proyector 3d de gir horitzontal

k·<( cos(s)(c_{1}+(-1)a_{1})+sin(s)(c_{3}+(-1)a_{3}) ),(c_{2}+(-1)a_{2}),...
...( (-1)sin(s)(c_{1}+(-1)a_{1})+cos(s)(c_{3}+(-1)a_{3}) )>=...
...i·<cos(s),0,(-1)sin(s)>+j·<0,1,0>+<sin(s),0,cos(s)>


k=( 1/(c_{3}+(-1)a_{3}) )
i=( (c_{1}+(-1)a_{1}) )/(c_{3}+(-1)a_{3})
j=( (c_{2}+(-1)a_{2}) )/(c_{3}+(-1)a_{3})


si s=0 ==>...
...k·<(c_{1}+(-1)a_{1}),(c_{2}+(-1)a_{2}),(c_{3}+(-1)a_{3})>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<x,0,0>+j·<0,y,0>+<0,0,z> ]


k=( 1/(c_{3}+(-1)a_{3}) )
i=( (c_{1}+(-1)a_{1}) )/(c_{3}+(-1)a_{3})
j=( (c_{2}+(-1)a_{2}) )/(c_{3}+(-1)a_{3})


si s=(pi/2) ==>...
...k·<(c_{3}+(-1)a_{3}),(c_{2}+(-1)a_{2}),(-1)(c_{1}+(-1)a_{1})>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<z,0,0>+j·<0,y,0>+<0,0,(-x)> ]


k=( (-1)/(c_{1}+(-1)a_{1}) )
i=( (-1)(c_{3}+(-1)a_{3}) )/(c_{1}+(-1)a_{1})
j=( (-1)(c_{2}+(-1)a_{2}) )/(c_{1}+(-1)a_{1})


si s=(-1)(pi/2) ==>...
...k·<(-1)(c_{3}+(-1)a_{3}),(c_{2}+(-1)a_{2}),(c_{1}+(-1)a_{1})>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<(-z),0,0>+j·<0,y,0>+<0,0,x> ]


k=( 1/(c_{1}+(-1)a_{1}) )
i=( (-1)(c_{3}+(-1)a_{3}) )/(c_{1}+(-1)a_{1})
j=( (c_{2}+(-1)a_{2}) )/(c_{1}+(-1)a_{1})










k·<f(s),g(s),d_{s}[f(s)]>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>


f(s) = cos(s)(c_{1}+(-1)a_{1})+sin(s)(c_{3}+(-1)a_{3})
d_{s}[f(s)] = (-1)sin(s)(c_{1}+(-1)a_{1})+cos(s)(c_{3}+(-1)a_{3})






si s=(pi/4) ==>...
...k·<(2^{(1/2)}/2)( (c_{1}+(-1)a_{1})+(c_{3}+(-1)a_{3}) ),...
...( (2^{(1/2)}/2)+(2^{(1/2)}/2) )(c_{2}+(-1)a_{2}),...
...(2^{(1/2)}/2)( (c_{3}+(-1)a_{3})+(-1)(c_{1}+(-1)a_{1}) )>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<(2^{(1/2)}/2)x+(2^{(1/2)}/2)z,0,0>+...
...j·<0,(2^{(1/2)}/2)y+(2^{(1/2)}/2)y,0>+...
...<0,0,(2^{(1/2)}/2)z+(2^{(1/2)}/2)(-x)> ]


k=( 1/( (c_{3}+(-1)a_{3})+(-1)(c_{1}+(-1)a_{1}) ) )
i=( ( (c_{1}+(-1)a_{1})+(c_{3}+(-1)a_{3}) ) )/( (c_{3}+(-1)a_{3})+(-1)(c_{1}+(-1)a_{1}) )
j=( 2(c_{2}+(-1)a_{2}) )/( (c_{3}+(-1)a_{3})+(-1)(c_{1}+(-1)a_{1}) )


