Numere complexe, Cuaternioni si Aplicatiimarcus/for_students/pres_marcus-quaternions.pdf · Cuprins Numere complexe CuaternioniOctonioniBibliogra e Octonioni John Baez The real numbers

Cuprins Numere complexe Cuaternioni Octonioni Bibliografie

Numere complexe, Cuaternioni si Aplicatii

Andrei Marcus

http://math.ubbcluj.ro/˜marcus/

Junior Summer University, 19 iulie 2011

Universitatea Babes-Bolyai Cluj-NapocaFacultatea de Matematica si Informatica

Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni


Cuprins I

1 Numere complexeDefinitieIstoricPlanul complexRotatii ın plan

2 CuaternioniDefinitie si proprietatiRotatie ın jurul unei axe

3 Octonioni

4 Bibliografie



Definitie

Numere complexe

x2 + 1 = 0; i2 = −1;

C = R× R

z = (a, b) = a + bi 7→(a −bb a

)(reprezentare matriceala)

conjugatul: z = a − bi



Istoric

Numere complexe

Niccolo Tartaglia (1499–1557); Scipione del Ferro (1465–1526)Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni


Istoric

Numere complexe

Girolamo Cardano (1501–1576); Rene Descartes (1596–1650)Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni


Planul complex

Planul complex

Diagrama Argand: r =√zz , sin θ = y/r , cos θ = x/r .

Formula lui Euler: e iθ = cos θ + i sin θ; e iπ = −1.Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni


Planul complex

Planul complex

Caspar Wessel (1745–1818); Jean-Robert Argand (1768–1822)



Planul complex

Euler (1707–1783)

Gauss (1777–1855)



Planul complex

Carl Friedrich Gauss ın 1828



Rotatii ın plan

Rotatia ın plan

(x ′

y ′

)=

(cos θ − sin θsin θ cos θ

)(xy

)



Definitie si proprietati

Cuaternioni

Hamilton 1843: H = R× R× R× Rq = (a, b, c , d) = a + bi + cj + dk , undei2 = j2 = k2 = ijk = −1;

Gauss: ınmultirea cuadruplelor;

Lagrange: Teorema celor 4 patrate.

conjugatul: q := a − bi − cj − dk ,

norma: |q| =√qq;

q−1 = q/|q|2; q′ = q/|q| are norma 1.




Cuaternioni

Sir William Rowan Hamilton (1805 – 1865)




Cuaternioni

Reprezentare matriceala

q 7→(

z −ww z

)7→

a −b −c −db a −d cc d a −b−d c b a




Cuaternioni: notatia vectoriala

q = s + ~v = [s, ~v ], unde s ∈ R, ~v = x~i + y~j + z~k




Cuaternioni: notatia vectoriala

Fie

qa = sa + ~va = [sa, ~va] = sa + xa~i + ya~j + za~k ,

qa = sb + ~vb = [sb, ~vb] = sb + xb~i + yb~j + zb~k .

Avem

qaqb = (sasb −~a · ~b) + sa~b + sb~a +~a × ~b,

unde

~va · ~vb = xaxb + yayb + zazb;

~va × ~vb =

∣∣∣∣∣∣~i ~j ~kxa ya zaxb yb zb

∣∣∣∣∣∣Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni


Rotatie ın jurul unei axe


Rotatie ın jurul unei axe; Olinde Rodrigues (1794–1851)





Fie q = s + λ~v un cuaternion de lungime 1,adica |~v | = 1, s2 + λ2 = 2.Notam λ = sin θ si s = cos θ.Fie p = [0, ~p] un cuaternion pur. Calculam conjugatul lui p:

qpq−1 = [s, λ~v ][0, ~p][s,−λ~v ]

= [−λ~v · ~p, s~p + λ~v × ~p][s,−λ~v ]

= [−λs~v · p + λs~p · ~v + λ2(~v × ~p) · ~v ,λ2(~v · ~p)~v + s2~p + λs~v × ~p − λs~p × ~v − λ2(~v × ~p)× ~v ]

= [λ2(~v × ~p) · ~v , λ2(~v · ~p)~v + s2~p + 2λs~v × ~p − λ2(~v × ~p)× ~v ].





Avem(~v × ~p) · ~v = 0

si(~v × ~p)× ~v = (~v · ~v)~p − (~p · ~v)~v = ~p − (~p · ~v)~v ,

deci

qpq−1 = [0, λ2(~v · ~p)~v + s2p + 2λs~v × ~p − λ2~p + λ2(~p · ~v)~v ]

= [0, 2λ2(~v · ~p)~v + (s2 − λ2)~p + 2λs~v × ~p].





Inlocuind s = cos θ si λ = sin θ obtinem:

qpq−1 = [0, 2 sin2 θ(~v · ~p)~v + (cos2 θ − sin2 θ)~p + 2 sin θ cos θ~v × ~p]

= [0, (1− cos 2θ)(~v · ~p)~v + cos 2θ~p + sin 2θ~v × ~p].

Observam ca luand q = [cos 12θ, sin 1

2θ~v ],

obtinem exact formula lui Rodrigues:

qpq−1 = [0, (1− cos θ)(~v · ~p)~v + cos θ~p + sin θ~v × ~p],

deci p′ = qpq−1 este chiar rotatia de unghi θ a vectorului ~p ınjurul vectorului ~v .




Rotatie – forma matriceala

Fieq = s + ~v ;~v = x~i + y~j + z~k ;|q|2 = s2 + |~v |2 = s2 + x2 + y 2 + z2 = 1;

~p = xp~i + yp~j + zp~k ;

Atuncip′ = qpq−1 =2(s2 + x2)− 1 2(xy − sz) 2(xz + sy)

2(xy + sz) 2(s2 + y 2)− 1 2(yz − sx)2(xz − sy) 2(yz + sx) 2(s2 + z2)− 1

xpypzp



Octonioni

O = H×H;

Definition

(p, q)(r , s) = (pr − sq, sp + qr)

x = (p, q) = x0e0 +x1e1 +x2e2 +x3e3 +x4e4 +x5e5 +x6e6 +x7e7e0 e1 e2 e3 e4 e5 e6 e7e1 −1 e3 −e2 e5 −e4 −e7 e6e2 −e3 −1 e1 e6 e7 −e4 −e5e3 e2 −e1 −1 e7 −e6 e5 −e4e4 −e5 −e6 −e7 −1 e1 e2 e3e5 e4 −e7 e6 −e1 −1 −e3 e2e6 e7 e4 −e5 −e2 e3 −1 −e1e7 −e6 e5 e4 −e3 −e2 e1 −1



Octonioni

John Thomas Graves (1806–1870); Arthur Cayley (1821–1895)Andrei Marcus http://math.ubbcluj.ro/˜marcus/ Cuaternioni


Octonioni

John Baez

The real numbers are the dependable breadwinner of thefamily, the complete ordered field we all rely on. The complexnumbers are a slightly flashier but still respectable youngerbrother: not ordered, but algebraically complete. Thequaternions, being noncommutative, are the eccentric cousinwho is shunned at important family gatherings. But theoctonions are the crazy old uncle nobody lets out of the attic:they are nonassociative.



Bibliografie I

John Vince: Quaternions for Computer Graphics,Springer-Verlag, New York 2011.

http://www-history.mcs.st-and.ac.uk/history/

http://en.wikipedia.org/


Documents

Numere complexe, Cuaternioni si Aplicatiimarcus/for_students/pres_marcus-quaternions.pdf · Cuprins Numere complexe CuaternioniOctonioniBibliogra e Octonioni John Baez The real numbers