(1,2, S)] (Fig.2) isex1,2 = sin [aex1,2(, S)] i Sex 1,2 = cos[Aex(1,2 ,S)] , (Fig.3) ;

Funciile radiale excentrice pot fi considerate ca module ale vectorilor de poziie aipunctelorW1,2de pe cercul unitate C(1,O), vectori exprimai prin relaiile , n carerad este vectorul unitate de direcie variabil, sau versorul /fazorul direciei drepteid+, a crui derivateste fazorul der =d(rad)/d i reprezin vectori perpendiculari pe direciile dreptelorOW1,2, tangenila cerc n punctele W1,2. Ei sunt denumii fazoriiradial centric i derivat centric.

Totodat, modulul funciei radeste corespondentul, n MC, a funcieirexpentrus = 0 = cndrex = 1iarder1,2 sunt versorii tangeni la cercul unitate n punctele W1,2.

Derivatele vectorilor de poziie ai punctelor W1,2 C, n funcie de timp,sunt vectorii vitez = . dex1,2. der = .[1 ] der, n care dex1,2 este FSM-CEdenumit derivat excentric de variabil excentric deoarece dex1,2 =

, iar inversa ei estefuncia de variabil centric

, deoarece Dex1,2

= d(

))/(d(1,2) .

Se poate observa c, introducerea fazorilor rad, rad i der, der ne scutete de scriereavectorilor cu o bar deasupra lor. Fazorii n funcie de , sau ai direciei , sunt defazai n avans fa defazorii n funcie de cu unghiul = arcsin[s.sin(-)] bex (Fig.8).

n figura 8 sunt reprezentate graficele FSM-CEbeta excentrice bex1,2: bex2sus i bex1 jos i se poate constata, facil, c suma lor este , adica 1+ 1= , sau bex1 + bex2 = .

Ele, ca i multe alte FSM-CE,sunt importante pentru c pot genera / reprezenta funcii periodicetriunghiulare simetrice, ca funcii de i n dini de ferestru, ca funcii de , pentru excentricitatea s = 1,fr serii Fourieri mult mai perfect / bine dect acestea.


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Fig. 9 FSM-CE derivat excentric dex1,2de variabil excentrici Dex1,2

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Fig. 10 FSM-Q cosinus cvadrilob coqi sinus cvadrilob siq

Dimensiunea de deformare s, deformeaz funciile cos i sin deplasndu-le punctele de

acelai y cu distana bex, pe direcia orizontal Ox, aa cum se poate constata n figura 2, transformndu-le n FSM-CE cexi, respectiv, sex. Ecartul 1, care este i domeniul de definiie al acestor funcii, sepstreaz intact. Nu i n cazul funciilor supermatematiceelevate (FSM-EL),lacare, deplasarea punctelor funciilor elevate, fa de cele circulare centrice, la creterea valorii dimensiuniide deformare s, are loc pe vertical, de unde provine i denumirea lor.

n micarea de rotaie pe cerc a punctelor W1,2, cu viteze de module variabile v1,2 = dex1,2,dreapta generatoaredse rotete n jurul excentruluiScu viteza unghiular.

Modulele vectorilor vitez au expresiile prezentate n continuare, prin FSM-CE derivatexcentric dex1,2 i Dex1,2. Expresiile funciilorSMCEdex1,2,derivat excentric de , sunt, totodati derivatele unghiurilor1,2() n funcie de variabila motoare sau independent, adic

dex1,2 =d1,2 ()/d = = , ca funcie de iDex1,2 = d/d1,2 = = , ca funcii de 1,2 .FSMCEdex1,2, prezentate n figura 9 i, respectiv Dex1,2 , iar jos sunt prezentate

n stare asamblat. Aceste funcii sunt, dup prearea autorului, cele mai frumoase funcii periodice ngeneral i cele mai frumoase FSM-CE n special, la fel de frumoase ca i funciile cvadrilobe FSM-Q(Fig.10), nu numai pentru c FSM-Q au fost introduse n Matematic de autor prin lucrarea [19].


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{s, -1 , 1} {s, -2 , 2}

Fig. 11FSM-CE amplitudine excentric(denumite, ca generalizare a dreptei, i strmbe plane)de variabil excentric aexi de variabil centric Aex

FSM-Q siq[,S] se aseamn destul de mult cu funcia eliptic Jacobi sinus eliptic sn(u,k) icoq[,S]cu cosinus eliptic cn(u,k), iarFSM-CE aex i Aex se aseamn cu funcia eliptic amplitudineam(u,k), sau amplitudinus, transformat n funcie periodic de perioada 2 cu ajutorul lui K(k).

FSM-CE amplitudine excentric prezint o importan deosebit deoarece ele generalizeaznoiunea de dreapt. Ele genereaz familii de strmbe i, pentru dimensiunea de deformare sauexcentricitatea numeric liniars = 0, se obine dreapta. n figura 11, dreapta este prima bisectoare.

Iar, pentru s = 1 se obine linia frnt,format din segmente de linii drepte.Aa cum rezult i din figura 11, FSM-CE de variabil excentric sunt continue numai n

domeniul s [-1,1], iar cele de variabil centric sunt continue pentru oricare valoare a excentricitii sie. Observaia este valabil pentru toate FSM-CE.

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]

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Fig.12 EXCENTRICE CIRCULARE

S-a demonstrat [23, 24] c, funciile SM-CE derivat excentric dex1,2 exprim funciile detransfer, sau raportul de transmitere de ordinul 1, sau ale vitezelor unghiulare, din teoria mecanismelor,

pentru toate (!) mecanismele plane cunoscute. Pentru detalii v.[23], 6.4 pag. 201 217.

Fig.13 EXCENTRICE CVADRILOBE

Funcia radial excentricrexexprimexact deplasarea mecanismului biel - manivelS =

2 1 1 2

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Fig.14,a RACHETE SUPERMATEMATICE ROMANETI

R.rex, a crui manivel motoare are lungimea r,egal cu excentricitatea reala e i lungimea bielei Leste egal cu raza cercului R, un mecanism att de cunoscut, pentru c intr n componena tuturorautoturismelor, cu excepia acelora cu motorWankel. i aplicaiile funciilor radiale excentrice ar puteacontinua, dar vom reveni la aplicaiile mai generale aleFSM-CE.


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Fig.14,b Ajutaje pentrurachetele romaneti

Concret, unicelor forme de cerc, ptrat, parabol, elips, hiperbol, diverse spirale, .m.a. din MC,grupate acum sub denumirea de centrice, denumire dat de regretatul matematician AntonHadnady, lecorespund o infinitate de forme excentrice, de acelai gen: excentrice circulare (Fig.12), ptratice(cuadrilobe Fig.13), spirale (Fig.15,bi Fig.15,d) sub form de elice (Fig.15,a, Fig.15,c i Fig.15,e),

parabolice, eliptice, hiperbolice [V. 24, SUPERMATEMATICA. FUNDAMENTE VOL.II].m.a. Cu uneledintre ele putndu-se reprezenta obiecte tehnice ca rachete, ajutaje (Fig.14) .m.a.