si s=(pi/6) ==>...
...k·<( (1/2)(c_{1}+(-1)a_{1})+(3^{(1/2)}/2)·(c_{3}+(-1)a_{3}) ),...
...( (1/2)+(3^{(1/2)}/2) )(c_{2}+(-1)a_{2}),...
...( (1/2)(c_{3}+(-1)a_{3})+(-1)(3^{(1/2)}/2)·(c_{1}+(-1)a_{1}) )>=...
...i·<1,0,0>+j·<0,1,0>+<0,0,1>
...[ i·<(1/2)x+(3^{(1/2)}/2)z,0,0>+...
...j·<0,(1/2)y+(3^{(1/2)}/2)y,0>+...
...<0,0,(1/2)z+(3^{(1/2)}/2)(-x)> ]


k=( 1/( (1/2)(c_{3}+(-1)a_{3})+(-1)(3^{(1/2)}/2)·(c_{1}+(-1)a_{1}) )
i=( ( (1/2)(c_{1}+(-1)a_{1})+(3^{(1/2)}/2)(c_{3}+(-1)a_{3}) ) )·...
(1/( (1/2)(c_{3}+(-1)a_{3})+(-1)(3^{(1/2)}/2)·(c_{1}+(-1)a_{1}) ) )
j=( ((1/2)+(3^{(1/2)}/2))(c_{2}+(-1)a_{2}) )/( (1/2)(c_{3}+(-1)a_{3})+(-1)(3^{(1/2)}/2)·(c_{1}+(-1)a_{1}) )




En ser un gir hi ha una component ortogonal a les y.

proyector 3d frontal amb rotació

k·<(c_{1}+(-1)a_{1}),(c_{2}+(-1)a_{2}),(c_{3}+(-1)a_{3})>=...
...i·<cos(s),sin(s),0>+j·<(-1)sin(s),cos(s),0>+<0,0,1>


k=( 1/(c_{3}+(-1)a_{3}) )
i=( (c_{1}+(-1)a_{1})cos(s)+(c_{2}+(-1)a_{2})sin(s) )/(c_{3}+(-1)a_{3})
j=( (c_{2}+(-1)a_{2})cos(s)+(c_{1}+(-1)a_{1})(-1)sin(s) )/(c_{3}+(-1)a_{3})

x·<1,0,0>=...
...( (c_{1}+(-1)a_{1})( cos(s) )^{2}+(c_{2}+(-1)a_{2})sin(s)cos(s) )/(c_{3}+(-1)a_{3})+...
...( (c_{2}+(-1)a_{2})(-1)sin(s)cos(s)+(c_{1}+(-1)a_{1})( sin(s) )^{2} )/(c_{3}+(-1)a_{3})


y·<0,1,0>=...
...( (c_{1}+(-1)a_{1})sin(s)cos(s)+(c_{2}+(-1)a_{2})( sin(s) )^{2} )/(c_{3}+(-1)a_{3})+...
...( (c_{2}+(-1)a_{2})( cos(s) )^{2}+(c_{1}+(-1)a_{1}(-1)sin(s)cos(s) )/(c_{3}+(-1)a_{3})

proyector frontal 3d

k·<c_{1}+(-1)a_{1},c_{2}+(-1)a_{2},c_{3}+(-1)a_{3}> =...
... i·<1,0,0>+j·<0,1,0>+<0,0,1>


k=( 1/(c_{3}+(-1)a_{3}) )
i=( (c_{1}+(-1)a_{1})/(c_{3}+(-1)a_{3}) )
j=( (c_{2}+(-1)a_{2})/(c_{3}+(-1)a_{3}) )


a_{i} = coordenades del observador.
c_{i} = coordenades del punt a observar.


