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{ |
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The first few Spherical Harmonics |
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|
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Feb 2005 G. Ward |
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} |
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{ Factorial (n!) } |
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fact(n) : if(n-1.5, n*fact(n-1), 1); |
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|
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{ Associated Legendre Polynomials 0-8 } |
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LegendreP2(n,m,x,s) : select(n+1, |
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select(m+1, 1), |
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select(m+1, x, s), |
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select(m+1, .5*(3*x*x - 1), 3*x*s, 3*(1-x*x)), |
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select(m+1, .5*x*(5*x*x-3), 1.5*(5*x*x-1)*s, 15*x*(1-x*x), 15*s*s*s), |
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select(m+1, |
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.125*(3 + x*x*(-30 + x*x*35)), |
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2.5*x*(-3 + x*x*7)*s, |
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7.5*(7*x*x-1)*(1-x*x), |
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105*x*s*s*s, |
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105*s*s*s*s), |
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select(m+1, |
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.125*x*(15 + x*x*(-70 + x*x*63)), |
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1.875*s*(1 + x*x*(-14 + x*x*21)), |
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52.5*x*(1-x*x)*(3*x*x-1), |
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52.5*s*s*s*(9*x*x-1), |
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945*x*s*s*s*s, |
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945*s*s*s*s*s), |
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select(m+1, |
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.0625*(-5 + x*x*(105 + x*x*(-315 + x*x*231))), |
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2.625*(5 + x*x*(-30 + x*x*33))*s, |
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13.125*s*s*(1 + x*x*(-18 + x*x*33)), |
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157.5*(11*x*x-3)*x*s*s*s, |
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472.5*s*s*s*s*(11*x*x-1), |
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10395*x*s*s*s*s*s, |
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10395*s*s*s*s*s*s), |
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select(m+1, |
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.0625*x*(-35 + x*x*(315 + x*x*(-693 + x*x*429))), |
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.4375*s*(-5 + x*x*(135 + x*x*(-495 + x*x*429))), |
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7.875*x*s*s*(15 + x*x*(-110 + x*x*143)), |
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39.375*s*s*(1 + x*x*(-18 + x*x*33)), |
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157.5*(11*x*x-3)*x*s*s*s, |
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472.5*s*s*s*s*(11*x*x-1), |
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10395*x*s*s*s*s*s, |
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10395*s*s*s*s*s*s), |
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select(m+1, |
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.0078125*(35 + x*x*(-1260 + x*x*(6930 + x*x*(-12012 + x*x*6435)))), |
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.5625*x*s*(-35 + x*x*(385 + x*x*(-1001 + x*x*715))), |
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19.6875*s*s*(-1 + x*x*(33 + x*x*(-143 + x*x*143))), |
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433.125*x*s*s*s*(3 + x*x*(-26 + x*x*39)), |
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1299.375*s*s*s*s*(1 + x*x*(-26 + x*x*65)), |
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67567.5*x*s*s*s*s*s*(5*x*x-1), |
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67567.5*s*s*s*s*s*s*(15*x*x-1), |
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2027025*x*s*s*s*s*s*s*s, |
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2027025*s*s*s*s*s*s*s*s) |
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); |
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{ Relation for Legendre with -M } |
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odd(n) : .5*n - floor(.5*n) - .25; |
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LegendreP(n,m,x) : if(m+.5, |
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LegendreP2(n,m,x,sqrt(1-x*x)), |
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if(odd(-m),-1,1)*fact(n+m)/fact(n-m) * |
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LegendreP2(n,-m,x,sqrt(1-x*x)) |
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); |
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{ SH normalization factor } |
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SHnormF(l,m) : sqrt(0.25/PI*(2*l+1)*fact(l-m)/fact(l+m)); |
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{ Spherical Harmonics theta function } |
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SHthetaF(l,m,theta) : SHnormF(l,m)*LegendreP(l,m,cos(theta)); |
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|
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{ Spherical Harmonic real portion } |
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SphericalHarmonicYr(l,m,theta,phi) : SHthetaF(l,m,theta)*cos(m*phi); |
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{ Spherical Harmonic imag. portion } |
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SphericalHarmonicYi(l,m,theta,phi) : SHthetaF(l,m,theta)*sin(m*phi); |
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|
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{ Ordered, real SH basis functions } |
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SH_B4(l,m,o,theta,phi) : if(m-.5, if(o, SphericalHarmonicYi(l,m,theta,phi), |
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SphericalHarmonicYr(l,m,theta,phi)), |
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SHthetaF(l,0,theta) ); |
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SH_B3(l,r,theta,phi) : SH_B4(l,floor((r+1.00001)/2),odd(r+1),theta,phi); |
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SH_B2(l,i,theta,phi) : SH_B3(l,i-l*l-1,theta,phi); |
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SphericalHarmonicB(i,theta,phi) : SH_B2(ceil(sqrt(i)-1.00001),i,theta,phi); |
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|
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{ Application of SH fitting coeff. f(i) } |
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SH_F2(n,f,theta,phi) : if(n-.5, f(n)*SphericalHarmonicB(n,theta,phi) + |
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SH_F2(n-1,f,theta,phi), 0); |
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SphericalHarmonicF(f,theta,phi) : SH_F2(f(0),f,theta,phi); |
