SO(2)
Rotacións no plano (1 GL)
Selector lineal p (entre 0 e 2π) s = sin(p) , c = cos(p)
Matriz de cambio
s11 = c s12 = -s s21 = s s22 = c
Eixes cartesiáns E1, E2)
E1= (s11, s21) , E2= (s12, s22)
segmentos
sE1=segment(O,E1) sE2 = segment(O,E2)
Punto xenérico (P):
p1 = x(P), p2=y(P)
Q = (s11*p1+s12*p2, s21*p1+s22*p2)
SO(3)
Rotacións no espazo 3d (3 GL)
Ángulo de xiro (selector lineal):
p (entre π/2 e 5π/2) s = sin(p) , c = cos(p)
cursor “perspectiva” : r (-0.1,0.1) oscilante e veloz , caixa “wiggle” para activalo
2 parámetros definitorios do plano de xiro: (entre -π/2 e π/2)
A libre, B interior ao cadrado de lado 0.96 (só visible cando p<1.6
a = tan((x(B)-x(A)-0.48)*pi) , b = tan((y(B)-y(A)-0.48)*pi )
Vectores ortogonais (módulos ao cadrado):
dv=1+a^2 , dw=1+a^2+b^2 , du=1+(a*b)^2+b^2+a^4 + 2a^2
Base ortonormal xenérica MT
v1 = 1/√dv , v2 = a/√dv , ( v3 = 0 )
w1 = - a/√dw , w2 = 1/√dw , w3 =b/√dw
u1 = (ab)/√du , u2 = -b/√du , u3 = (1+a^2)/√du
Matriz de cambio
s11 = v1 ( c*v1+s*v2 ) + v2 (-s*v1+c*v2)
s12 = w1 ( c*v1+s*v2 ) + w2 (-s*v1+c*v2)
s13 = u1 ( c*v1+s*v2 ) + u2 (-s*v1+c*v2)
s21 = v1 ( c*w1+s*w2 ) + v2 (-s*w1+c*w2)
s22 = w1 ( c*w1+s*w2 ) + w2 (-s*w1+c*w2) + w3*w3
s23 = u1 ( c*w1+s*w2 ) + u2 (-s*w1+c*w2) + u3*w3
s31 = v1 ( c*u1+s*u2 ) + v2 (-s*u1+cu2)
s32 = w1 ( c*u1+s*u2 ) + w2 (-s*u1+c*u2) + w3*u3
s33 = u1 ( c*u1+s*u2 ) + u2 (-s*u1+c*u2) + u3*u3
Eixes cartesiáns E1, E2, E3
E1= (s11+ r*s31, s21) E2= (s12+ r*s32, s22) E3= (s13+ r*s33, s23)
segmentos
sE1=segment(O,E1) sE2 = segment(O,E2) sE3 = segment(O,E3)
Bivector de P
bP = cónica(E1, E2, -E1, -E2, (E1+E2)/√2)
selectores: eixes, bivector12, wiggle, xiros (oculta o cadro)
SO(4)
3 selectores (en dous cadrados de lado 0.96): vértices A , B
en A: 2 ángulos de xiro (cada un nun plano ortogonal): P entre (0, 2π)
en B: 4 parámetros definitorios dun dos planos: M, N (entre -π/2+0.1 e π/2-0.1)
cursor “perspectiva” : r (-0.1,0.1) oscilante e veloz , caixa “wiggle” para activalo
s = sin(x(P-A)*2π) , c = cos(x(P-A)*2π)
d = cos((y(P-A)+1/4)*2π) , f = sin((y(P-A)+1/4)*2π)
poderase manter unha “visión recta” inicial?
a=tan((x(M-B)-4/9)*π) , b=tan((y(M-B)-4/9)*π)
m=tan((x(N-B)-4/9)*π) , n=tan((y(N-B)-4/9)*π)
base ortogonal xenérica (módulos)
dv = 1+a^2+b^2 h = (a*m+ b*n) / dv
dw = 1+h^2+m^2+(a*h)^2 -2a*m*h+(b*h)^2+n^2-2b*n*h)
dt = 1+a^2+m^2 k = (a*b+ m*n) / dt
du = 1+k^2+b^2+(a*k)^2-2*b*a*k+n^2+(m*k)^2-2*n*m*k
Base ortonormal xenérica
v1 = 1/√dv , ( v2 = 0 ) , v3 = a/√dv , v4 = b/√dv
w1 = -h/√dw , w2 = 1/√dw , w3 = (m-a*h)/√dw , w4 = (n-b*h)/√dw
t1 = -a/√dt , t2 = -m/√dt , t3 = 1/√dt , ( t4 =0 )
u1 = (-b+a*k)/√du , u2 = (-n+m*k)/√du , u3 = -k/√du , u4 = 1/√du
Matriz de rotación
s11 = v1 ( c*v1) + v3 (d*v3+f*v4) + v4 (-f*v3+d*v4 )
s12 = w1 ( c*v1) + w2 (-s*v1) + w3 (d*v3+f*v4) + w4 (-f*v3+d*v4 )
s13 = t1 ( c*v1) + t2 (-s*v1) + t3 (d*v3+f*v4)
s14 = u1 ( c*v1 ) + u2 (-s*v1) + u3 (d*v3+f*v4) + u4 (-f*v3+d*v4 )
s21 = v1 ( c*w1+s*w2 ) + v3 (d*w3+f*w4) + v4 (-f*w3+d*w4 )
s22 = w1 ( c*w1+s*w2 ) + w2 (-s*w1+c*w2) + w3 (d*w3+f*w4) + w4 (-f*w3+d*w4 )
s23 = t1 ( c*w1+s*w2 ) + t2 (-s*w1+c*w2) + t3 (d*w3+f*w4)
s24 = u1 ( c*w1+s*w2 ) + u2 (-s*w1+c*w2) + u3 (d*w3+f*w4) + u4 (-f*w3+d*w4 )
s31 = v1 ( c*t1+s*t2 ) + v3 (d*t3) + v4 (-f*t3)
s32 = w1 ( c*t1+s*t2 ) + w2 (-s*t1+c*t2) + w3 (d*t3) + w4 (-f*t3)
s33 = t1 ( c*t1+s*t2 ) + t2 (-s*t1+c*t2) + t3 (d*t3)
s34 = u1 ( c*t1+s*t2 ) + u2 (-s*t1+c*t2) + u3 (d*t3) + u4 (-f*t3)
s41 = v1 ( c*u1+s*u2 ) + v3 (d*u3+f*u4) + v4 (-f*u3+d*u4 )
s42 = w1 ( c*u1+s*u2 ) + w2 (-s*u1+c*u2) + w3 (d*u3+f*u4) + w4 (-f*u3+d*u4 )
s43 = t1 ( c*u1+s*u2 ) + t2 (-s*u1+c*u2) + t3 (d*u3+f*u4)
s44 = u1 ( c*u1+s*u2 ) + u2 (-s*u1+c*u2) + u3 (d*u3+f*u4) + u4 (-f*u3+d*u4 )
Eixes cartesiáns transformados: E1, E2, E3, E4
E1= (s11, s21)+r*(s31, s41) E2= (s12, s22)+r*(s32, s42) azuis
E3= (s13, s23) +r*(s33, s43) E4= (s14, s24)+r*(s34, s44) vermellos
segmentos dende O
Bivectores de E1, E2
b12 = cónica(E1, E2, -E1, -E2, (E1+E2)/√2) azul
b34= cónica(E3, E4, -E3, -E4, (E3+E4)/√2) vermella