k=( 1/( (c_{3}+d_{3})+(-1)(a_{3}+d_{3}) ) )
i=( ( (c_{1}+d_{1})+(-1)(a_{1}+d_{1}) )/( (c_{3}+d_{3})+(-1)(a_{3}+d_{3}) ) )
j=( ( (c_{1}+d_{2})+(-1)(a_{1}+d_{2}) )/( (c_{3}+d_{3})+(-1)(a_{3}+d_{3}) ) )

suma y intesecció de espais vectorials

E = k·<1,(-1)>
F = s·<(-1),1>
G = i·<1,0>+j·<0,1>


La suma
E+F+G = (i+k+(-s))·<1,0>+(j+s+(-k))·<0,1>
si ( i=0 & j=0 & s=0 ) ==> F+G = k·<1,(-1)>
si ( i=0 & j=0 & k=0 ) ==> F+G = s·<(-1),1>
si ( k=0 & s=0 ) ==> F+G = i·<1,0>+j·<0,1>


La intersecció
k·<1,(-1)> =  i·<1,0>+j·<0,1>
k·<1,(-1)> =  k·<1,0>+(-k)·<0,1>
k·<1,(-1)> =  k·<1,(-1)>
E[M]G = k·<1,(-1)>
La intersecció
s·<(-1),1> =  i·<1,0>+j·<0,1>
s·<(-1),1> =  (-s)·<1,0>+s·<0,1>
s·<(-1),1> =  s·<(-1),1>
F[M]G = s·<(-1),1>


La intersecció
k·<1,(-1)> =  s·<(-1),1>
k·<1,(-1)> =  (-k)·<(-1),1>
k·<1,(-1)> =  k·<1,(-1)>
E[M]F = k·<1,(-1)>


La intersecció
s·<(-1),1> =  k·<1,(-1)>
s·<(-1),1> =  (-s)·<1,(-1)>
s·<(-1),1> =  s·<(-1),1>
E[M]F = s·<(-1),1>


E[M]F[M]G = s·<(-1),1> = (-k)·<1,(-1)>
E[M]F[M]G = k·<1,(-1)> = (-s)·<(-1),1>


dim(E+F+G) = dim(E)+dim(F)+dim(G)+...
...+(-1)dim(E[M]F)+(-1)·dim(E[M]G)+(-1)·dim(F[M]G)+dim(E[M]F[M]G)


E = k·<2n,2n+1>
F = s·<2n+1,2n>
G = i·<1,0>+j·<0,1>


La suma
E+F+G = (i+2nk+(2n+1)s)·<1,0>+(j+(2n+1)k+2ns)·<0,1>
si ( i=0 & j=0 & s=0 ) ==> F+G = k·<2n,2n+1>
si ( i=0 & j=0 & k=0 ) ==> F+G = s·<2n+1,2n>
si ( k=0 & s=0 ) ==> F+G = i·<1,0>+j·<0,1>


La intersecció
k·<2n,2n+1> =  i·<1,0>+j·<0,1>
k·<2n,2n+1> =  2nk·<1,0>+(2n+1)k·<0,1>
k·<2n,2n+1> =  k·<2n,2n+1>
E[M]G = k·<2n,2n+1>
La intersecció
s·<2n+1,2n> =  i·<1,0>+j·<0,1>
s·<2n+1,2n> =  (2n+1)s·<1,0>+2ns·<0,1>
s·<2n+1,2n> =  s·<2n+1,2n>
F[M]G = s·<2n+1,2n>

suma y intersecció de espais vectorials

E = k·<1,1>
F = i·<1,0>+j·<0,1>


k·<1,1>+j·<0,1> = k·<1,0>+(j+k)·<0,1>
k·<1,1>+i·<1,0> = k·<0,1>+(i+k)·<1,0>


La suma
E+F = (i+k)·<1,0>+(j+k)·<0,1>
si k=0 ==> F+G = i·<1,0>+j·<0,1>
si ( i=0 & j=0 ) ==> F+G = k·<1,1>