ParametricPlot3D[{{0.3 Cos[t] Exp[0.2 (0.25 t-ArcSin[1 Sin[0.25 t]])], 0.3 Sin[t] Exp[0.2

(0.25 t-ArcSin[1 Sin[0.25 t]])],

0.5 (0.25 t-ArcSin[Sin[0.25 t]])} }, {t,0,26}]

Cu FSM-CEamplitudine excentric (aex) de variabil excentric , de excentricitate numeric liniar s = 1 i unghiular = 0

Fig.15,a ELICEA SUPERMATEMATIC


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PolarPlot[{0.3 Exp[0.2 (t/4-ArcSin[ Sin[t/4]])]}, {t,0,10 Pi}]ParametricPlot[{{0.3 Cos[t] Exp[0.2(0.25t-ArcSin[1Sin[0.25 t]])],

0.3Sin[t] Exp[0.2 (0.25t- ArcSin[1 Sin[0.25 t]])]}},{t,0,26}]

Fig.15,b SPIRALE SUPERMATEMATICE

Ecuaii parametrice n 2D cu FSM-CEamplitudine excentric aex

www.supermathematica.com ; www.supermatematica.ro

ParametricPlot3D[{(0.3 Cos[t]/Sqrt[1-(0.9 Sin[t])^2]) Exp[0.2 (0.25 t-ArcSin[1 Sin[0.25 t]])], (0.3 Sin[t]/Sqrt[1-(0.9 Cos[t])^2]) Exp[0.2

(0.25 t-ArcSin[1 Sin[0.25 t]])], 0.5 (0.25 t-ArcSin[Sin[0.25 t]])} ,{t,0,26}]

Fig.15,cELICE SUPERMATEMATICEPTRATEde excentricitate numeric s = 1, n care FCCcosi sin sunt nlocuite cu FSM cvadrilobe

cosinus coqi sinus siqcvadrilobe (n englez quadrlobics*)ParametricPlot[{{(0.3 Cos[t]/Sqrt[1-(0.9 Sin[t])^2]) Exp[0.2 (0.25 t-ArcSin[1 Sin[0.25 t]])], (0.3 Sin[t]/Sqrt[1-(0.9 Cos[t])^2]) Exp[0.2

(0.25 t-ArcSin[1 Sin[0.25 t]])]} },{t,0,26}]
http://www.supermathematica.com/http://www.supermathematica.com/http://www.supermathematica.com/


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totodat, icurba ei generatoare. De aceea, nsi MC aparine ME, pentru unicul caz (s = e = 0), dininfinitatea de cazuri posibile n care poate fi plasat, n plan, un punct denumit excentruE(e, ). Caz n

care, E se suprapune peste unul sau dou puncte denumite centru: originea O(0,0) a unui reper,considerat originea O(0,0) a sistemului referenial i / sau centrul C(0,0) al cercului unitate, pentrufuncii circulare, respectiv, centrul de simetrie al celor dou ramuri ale hiperbolei echilaterale, pentrufuncii hiperbolice centrice i excentrice.

Fig.15,dSPIRALE SUPERMATEMATICE

A fost suficient ca un punct Es fie expulzat din centru (Oi/sau C), pentru ca, din lumea MC saparo nou lume aME, iar reuniunea celor dou lumi s dea natere lumiiSM.i aceast apariie, a avut loc n oraul revoluiei romne, din 1989, Timioara, acelai ora n

care, la 3 noiembrie 1823, Janos Bolyay scria: "Di nnimicam creat o nou lume". Cu aceste cuvinte aanunat descoperirea formulei fundamentale a primeigeometrii neeuclidiene.

El din nimic, eu din efortul colectiv de multiplicare a funciilor periodice, funcii necesareINGINERULUI pentru a descrie anumite fenomene periodice, am completat matematica cu noi funcii, cunoi obiecte, n general, cu o infinitate de entiti matematice complet noi (Fig. 15).

Dac Euler, la definirea funciilor trigonometrice, ca funcii circulare directe, n-ar fi ales treipuncte confundate: originea O, centrul cercului Ci S ca pol al unei semidrepte, cu care a intersectatcercul trigonometric/unitate,FSM-CE puteau fi cunoscute demult, eventual sub o alt denumire.

n funcie de modul n care se spliteaz(separ cte un punct, din cele suprapuse, sau toate),apar urm

toarele tipuri de FSM:

O C S Funcii Centrice, aparinnd MC; iar cele aparinndMEsuntO C S Funcii Supermatematice Circulare Excentrice (FSM-CE);O C S Funcii Supermatematice Circulare Elevate (FSM-CEL);O C S Funcii Supermatematice Circulare Exotice (FSM-CEX);Aceastecomplemente noi de matematici,reunite sub denumirea provizorie deSM, sunt unelte,

sau instrumente, deosebit de utile, de mult ateptate, dovad fiind numrul mare i diversitatea funciilorperiodice introduse n matematic i modul, uneori complicat, de a se ajunge la ele, ncercndu-sesubstituirea cercului cu alte curbe, n majoritate lor nchise.


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ParametricPlot3D[{1/13 (2 t) Cos[t] (5+Cos[(2 t)/13+u]), 1/13 (2 t) Sin[t] (5+Cos[(2

t)/13+u]),

8 (0.25 t+1/13 0.3 (2 t) Sin[(2 t)/13+u]-ArcSin[Sin[0.25 t]])},{t, 0, 26},{u, 0, 2}]

CIRCULAR PTRAT

sx = 0,4; sy = 0; sz= 0,25 TRIUNGHIULARE sx = 0,9; sy = 0; sz = 0,25ParametricPlot3D[{1/13 (2 t) Cos[t-ArcSin[0.9 Sin[t]]] (5+Cos[(2 t)/13+u]), 1/13 (2 t)

Sin[t] (5+Cos[(2 t)/13+u]),8 (0.25 t+1/13 0.3 (2 t) Sin[(2 t)/13+u]-ArcSin[Sin[0.25 t]])},{t, 0, 26},{u, 0, 2}

Fig.15,eELICE SUPERMATEMATICE



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Fig.16 ELICE:ARCURI SPIRALE DE DIVERSE SECIUNI



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Fig.17 SFERA-CUB CONOPIRAMIDA I PIRAMIDA CONIC


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Fig.18Miriapozi i cvadripozi. Rampe suport pentru lansarea rachetelor romneti


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Fig.19 TOR CENTRIC I TOR EXCENTRIC

Pentru obinerea unor funcii speciale i periodice noi, s-a ncercat nlocuirea cerculuitrigonometric cu ptratul sau cu rombul, aa cum a procedat fostul ef al Catedrei de Matematic de laFacultatea de Mecanica din Timioara, profesorul universitar Dr. mat. Valeriu Alaci, descoperindfunciile trigonometrice ptraticei rombice. Apoi, profesorul de matematic timioreanEugenViaa ntrodus funciile pseudo-hiperbolice, iar profesorul de matematici M. O.Enculescu a definit funciilepoligonale, nlocuind cercul cu un poligon cu n laturi; pentru n = 4 obinnd funciile trigonometrice

ptratice Alaci.De curnd, matematiciana american, de origine romn, Prof. Malvina Baica de la Universitatea

Wisconsin, impreun cu Mircea Crdu au completat spaiul dintre funciile circulare Euler i funciileptratice Alaci cu funciile transtrigonometrice (Periodic Transtrigonometric Functios),infratrigonometrice.m.a.

