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{ |
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He-Torrance Reflectance Model (Siggraph 1991) |
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|
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Complete spectral version (8/17/95) |
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|
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The primitive for this function should look something like: |
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|
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void BRTDfunc name |
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10 |
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s`r s`g s`b |
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0 0 0 |
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dd`r dd`g dd`b |
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He3.cal |
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0 |
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17 amb`r amb`g amb`b |
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amb`r amb`g amb`b |
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0 0 0 |
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sigma0 tau |
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n_real`r n_imag`r |
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n_real`g n_imag`g |
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n_real`b n_imag`b |
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} |
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|
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{ Constants } |
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lambda`r : .67; { red wavelength (microns) } |
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lambda`g : .55; { green wavelength (microns) } |
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lambda`b : .43; { blue wavelength (microns) } |
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z0err : .0001; { accepted error in value of z0 } |
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Dsumlim : .000001; { last term of D summation } |
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Dsummax : 200; { maximum terms in D summation } |
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|
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{ Parameters } |
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sigma0 = arg(10); { surface height deviation (microns) } |
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tau = arg(11); { correlation distance (microns) } |
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n_real`r = arg(12); { red real part of index of refraction } |
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n_imag`r = arg(13); { red imaginary part of index of refraction } |
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n_real`g = arg(14); { green real part of index of refraction } |
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n_imag`g = arg(15); { green imaginary part of index of refraction } |
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n_real`b = arg(16); { blue real part of index of refraction } |
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n_imag`b = arg(17); { blue imaginary part of index of refraction } |
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{ Derived parameters } |
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n_k`r = n_imag`r/n_real`r; |
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n_k`g = n_imag`g/n_real`g; |
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n_k`b = n_imag`b/n_real`b; |
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|
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{ Repeated formulas } |
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cotexp(t) = tau/sigma0/2/tan(t); |
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shadowf2(et,erfcet) = (1-.5*erfcet) / |
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((Exp(-sq(et))/sqrt(PI)/et - erfcet)/2 + 1); |
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shadowf1(t) = or(FTINY-sigma0, .01-abs(t)); |
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shadowf0(t) = abs(t) - (PI/2-.0001); |
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shadowf(t) = if(shadowf0(t), 0, if(shadowf1(t), 1, |
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shadowf2(cotexp(t), erfc(cotexp(t))))); |
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K(t) = if(abs(t)-FTINY, tan(t) * erfc(cotexp(t)), 0); |
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fuvA`r(ct) = sq(n_real`r)*(1-sq(n_k`r)) - (1-sq(ct)); |
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fuvB`r(ct) = sqrt(sq(fuvA`r(ct)) + 4*sq(sq(n_real`r)*n_k`r)); |
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fu2`r(ct) = (fuvA`r(ct) + fuvB`r(ct))/2; |
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fv2`r(ct) = (-fuvA`r(ct) + fuvB`r(ct))/2; |
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fperp2`r(ct) = (sq(ct-sqrt(fu2`r(ct))) + fv2`r(ct)) / |
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(sq(ct+sqrt(fu2`r(ct))) + fv2`r(ct)); |
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fpara2`r(ct) = (sq(sq(n_real`r)*(1-sq(n_k`r))*ct - sqrt(fu2`r(ct))) + |
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sq(2*sq(n_real`r)*n_k`r*ct - sqrt(fv2`r(ct)))) / |
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(sq(sq(n_real`r)*(1-sq(n_k`r))*ct + sqrt(fu2`r(ct))) + |
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sq(2*sq(n_real`r)*n_k`r*ct + sqrt(fv2`r(ct)))); |
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fresnel2`r(ct) = (fperp2`r(ct) + fpara2`r(ct))/2; |
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fuvA`g(ct) = sq(n_real`g)*(1-sq(n_k`g)) - (1-sq(ct)); |
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fuvB`g(ct) = sqrt(sq(fuvA`g(ct)) + 4*sq(sq(n_real`g)*n_k`g)); |
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fu2`g(ct) = (fuvA`g(ct) + fuvB`g(ct))/2; |