La intersecció
k·<1,1> =  i·<1,0>+j·<0,1>
k·<1,1> =  k·<1,0>+k·<0,1>
k·<1,1> =  k·<1,1>
E[M]F = k·<1,1>


dim(E+F) =dim(E)+dim(F)+(-1)·dim(E[M]F)


E = k·<1,0>
F = s·<0,1>
G = i·<1,0>+j·<0,1>


La suma
E+F+G = (i+k)·<1,0>+(j+s)·<0,1>
si ( i=0 & j=0 & s=0 ) ==> F+G = k·<1,0>
si ( i=0 & j=0 & k=0 ) ==> F+G = s·<0,1>
si ( k=0 & s=0 ) ==> F+G = i·<1,0>+j·<0,1>


La intersecció
k·<1,0> =  i·<1,0>+j·<0,1>
k·<1,0> =  k·<1,0>+0·<0,1>
k·<1,0> =  k·<1,0>
E[M]G = k·<1,0>
La intersecció
s·<0,1> =  i·<1,0>+j·<0,1>
s·<0,1> =  0·<1,0>+s·<0,1>
s·<0,1> =  s·<0,1>
F[M]G = s·<0,1>


dim(E+F+G) = dim(E)+dim(F)+dim(G)+...
...+(-1)dim(E[M]F)+(-1)·dim(E[M]G)+(-1)·dim(F[M]G)+dim(E[M]F[M]G)

producte escalar

<k,0>[o]<0,k> = 0
<(-k),0>[o]<0,(-k)> = 0


<k,0>[o]<0,(-k)> = 0
<(-k),0>[o]<0,k> = 0


<k,k>[o]<k,(-k)> = ( k^{2}+(-1)k^{2}) = 0
<k,k>[o]<(-k),k> = ( (-1)k^{2}+k^{2}) = 0


<(-k),(-k)>[o]<k,(-k)> = ( (-1)k^{2}+k^{2}) = 0
<(-k),(-k)>[o]<(-k),k> = ( k^{2}+(-1)k^{2}) = 0


<k,k>[o]<k,k> = ( k^{2}+k^{2} ) = 2k^{2}
<k,k>[o]<(-k),(-k)> = (-1)( k^{2}+k^{2} ) = (-2)k^{2}


<k,k>[o]<0,k> = k^{2}
<k,k>[o]<k,0> = k^{2}


<(-k),(-k)>[o]<0,k> = (-1)k^{2}
<(-k),(-k)>[o]<k,0> = (-1)k^{2}


<k,k>[o]<0,(-k)> = (-1)k^{2}
<k,k>[o]<(-k),0> = (-1)k^{2}


<(-k),(-k)>[o]<0,(-k)> = k^{2}
<(-k),(-k)>[o]<(-k),0> = k^{2}

integrals circulars de límit zero

lim[s-->0]int[z=se^{xi}+a][ f(z)/(z+(-a)) ]d[z]·(1/2pi) = f(a)
z=se^{xi}+a


lim[s-->0]int[( ln(z) )^{n}=se^{xi}+a][ f(z)/( ( ln(z) )^{n}+(-a) ) ]d[z]·(1/2pi) = ...
...= f( e^{a^{(1/n)}} )·e^{a^{(1/n)}}·(1/n)·a^{((1/n)+(-1))}
z=e^{( se^{xi}+a )^{(1/n)}}


lim[s-->0]int[( ln(z)+c )^{n}=se^{xi}+a][ f(z)/( ( ln(z)+c )^{n}+(-a) ) ]d[z]·(1/2pi) = ...
...= f( e^{a^{(1/n)}+(-c)} )·e^{a^{(1/n)+(-c)}}·(1/n)·a^{((1/n)+(-1))}
z=e^{( se^{xi}+a )^{(1/n)}+(-c)}


lim[s-->0]int[( e^{x} )^{n}=se^{xi}+a][ f(z)/( ( e^{x} )^{n}+(-a) ) ]d[z]·(1/2pi) = ...
...= f( ln( a^{(1/n)} ) )·(1/n)·(1/a)
z=ln( ( se^{xi}+a )^{(1/n)} )