Matematicianul sovietic Marcuevici a descris, n lucrarea Funcii sinus remarcabile funciiletrigonometrice generalizatei funciile trigonometricelemniscate.

nc din anul 1877, matematicianul german Prof. Dr. August Biehringer, substituind triunghiuldreptunghic cu unul oarecare, a definit funciile trigonometricenclinate.

Savantul englez, de origine romn, ing. George (Gogu) Constantinescu a nlocuit cercul cuevolventa i a definit funciile trigonometrice ramneti: cosinus romnesc i sinusul romnesc,exprimate de funciile Cor i Sir cu care a soluionat exact unele ecuaii difereniale neliniare aleteoriei sonicitii, creat de el. i ce puin cunoscute sunt, toate aceste funcii, chiari n Romnia !Ce simple pot deveni i, de fapt, sunt lucrurile complicate! Acest paradox sugereaz c, prin simpla

deplasare / expulzare a unui punct dintr-un centru i prin apariia excentrului, poate s apar o noulume, lumeaMEi, totodat, un nou univers, universulSM.Apropo de paradox. Cel care l-a contrazis pe Albert Einstein, Prof. Dr. Math. Florentin Smarandacheeste i iniiatorul curentului de avangard numit paradoxism,la care particip peste 300 de scriitori de peglob. Pentru introducerea n Matematic a SFEREI PTRATE i a CUBULUI CIRCULAR (vezifigura 17), autorul acestui ALBUM a fost admis n Asociaia Internaional de Paradoxism, ca membrude onoare (cu diplom), cu deviza, referitoare la Supermatematic Orice este posibil, chiar iimposibilul, iar Universitatea Gallup, din Now Mexico, i-a acordat un CERTIFICAT DE APRECIERE

pentru contribuiile aduse la dezvoltarea Matematicii.i, fiindc suntem n zona aprecierilor, nici


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Fig.20 CLEPSIDRE SUPERMATEMATICE

Academia Romniei nu s-a lsat mai prejos i, n anul 1983, i-a acordat autorului Premiul Traian Vuiapentru automatizri, pe anul 1981 (!), pentru primul robot industrial romnesc REMT-1 i prima linieautomat robotizat de la Electromotor din Timioara. Premiul a fost de 0 lei, 0 bani!.i, n acelevremuri, se tia, doar teoretic, ce-i criza economic mondial. Autorul a mai conceput, proiectat i realizat,primul robot romnesc (didactic), din Laboratorul su de Proiectarea Dispozitivelor, Dispozitive de

Automatizare a Proceselor de Producie i Roboi Industriali , primul robot industrial pur pneumaticVoinicel I, care i-a pierdut braul sub berbecul unei prese vechi cu friciune, n procesul de produciede la Ambalajul Metalic din Timioara i a conceput i proiectat n 1985, pentru URSS, n cadrul unuicontract internaional, robotul de deservire a preselor de materiale plastice ROMAPET (RObotMAteriale Plastice Electrotimi Timioara). n paragraful de laude poate fi acceptat i nfiinarea,niial la Catedra de TCM i apoi la Facultatea de Mecanic, a Universitii POLITEHNICA dinTimioara a primei specializri din domeniul MECATRONICII din Romnia.


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Fig. 21 CUB N CAROURI I CUB CIOBIT

Fig.22 CUBUL ROMNESC, cel mai uor cub din lume (V = 0),format din 6 piramide cu vrful comun, fr suprafeele lor de baz.


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Noiuni ca Supermathematics Functionsi Funcii circulare excentrice au aprut pe celemai utilizate motoare de cutare ca Google, Yahoo, Altavista .a. nc de la apariia Internetului.

cel, pentru S(s [0,1], = 0) sel, pentru S(s [0,1], = 0)

cel, pentru S(s [0,1], = 1) sel, pentru S(s [0,1], = 0)Fig.23 FSM ELEVATE directei inverse

Noile noiuni, cum ar fi cea decuadrilobe quadrilobas , cu care sunt numiteexcentricelecareumplu continuu spaiul dintre un cerci unptrat, circumscris cercului, au fost incluse i n dicionarul dematematic. Intersecia cudrilobei cu drepta d genereaz noile funcii denumite cosinus cuadrilob isinus cuadrilob. Cu uneleforme matematice noi, cacele din figura 18, se mndresc i o serie de web-site-uri care creaz i distribuie programe de matematic performante. Acelai lucru se ntmpl i cu torul(Fig.18) care, din tor circular poate deveni ptrat, ca form i/sau n seciune. Diferena consist n faptulc supermatematicapoate s fac acest lucru simplu, cu FSM derivat excentric, sau cu funciicvadrilobe i chiar cu funcii centrice, pe cnd restul omenirii are nevoie de programe de matematicspecial realizate n acest scop. Chiar dac se scrie doar Cub, Thor, Sfer, .a.,n spatele lorse afl

programe elaborate de matematic, uneori stufoase, realizate cu mare efort i multe cunotinte dematematic i, mai ales, de programare pe calculatoare numerice.

Beneficiile pe careSMle aduce, n tiini n tehnologie, sunt mult prea numeroase pentru a fietalate aici. Dar, ne face o deosebit plcere s amintim cSMterge graniele dintre liniarineliniar;

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Mircea Eugen elariuliniarul aparinnd MC, iar neliniarul fiind apanajul ME, ca i dintre ideali real, sau dintreperefeciuneiimperfectiune.