lim[s-->0]int[( e^{x}+c )^{n}=se^{xi}+a][ f(z)/( ( e^{x}+c )^{n}+(-a) ) ]d[z]·(1/2pi) = ...
...= f( ln( a^{(1/n)}+(-c) ) )·( 1/(a^{(1/n)}+(-c)) )·(1/n)·a^{(1/n)+(-1))}
z=ln( ( se^{xi}+a )^{(1/n)}+(-c) )


lim[s-->0]int[( x+c )^{n}=se^{xi}+a][ f(z)/( ( x+c )^{n}+(-a) ) ]d[z]·(1/2pi) = ...
...= f( a^{(1/n)}+(-c) )·(1/n)·a^{(1/n)+(-1))}
z=( se^{xi}+a )^{(1/n)}+(-c)


lim[s-->0]int[( x^{p}+c )^{n}=se^{xi}+a][ f(z)/( ( x^{p}+c )^{n}+(-a) ) ]d[z]·(1/2pi) = ...
...= f( ( a^{(1/n)}+(-c) )^{(1/p)} )·(1/p)·(a^{(1/n)+(-c)})^{((1/p)+(-1))}·(1/n)·a^{(1/n)+(-1))}
z=( ( se^{xi}+a )^{(1/n)}+(-c) )^{(1/p)}

series de Fourier

x^{2} = (pi^{2}/3)+4·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{2} ) = ( pi^{2}/6 )
si x=0 ==> ∑ ( (-1)^{k}/k^{2} ) = (-1)( pi^{2}/12 )


x^{4} = (pi^{4}/5)+8pi^{2}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+..
...+(-48)·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{4} ) = ( pi^{4}/90 )
si x=0 ==> ∑ ( (-1)^{k}/k^{4} ) = (-1)( 7pi^{4}/720 )


x^{6} = (pi^{6}/7)+12pi^{4}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-240)pi^{2}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+1440·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{6} ) = ( pi^{6}/945 )
si x=0 ==> ∑ ( (-1)^{k}/k^{6} ) = (-1)( 31pi^{4}/30240 )


x^{8} = (pi^{8}/9)+16pi^{6}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-672)pi^{4}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+13440pi^{2}·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )+...
...+(-80640)·∑ ( ( 1/k^{8} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{8} ) = ( pi^{8}/9450 )


x^{10} = (pi^{10}/11)+20pi^{8}·∑ ( ( 1/k^{2} )·cos(k·pi)·cos(kx) )+...
...+(-1440)pi^{6}·∑ ( ( 1/k^{4} )·cos(k·pi)·cos(kx) )+...
...+60480pi^{4}·∑ ( ( 1/k^{6} )·cos(k·pi)·cos(kx) )+...
...+(-1209600)pi^{2}·∑ ( ( 1/k^{8} )·cos(k·pi)·cos(kx) )+...
...+7257600·∑ ( ( 1/k^{10} )·cos(k·pi)·cos(kx) )
si x=pi ==> ∑ ( 1/k^{10} ) = ( pi^{10}/93555 )

moments geométrics

sum(  k^{0}x^{k} ) =( ( x^{n+1}+(-1) )/( x+(-1) ) )
sum(  k^{1}x^{k} ) =( ( (n+1)x^{n+1}+1 )/( x+(-1) ) ) + (-1)( (x^{n+2}+(-1))/(x+(-1))^{2} )

moments de la exponencial

sum( k^{0}·(x^{k}/k!) ) = e^{x}
sum( k^{1}·(x^{k}/k!) ) = xe^{x}
sum( k^{2}·(x^{k}/k!) ) = (x^{2}+x)e^{x}
sum( k^{3}·(x^{k}/k!) ) = (x^{3}+3x^{2}+x)e^{x}