ParametricPlot3D[{{0.4 u Cos[t]+8,0.1 u Sin[t-ArcSin[0.8 Sin[t]]],9.5-4.5 u},{3.5 u-

2,((0.7-0.1 u)Cos[t])/,((1-0.1 (u+0.25 u2))Cos[t+/2])/},

{4-2.1 u, (1.4-0.4 u) Cos[t], (1.2-0.4 u) Sin[t]},{2.5 u-0.5,0.3 Cos[t], 0.3 Sin[t]+0.8},

{1+4 u,(0.7-0.1 u) Cos[t], (0.8-0.2 u) Sin[t]},{0.8 u Cos[t] +3.5,-4.5+2 u,

0.2 u Sin[t-ArcSin[0.8 Sin[t-5.8]]]},{0.8 u Cos[t]+3.5,4.5-2 u,

0.2 u Sin[t-ArcSin[0.8 Sin[t-5.8]]]}},{t,0,2 },{u,1,2}]

Fig.24 AVION SUPERMATEMATIC

Se afirm cTopologia este o parte a matematicii care nu face deosebire dintre un covrig i oceac. Ambele au cte un orificiu perforat. Ei bine,SMnu face distincie dintre un cerc (e = 0) i unptrat perfect (s = 1), dintre un cerci untriunghi perfect, dintre elipsi undreptunghi perfect,

dintre o sferi un cub perfect.m.a; cu aceleai ecuaii parmetrice obinndu-se att formele ideale aleMC (cerc, elips, sfer.m.a) ct i cele reale (ptrat, dreptunghi, cub .m.a.), care nu aveau, pn decurnd, adic, pn la apariia supermatematicii, ecuaii matematice de definiie.

Pentru s [-1,1], n cazul funciilor de variabil excentric de , ca i n cazul funciilor devariabil centric , pentru s [-, +], se obin o infinitate de forme intermediare, ca de exemplu,

ptrat, dreptunghi sau cub cu coluri rotunjite i cu laturi i, respectiv, fee din ce n ce mai curbate, odatcu creterea excentricitaii s. Ceea ce faciliteaz utilizarea noilor funciiSMla desenarea i reprezentareaunor piese tehnice, cu muchii rotunjite sau teite, n programeleSM-CAD / CAM, care nu mai utilizeazcomputerul ca pe o planet de desen, ci realizeaz obiectele tehnice dintr-odat, prin ecuaii parametrice,


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cu consecine remarcabile n economia de memorare a acestora; memorate fiind ecuaiile i nu imensitateade pixeli care definesc / mrginesc o pies tehnic. A se vedea figura 24.

Numeroasele funcii prezentate, fiind pentru ntia dat introduse n matematic, pentru fixarea lor

n memorie, autorul a considerat necesar o prezentare a ecuaiilor lor, astfel nct, cei ce doresc scontribuie la extinderea aplicaiilor lor, s o poat face.Funciile SM circulare elevate (FSM-CEL), denumite astfel, pentru c, prin modificarea

excentricitii numerice s,punctele curbelor funciilor sinus elevatsel ca i a funciei circulare elevatecosinus elevat cel se eleveaz, adic se ridic pe vertical ieind din ecartul de [-1, +1] al celorlaltefuncii sinus i cosinus centrice iexcentrice.

Graficele funciilor celi selpot fi simplu reprezentate prin produsele :cel1,2 = rex1,2.cos i Cel1,2= Rex1,2.cossel1,2 = rex1,2.sin i Sel1,2 = Rex1,2.sin

i sunt prezentate, mpreun, n figura 23, numai cele directe i cele inverse, de variabil excentric .Cele mai generale funcii SM sunt funciile circulare exotice, care sunt definite pe un cerc

unitate, ne centrat n originea sistemului de axe xOy i nici n excentrul S,ci ntr-un punct oarecare C(c,

), din planul cercului unitate, de coordonate polare(c, ),n reperulxOy.

Foarte multe dintre planele cuprinse n ALBUM sunt realizate cuFSM-CEde excentru variabili de arce care sunt multiplin de (n.).Relaiile folosite, pentru fiecare caz n parte, sunt prezentate explicit, n majoritatea cazurilor

utilizndu-se funciile matematice centrice, prin care, aa cum s-a vzut, pot fi exprimate toate funciileSM, mai ales atunci cnd programele de vizualizare a graficelor nu dispun deFSM.


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Mircea Eugen elariuPrimele desene n 3D, ale funciei rex[, s], au fost reprezentate, cu muli ani n urm, de

regretatul Prof. Dr. Ing. Victor Ancua, prin deplasarea manual a hrtiei n imprimanta Hp, unic nTimioara la acea vreme, cu cte un pas, pentru fiecare valoare atribuit excentricitiis [0, 2].

Primul program de (super)matematic, de vizualizare a FSM-CE, fr scrierea explicit arelaiilor lor de definiie, ci scriind doarcex(, e, ), sex(, e, ), rex(, e, ), dex(, e, ) .a.m.d. a fostrealizat, sub denumirea comercial de Realan10, Realan11, Realan12, de programatorul american deexcepie, de origine romn, Dr. ing. Dan Mican cadrul Proiectului de Diplom de absolvire a Secieide TCM (v. imaginile dinspre parc a unora dintre laboratoarele seciei), a Facultaii de Mecanic, dincadrul Universitii POLITEHNICA din Timioara, promoia 1991.

Primul program de vizualizare a FSM-CE, de excentre variabile, adic funcii i nu constante, afost realizat apoi de Prof. dr. ing.Dnu odean, atunci asistent, acumeful Catedrei de TCM.

Ceea ce nu nseamn c, n viitor, computerele nu vor avea implementate noile complemente dematematic, pentru a le lrgi vast domeniul lor de utilizare. Microsoft a zis c mai cuget asupraavantajelor i dezavantajelor acestei aciuni. Cuget, ardelenete, de peste 10 ani !

i nici specialitii n realizarea de programe de proiectare, asistate de calculatorCAD/CAM/CAE, nu vor ntrzia prea mult n realizarea noilor programe, fundamental diferite, prin careobiectele tehnice sunt realizate cu FSM circulare sau hiperbolice parametrice, aa cum suntexemplificate unele realizri ca avioane (Fig. 24), case .a. nhttp://www.eng.upt.ro/~mselariu i cum oaib poate fi reprezentata ca o excentric toroidal(sau ca un tor excentric) ptrat sau dreptunghiularntr-o seciune axiali, respectiv, o plac ptrat cu un orificiu central ptrat poate fi un tor ptrat deseciune ptrat. Toate acestea, deoareceSM nu face distincie dintre cerc i ptrat, sau dintre elipsidreptunghi, aa cum s-a mai afirmat.
http://www.eng.upt.ro/~mselariuhttp://www.eng.upt.ro/~mselariuhttp://www.eng.upt.ro/~mselariuhttp://www.eng.upt.ro/~mselariu


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Dar, cele mai importante realizri pot fi obinute n tiin prin soluionarea unor problemeneliniare, deoareceSMreunete, ntr-un tot unitar, cele dou domenii att de diferite n trecut, dintre caredomeniul neliniar necesita ingenioase abordri pentru rezolvarea fiecrei probleme n parte.

Astfel, n domeniul vibraiilor, caracteristici elastice statice (CES) neliniare moi (regresive) sautari (progresive) se pot obine foarte simplu scriind y = m.x, numai cm nu mai este m = tan, ca n cazulliniar (s = 0), ci m = tex1,2i n funcie de semnul excentricitii numerices, pozitiv sau negativ, sau

pentru Splasat pe axa x negativ ( = ) sau pe axa x pozitiv ( = 0), se obin cele dou tipuri decaracteristici elastice neliniare i, evident, pentrus = 0 se va obine CES liniar.

Deoarece, funciilecexisexca iCexiSexi combinaiile lor, sunt soluii ale unor ecuaiidifereniale de ordinul doi, cu coeficieni variabili, s-a constatat ci pentrus = 1, inu numai pentrus =0, se obin sisteme liniare (Cebev). La acestea, masa (punctul M) se rotete pe cerc cu o vitezunghiulara = 2. = constant, dubl (fa de a sistemului liniar des = 0 de = = constant), dar serotete numai o jumtate de perioad, iar n cealalt jumtate de perioad stagneaz n punctul A(R,0),

pentrue = sR= R sau = 0i n punctul A( R, 0), pentrue = s.R= 1, sau = . n acest fel,perioada de oscilaie T, a celortrei sisteme liniare, este aceeai i egal cu T = / 2. Pentru celelalte

valori, intermediare, ale luisi e se obin sisteme de CES neliniare.Proiecia, pe oricare direcie, a micrii de rotaie a punctului M pe cercul de raza R, egal cuamplitudinea oscilaiei, cu viteza unghiular = .dex variabil (dup funciadex) este o micareoscilantneliniar.

Apariia funciei rege rex i a proprietilor ei a facilitat apariia unei metode hibride(analitico-numeric) prin care s-a obinut o relaie simpl, cu numai doi termeni, de calcul a integraleieliptice complete de prima speK(k), cu o precizie incredibil de mare, de minimum 15 zecimale exacte,dup numai 5 pai. Realizarea pailor urmtori, poate conduce la obinerea unei noi relaii de calcul a luiK(k), cu precizie considerabil mai mare i cu posibiliti de extindere i la alte integrale eliptice i nunumai. Relaia lui E(k), dup 6 pai, are aceeai precizie de calcul [23], [24].

ApariiaFSMa facilitat apariia unei noi metode de integrare, denumitaintegrare prin divizareadiferenialei [25]. Cunete avantaje, ce rezult din soluionarea simpl, n domeniul real, al unor integralerezolvabile n domeniul complex prin teorema reziduurilor.

SM nu este o lucrare ncheiata ci, de abia o introducere n acest domeniu vast, un prim pas, unpas mic al autorului i un pas uria al matematicii.

Ne oprim aici, pentru a nu va rpi din plcerea de-a v delecta privirea cu planele prezentuluiALBUM.

Vizionare plcut !Autorul

e-mail :[email protected]@gmail.com;

www.supermathematica.comwww.supermatematica.ro

www.eng.upt.ro/~mselariu

www.cartiAZ.ro
mailto:[email protected]:[email protected]:[email protected]://www.supermathematica.com/http://www.supermathematica.com/http://www.supermatematica.ro/http://www.supermatematica.ro/http://www.supermatematica.ro/http://www.supermathematica.com/mailto:[email protected]


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elariu Mircea Eugen

elariuM

irceaEugen

TECHN

O-ARTOFSELARIUSUPERMATH

EMATICSFUNCTIONS

ISBN

TECHNO-ART OF ELARIU

SUPERMATHEMATICS

FUNCTIONS

2-th Edition 2012

Editura Nexia Oradea


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Mircea Eugen elariu

MirceaEugenelariu

TECHN

O-ARTOFSELARIUSUPERMATH

EMATICSFUNCTIONS

ISBN

TECHNO-ART OF ELARIU

SUPERMATHEMATICS

FUNCTIONS

2-th Edition 2012

Editura Nexia Oradea


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D E S E N E D I N P L A C I


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BUM

Mircea Eugen elariu

MirceaEugenelariu

ALBUM

DEDESENE

REALIZATECUFUNCI

ISUPERMATEMATICE

REALIZATE CU FUNCTII

SUPERMATEMATICE ELARIU

DE DESENE

2012


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0. FOREWORD(FOR SUPERMATHEMATICS FUNCTIONS)

In this album we include the so called Supermathematics functions (SMF), which

constitute the base for, most often, generating technical, neo-geometrical objects, therefore less

artistic.

These functions are the results of 38 years of research, which began at University of Stuttgart in

1969. Since then, 42 related works have been published, written by over 19 authors, as shown in the

References.

The name was given by the regretted mathematician Professor Emeritus Doctor Engineer

Gheorghe Silas who, at the presentation of the very first work in this domain, during the First

National Conference of Vibrations in Machine Constructions, Timioara, Romania, 1978, named

CIRCULAR EX-CENTRIC FUNCTIONS, declared: Young man, you just discovered not only

some functions, but a new mathematics, a supermathematics! I was glad, at my age of 40, like a

teenager. And I proudly found that he might be right!

The prefix super is justified today, to point out the birth of the new complements in

mathematics, joined together under the name of Ex-centric Mathematics (EM), with much more

important and infinitely more numerous entities than the existing ones in the actual mathematics,

which weare obliged to call it Centric Mathematics (CM.)

To each entity from CM corresponds an infinity of similar entities in EM, therefore the

Supermathematics (SM) is the reunion of the two domains: SM = CM EM, where CM is a

particular case of null ex-centricity ofEM. Namely, CM = SM(e = 0). To each known function inCM corresponds an infinite family of functions in EM, and in addition, a series of new functions

appear, with a wide range of applications in mathematics and technology.

In this way, to x = cos corresponds the family of functions x = cex = cex(, s, ) where s= e/Rand are the polar coordinates of the ex-center S(s,), which corresponds to theunity/trigonometric circle orE(e, ),which corresponds to a certain circle of radius R, considered aspoleof a straight line d, which rotates around EorS with the position angle , generating in thisway the ex-centric trigonometric functions, or ex-centric circular supermathematics functions (EC-

SMF), by intersecting d with the unity circle (see.Fig.1). Amongst them the ex-centriccosineof,denoted cex = x, where x is the projection of the point W, which is the intersection of the straightline with the trigonometric circle C(1,O), or the Cartesian coordinates of the point W. Because a

straight line, passing through S, interior to the circle (s 1eR), intersects the circle in twopoints W1 and W2, which can be denoted W1,2, it results that there are two determinations of the

ex-centric circular supermathematics functions (EC-SMF): a principal one of index 1cex1, and asecondary one cex2, of index 2, denotedcex1,2.EandS were named ex-centersbecause theywere excluded from the centerO(0,0). This exclusion leads to the apparition of EM and implicitly

of SM. By this, the number of mathematical objects grew from one to infinity: to a unique function

from CM, for example cos, corresponds aninfinity of functionscex, due to the possibilities ofplacing theex-center Sand/orE anywhere in the plane.

S(e, )can take an infinite number of positions in the plane containing the unity ortrigonometric circle. For each position ofS and E we obtain a functioncex. IfS is a fixed point,then we obtain the ex-centric circularSM functions (EC-SMF), with fixed ex-center, or with


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constants and . But S orE can take different positions, in the plane, by various rules or laws, whilethe straight line which generates the functions by its intersection with the circle, rotates with the

angle aroundSandE.

Fig.1 Definition of Ex-centric Circular Supermathematics Functions (EC-SMF)

In the last case, we have an EC-SMF of ex-center variable point S/E, which meanss = s ()and/or = (). If the variable position ofS/E is represented also by EC-SMF of the same ex-centerS(s, ) or by another ex-center S1[s1 = s1(), 1= 1 ()], then we obtain functions of double ex-

centricity. By extrapolation, well obtain functions of triple, and multiple ex-centricity. Therefore,EC-SMFare functions of as many variables as we want or as many as we need.

If the distances from O to the points W1,2on the circle C(1,O) areconstant and equal to the

radius R = 1 of the trigonometric circle C, distances that will be named ex-centric radiuses, the

distances from S to W1,2 denoted by r1,2 are variables and are named ex-centric radiuses of the unity

circle C(1,O) and represent, in the same time, new ex-centric circular supermathematics functions

(EC-SMF), which were named ex-centric radial functions, denoted rex1,2, if are expressed infunction of the variablenamed ex-centric and motor, which is the angle from the ex-centerE.Or, denoted Rex1,2, if it is expressed in function of the angle orthe centric variable, the angleat O(0,0). The W1,2 are seen under the angles 1,2from O(0,0) and under the angles and + from S(e, )andE. The straight line d is divided by S din the twosemi-straight lines, one

M1

W1

x

y

O

SE

cex1

sex1

cex2

sex

OS = s OE = e

OW1 = OW2 = 1

OM1 = OM2 = R

SW1 = r1 = rex1

SW2 = r2 = rex2

EM1 = R.r1 = R.rex1

EM2 = R.r2 = R.rex2

W1OA = 1W2OA = 2

SOA=

S(s,)E(e,)

M1,2 (R, 1,2)

W1,2 (1, 1,2)

A

aex1,2 = 1,2 () = 1,2()

= bex1,2 == arcsin[s.sin(-)]cex1,2 = cos 1,2

sex1,2 = sin 1,2

W2

dex1,2 =

d

d 2,1=

=1 -

)(sin1

)cos(.

22

s

s

Dex 1,2 =21

d

d


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positived + and the other negative d. For this reason, we can considerr1 = rex1a positiveoriented segment on d ( r1 > 0) andr2= rex2 a negative orientedsegmenton d(r2 < 0) in thenegative sense of the semi-straight line d .

Using simple trigonometric relations, in certain triangles OEW1,2, or, more precisely,

writing the sine theorem (as function of) and Pitagoras generalized theorem (for the variables1,2) in these triangles, it immediately results the invariant expressions of the ex-centric radialfunctions:

r1,2 () = rex1,2 = s.cos( ) )(sin122

s

and

r1,2(1,2) = Rex1,2 = )cos(..212

ss .

All EC-SMF have invariant expressions, and because of that they dont need to be

tabulated, tabulated being only the centric functions from CM, which are used to express them. Inall of their expressions, we will always find one of the square roots of the previous expressions, of

ex-centric radial functions.

Finding these two determinations is simple: for + (plus) in front of the square roots we

always obtain the first determination (r1 > 0) and for the (minus) sign we obtain the seconddetermination (r2 < 0). The rule remains true for all EC-SMF. By convention, the first

determination, of index 1, can be used or written without index.

Some remarks about these REX(King) functions:

The ex-centric radial functions are the expression of the distance between two points,in the plane, in polar coordinates: S(s, ) and W1,2 (R=1, 1,2), on the direction of the

straight line d, skewedat an angle in relation to Ox axis;

Therefore, using exclusively these functions, we can express the equations of allknown plane curves, as well as of other new ones, which surfaced with the

introduction ofEM.An example is represented by Booths lemniscates (see Fig. 2,a, b, c), expressed, in polar coordinates, by the equation:

() = R(rex1+rex 2) = 2 s.Rcos( - ) forR=1, = 0 ands [0, 3]

Fig.2,a BoothsLemniscates forR = 1andnumerical ex-centricity e [1.1, 2]

Fig. 2,b BoothsLemniscates forR = 1andnumerical ex-centricity e [2.1, 3]

-1 -0.5 0.5 1

-0.4

-0.2

0.2

0.4

-1 -0.5 0.5 1

-0.2

-0.1

0.1

0.2


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Another consequence is the generalization of the definition of a circle:The Circle is the plane curve whose points M are at the distances r() = R.rex =R.rex [, E(e, )] in relation to a certainpoint from the circles planeE(e, ).IfS O(0,0), thens = 0 and rex = 1 = constant, andr() = R= constant, we obtainthe circles classical definition: the points situated at the same distance R from a

point, the center of the circle.

Fig. 2,c


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The functionsrex and Rex expresses the transfer functions of zero degree, or ofthe position of transfer, from the mechanism theory, and it is the ratio between the

parameter R(1,2), which positions the conducted element OM1,2 and parameterR.r1,2(), which positions the leader elementEM1,2.Between these two parameters, there are the following relations, which can be

deduced similarly easy from Fig. 1 that defines EC-SMF.

Between the position angles of the two elements, leaded and leader, there are the

following relations:

1,2 = arcsin[e.sin()] = 1,2() =aex1,2

and

= 1,2 1,2(1,2 )= 1,2 arcsin[)cos(..21

)sin(.

2,1

2

2,1

ss

s ] = Aex (1,2 ).

The functions aex 1,2 and Aex 1,2 are EC-SMF, called ex-centric amplitude,because of their usage in defining the ex-centric cosine and sine from EC-SMF, in

the same manner as the amplitude function or amplitudinus am(k,u) is used for

defining the elliptical Jacobi functions:

sn(k,u) = sin[am(k,u)], cn(k,u) = cos[am(k,u)],

or:

cex1,2 = cos(aex1,2) , Cex1,2 = cos(Aex1,2)andsex 1,2 = sin (aex1,2), Sex 1,2 = cos (Aex 1,2)

The radial ex-centric functions can be considered as modules of the position vectors

2,1r for the W1,2 on the unity circle C (1,O). These vectors are expressed by the

following relations:

radrexr .2,12,1

,

where rad is the unity vector of variable direction, or the versor/phasor of thestraight line directiond

+, whose derivative is the phasorder =d(rad)/dandrepresents normal vectors on the straight lines OW1,2, directions, tangent to the circle

in the W1,2. They are named the centric derivative phasors. In the same time, the

modulus rad function is the corresponding, in CM, of the function rex fors = 0 = when rex = 1 andder 1,2 are the tangent versors to the unity circle inW1,2.

The derivative of the 2,1r vectors are the velocity vectors:


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2,12,1

2,1

2,1 .

derdex

d

rdv

of the W1,2 C points in their rotating motion on the circle, with velocities ofvariable modulus v1,2 =dex1,2, when the generating straight line d rotates aroundthe ex-centerS with a constant angular speed and equal to the unity, namely = 1.The velocity vectors have the expressions presented above, where der 1,2 are the

phasors of centric radiuses R1,2 of module 1 and of1,2 directions. The expressionsof the functions EC-SMdex1,2 ,ex-centric derivativeof, are, in the same time,also the1,2() angles derivatives, as function of the motor or independent variable, namely

dex1,2 = d1,2 ()/d = 1

)(sin.1

)cos(.

22

s

s

as function of, and

Dex 1,2 = d()/d1,2 =2,1

2

2,1

2,1

2

2,1

Re

)cos(.1

)cos(..21

)cos(.1

x

s

ss

s

,

as functions of 1,2 .It has been demonstrated that the ex-centric derivative functionsEC-SMFexpress

the transfer functions of the first order, or of the angular velocity, from the

Mechanisms Theory, forall (!) known plane mechanisms.

The radial ex-centric function rex expresses exactly the movement of push-pullmechanism S = R. rex, whose motor connecting rod has the length r, equal with ethe real ex-centricity, and the length of the crank is equal to R, the radius of thecircle, a very well-known mechanism, because it is a component of all automobiles,

except those with Wankel engine.

The applications of radial ex-centric functions could continue, but we will concentrate now

on the more general applications ofEC-SMF.

Concretely, to the unique forms as those of the circle, square, parabola, ellipse, hyperbola,

different spirals, etc. from CM, which are now grouped under the name ofcentrics, correspond an

infinity of ex-centrics of the same type: circular, square (quadrilobe), parabolic, elliptic,

hyperbolic, various spirals ex-centrics, etc. Any ex-centric function, with null ex-centricity (e =

0), degenerates into a centric function, which represents, at the same time its generating curve.

Therefore, the CM itself belongs to EM, for the unique case (s = e = 0), which is one case from an

infinity of possible cases, in which a point named eccenterE(e, )can be placed in plane. In thiscase, E is overleaping on one or two points named center: the origin O(0,0) of a frame, considered

the origin O(0,0) of the referential system, and/or the centerC(0,0) of the unity circle for circular

functions, respectively, the symmetry center of the two arms of the equilateral hyperbola, for

hyperbolic functions.

It was enough that a point E be eliminated from the center (O and/orC) to generate from

the old CM a new world ofEM. The reunion of these two worlds gave birth to theSM world.

This discovery occurred in the city of the Romanian Revolution from 1989, Timioara,which is the same citywhere on November 3

rd, 1823 Janos Bolyai wrote: FromnothingIve created anew world. With these words, he announced the discovery of the fundamental formula of the first non-Euclidean geometry.


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Hefrom nothing, Iin a joint effort, proliferated the periodical functions which are so helpful to

engineers to describe some periodical phenomena. In this way, I have enriched the mathematics with new

objects.When Euler defined the trigonometric functions, as direct circular functions, if he wouldnt have

chosen three superposed points: the origin O, the center of the circle C and S as a pole of a semi straightline, with which he intersected the trigonometric/unity circle, the EC-SMF would have been discovered

much earlier, eventually under another name.Depending on the way of the split (we isolate one point at the time from the superposed ones, or

all of them at once), we obtain the following types ofSMF:

O C S Centric functions belonging to CM;and those which belong toEM are:

O C S Ex-centric Circular SupermathematicsFunctions (EC-SMF);O C S Elevated Circular SupermathematicsFunctions (ELC-SMF);O C S Exotic Circular SupermathematicsFunctions (EXC-SMF).These new mathematics complements, joined under the temporary name of SM, are extremely

useful tools or instruments, long awaited for. The proof is in the large number and the diversity of periodicalfunctions introduced in mathematics, and, sometimes, the complex way of reaching them, by trying thesubstitution of the circle with other curves, most of them closed.

To obtain new special, periodical functions, it has been attempted the replacement of thetrigonometric circle with the square or the diamond. This was the proceeding of Prof. Dr. Math. ValeriuAlaci, the former head of the Mathematics Department of Mechanics College from Timisoara, whodiscovered the square and diamond trigonometric functions. Hereafter, the mathematics teacher Eugen

Visa introduced the pseudo-hyperbolic functions, and the mathematics teacher M. O. Enculescu definedthe polygonal functions, replacing the circle with an n-sides polygon; for n = 4 he obtained the square Alacitrigonometric functions. Recently, the mathematician, Prof. MalvinaBaica, (of Romanian origin) from the

University of Wisconsin together with Prof. Mircea Crdu, have completed the gap between the Eulercircular functions and Alaci square functions, with the so-called Periodic Transtrigonometric functios.

The Russian mathematician Marcusevici describes, in his workRemarcable sine functions thegeneralized trigonometric functions and the trigonometric functions lemniscates.

Even since 1877, the German mathematician Dr. Biehringer, substituting the right triangle with an

oblique triangle, has defined the inclined trigonometric functions. The British scientist of Romanian originEngineerGeorge (Gogu) Constantinescu replaced the circle with the evolventand defined the Romanian

trigonometric functions: Romanian cosine and Romanian sine, expressed by Cor and Sir functions,which helped him to resolve some non-linear differential equations of the Sonicity Theory, which he created.

And how little known are all these functions even in Romania!Also, the elliptical functions are defined on an ellipse. A rotated one, with its main axis along Oy

axis.

How simple the complicated things can become, and as a matter of fact they are! This

paradox(ism) suggests that by a simple displacement/expulsion of a point from a center and by the

apparition of the notion of the eccenter, a new world appeared, the world of EMand, at the same

time, a new Universe, the SMUniverse.

Notions like Supermathematics Functions and Circular Ex-centric Functions

appeared on most search engines like Google, Yahoo, AltaVista etc., from the beginning of the

Internet. The new notions, like quadrilobe quadrilobas, how the ex-centrics are named, and

which continuously fill the space between a square circumscribed to a circle and thecircle itself


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were included in the Mathematics Dictionary. The intersection of the quadriloba with the straight

line d generates the new functions called cosinequadrilobe-ic andsine quadrilobe-ic.

The benefits ofSM in science and technology are too numerous to list them all here. But weare pleased to remark that SM removes the boundaries between linear and non-linear; the linear

belongs to CM, and the non-linear is the appanage ofEM, as between ideal and real, or as between

perfection and imperfection.

It is known that the Topology does not differentiate between a pretzel and a cup of tea.

Well, SM does not differentiate between a circle(e = 0) and a perfect square (s = 1), between a

circle and a perfect triangle, between an ellipse and a perfect rectangle, between a sphere and a

perfectcube, etc. With the same parametric equations we can obtain, besides the ideal forms of

CM (circle, ellipse, sphere etc.), also thereal ones (square, oblong, cube, etc.). Fors [-1,1], in the

case of ex-centric functions of variable, as in the case of centric functions of variable , fors[-

,+], it can be obtained an infinity of intermediate forms, for example, square, oblong or cubewith rounded corners and slightly curved sides or, respectively, faces. All of these facilitate the

utilization of the new SM functions for drawing and representing of some technical parts, with

rounded or splayed edges, in the CAD/ CAM-SM programs, which dont use the computer as

drawing boards any more, but create the technical object instantly, by using the parametric

equations, that speed up the processing, because only the equations are memorized, not the vast

number of pixels which define the technical piece.

The numerous functions presented here, are introduced in mathematics for the first time,

therefore, for a better understanding, the author considered that it was necessary to have a short

presentation of their equations, such that the readers, who wishes to use them in their applications

development, to be able to do it.

SM is not a finished work; its merely an introduction in this vast domain, a first step, theauthors small step, and a giant leap for mathematics.

The elevated circular SM functions (ELC-SMF), named this way because by the

modification of the numerical ex-centricity s the points of the curves of elevated sine functions sel as of the elevated circular function elevated cosine cel is elevatingin other words it rises on thevertical, getting out from the space {-1, +1] of the other sine and cosine functions, centric or ex-

centric. The functionscex and sex plots are shown in Fig. 3, where it can be seen that thepoints of these graphs get modified on the horizontal direction, but all remaining in the space

[-1,+1], named the existence domain of these functions.

The functions cel and sel plots can be simply represented by the products:

cel 1,2 = rex1,2 . cos and Cel1,2= Rex 1,2. cossel 1,2 = rex 1,2 . sin and Sel 1,2 = Rex 1,2. sin

and are shown Fig. 4.

The exotic circular functionsare the most general SM, and are defined on the unity circle

which is not centered in the origin of the xOy axis system, neither in the eccenterS,but in a certain

point C (c,) from the plane of the unity circle, of polar coordinates (c, ) in the xOy coordinatesystem.

Many of the drawings from this album are done with EC-SMF of eccenter variable and with

arcs that are multiples of (n.). The used relations for each particular case are explicitly shown, inmost cases using the centric mathematical functions, with which, as we saw, we could express all


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SM functions, especially when the image programs cannot use SMF. This doesnt mean that, in the

future, the new math complements will not be implemented in computers, to facilitate their vast

utilization.

Fig. 3,a The ex-centric circular

supermathematicsfunction

(EC-SMF) ex-centric cosine of cex

for = 0, [0, 2]

Fig. 3,b The ex-centric circular

supermathematicsfunction

(EC-SMF) ecentric sine of sex for

= 0, [0, 2]

Numerical ex-centricity s = e/R [ -1, 1]

The computer specialists working in programming the computer assisted design software

CAD/CAM/CAE, are on their way to develop these new programs fundamentally different,

because the technical objects are created with parametric circular orhyperbolicSMFs, as it has

been exemplified already with some achievements such as airplanes, buildings, etc. in

http://www.eng.upt.ro/~mselariuand how a washer can be represented as a toroid ex-centricity (or

as an ex-centric torus), square oroblong in an axial section, and, respectively, a square plate with

a central square hole can be a square torus of square section. And all of these, because SM

doesnt make distinction between a circle and a square or between an ellipse and a rectangle, as we

mentioned before.
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But the most important achievements in science can be obtained by solving some non-linear

problems, because SM reunites these two domains, so different in the past, in a single entity.

Among these differences we mention that the non-linear domain asks for ingenious approaches foreach problem. For example, in the domain of vibrations, static elastical characteristics (SEC) soft

non-linear (regressive) or hard non-linear (progressive) can be obtained simply by writing y = m. x,

where m is not anymorem = tan as in the linear case (s = 0 ), but m = tex1,2 and depending onthe numericalex-centricityssign, positive or negative, or forS placed on the negative x axis ( =)or on the positive x axis ( = 0), we obtain the two nonlinear elastic characteristics, and obviously

for s=0 well obtain the linear SEC.

Fig. 4,aELC-SMF elevated cosine of - cel, fors [-1, +1], = 0, [0, 2].

Fig. 4,b ELC-SMFelevated sine of - sel,fors[-1, +1], = 1, [0, 2].

Due to the fact that the functions cex and sex , as well Cex and Sex and theircombinations, are solutions of some differential equations of second degree with variable

coefficients, it has been stated that the linear systems (Tchebychev) are obtained also fors = 1,and not only for s = 0. In these equations, the mass ( the point M) rotates on the circle with a

double angular speed = 2. (reported to the linear system where s = 0 and = = constant)in a half of a period, and in the other half of period stops in the point A(R,0) fore = sR= R or =

0and in A(R, 0) for e =s.R= 1, or = .Therefore, the oscillation period T of the threelinear systems is the same and equal with T= / 2. The nonlinear SEC systems are obtained forthe others values, intermediates, ofs and e. The projection, on any direction, of the rotating motion

ofM on the circlewith radius R, equal to the oscillation amplitude, of a variable angular speed =.dex ( afterdex function) is an non-linear oscillating motion.

The discovery of kingfunction rex, with its properties, facilitated the apparition of ahybrid method (analytic-numerical), by which a simple relation was obtained, with only two

1 2 3 4 5 6

-1

-0.5

0.5

1

1.5

2

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-2

-1.5

-1

-0.5

0.5

1

1 2 3 4 5 6

-1

-0.5

0.5

1

1.5

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-1.5

-1

-0.5

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1

s [0, 1]

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terms, to calculate the first degree elliptic complete integral K(k), with an unbelievable precision,

with a minimum of15 accurate decimals, after only 5 steps. Continuing with the next steps, can

lead us to a new relation to compute K(k), with a considerable higher precision and withpossibilities to expand the method to other elliptic integrals, and not only to those. After 6 steps, the

relation ofE (k) has the same precision of computation.

The discovery of SMF facilitated the apparition of a new integration method, named

integration through the differential dividing.

We will stop here, letting to the readers the pleasure to delight themselves by viewing the

drawings from this album.

The Author

[email protected]
mailto:[email protected]:[email protected]:[email protected]

album de desene realizate cu functii supermatematice selariu – 2012 - supliment

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