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fv2`g(ct) = (-fuvA`g(ct) + fuvB`g(ct))/2; |
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fperp2`g(ct) = (sq(ct-sqrt(fu2`g(ct))) + fv2`g(ct)) / |
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(sq(ct+sqrt(fu2`g(ct))) + fv2`g(ct)); |
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fpara2`g(ct) = (sq(sq(n_real`g)*(1-sq(n_k`g))*ct - sqrt(fu2`g(ct))) + |
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sq(2*sq(n_real`g)*n_k`g*ct - sqrt(fv2`g(ct)))) / |
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(sq(sq(n_real`g)*(1-sq(n_k`g))*ct + sqrt(fu2`g(ct))) + |
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sq(2*sq(n_real`g)*n_k`g*ct + sqrt(fv2`g(ct)))); |
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fresnel2`g(ct) = (fperp2`g(ct) + fpara2`g(ct))/2; |
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fuvA`b(ct) = sq(n_real`b)*(1-sq(n_k`b)) - (1-sq(ct)); |
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fuvB`b(ct) = sqrt(sq(fuvA`b(ct)) + 4*sq(sq(n_real`b)*n_k`b)); |
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fu2`b(ct) = (fuvA`b(ct) + fuvB`b(ct))/2; |
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fv2`b(ct) = (-fuvA`b(ct) + fuvB`b(ct))/2; |
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fperp2`b(ct) = (sq(ct-sqrt(fu2`b(ct))) + fv2`b(ct)) / |
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(sq(ct+sqrt(fu2`b(ct))) + fv2`b(ct)); |
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fpara2`b(ct) = (sq(sq(n_real`b)*(1-sq(n_k`b))*ct - sqrt(fu2`b(ct))) + |
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sq(2*sq(n_real`b)*n_k`b*ct - sqrt(fv2`b(ct)))) / |
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(sq(sq(n_real`b)*(1-sq(n_k`b))*ct + sqrt(fu2`b(ct))) + |
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sq(2*sq(n_real`b)*n_k`b*ct + sqrt(fv2`b(ct)))); |
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fresnel2`b(ct) = (fperp2`b(ct) + fpara2`b(ct))/2; |
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|
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{ Formulas dependent only on reflected direction } |
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theta_r = acos(RdotP); |
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shadowf_r = shadowf(theta_r); |
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K_r = K(theta_r); |
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srx = Dy*NzP - Dz*NyP; sry = Dz*NxP - Dx*NzP; srz = Dx*NyP - Dy*NxP; |
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srn2 = sq(srx) + sq(sry) + sq(srz); |
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prx = sry*Dz - srz*Dy; |
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pry = srz*Dx - srx*Dz; |
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prz = srx*Dy - sry*Dx; |
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s`r = fresnel2`r(RdotP)*Exp(-g`r(RdotP))*sq(shadowf_r); |
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s`g = fresnel2`g(RdotP)*Exp(-g`g(RdotP))*sq(shadowf_r); |
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s`b = fresnel2`b(RdotP)*Exp(-g`b(RdotP))*sq(shadowf_r); |
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|
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{ Formulas dependent on incident direction } |
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{ z0 } |
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z0d(Ki,z) = -(Ki+K_r)/(4*sigma0)*z*Exp(-sq(z/sigma0)/2) - sqrt(PI/2); |
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z0lim(x) = if(x, max(x,z0err), min(x,-z0err)); |
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z0off(Ki,z) = (sigma0/4*(Ki+K_r)*Exp(-sq(z/sigma0)/2)-sqrt(PI/2)*z)/ |
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z0lim(z0d(Ki,z)); |
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z0root(Ki, x0, x1, i) = if(i, |
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if(z0err-abs(x1-x0), |
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x1, |
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z0root(Ki,x1,x1-z0off(Ki,x1),i-1)), |
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0); |
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z0(ti) = z0root(K(ti), .1, -z0off(K(ti),.1), 100); |
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{ sigma } |
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sigma(ti) = if( FTINY-sigma0, sigma0, |
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sigma0/sqrt(1+sq(z0(ti)/sigma0)) ); |
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{ g } |
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g`r(cti) = sq(2*PI/lambda`r*sigma(Acos(cti))*(cti+RdotP)); |
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g`g(cti) = sq(2*PI/lambda`g*sigma(Acos(cti))*(cti+RdotP)); |
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g`b(cti) = sq(2*PI/lambda`b*sigma(Acos(cti))*(cti+RdotP)); |
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{ |F|^2 } |
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fresnel2dd`r(kix,kiy,kiz) = fresnel2`r(sqrt(sq(kix-Dx) + sq(kiy-Dy) + |
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sq(kiz-Dz))/2); |
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fresnel2dd`g(kix,kiy,kiz) = fresnel2`g(sqrt(sq(kix-Dx) + sq(kiy-Dy) + |
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sq(kiz-Dz))/2); |
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fresnel2dd`b(kix,kiy,kiz) = fresnel2`b(sqrt(sq(kix-Dx) + sq(kiy-Dy) + |
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sq(kiz-Dz))/2); |
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{ G } |
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G(kix,kiy,kiz) = sq( (sq(kix-Dx)+sq(kiy-Dy)+sq(kiz-Dz)) / |
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(NxP*(kix-Dx)+NyP*(kiy-Dy)+NzP*(kiz-Dz)) ); |
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{ D } |
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Dsum2(m,lt,c,t,e,g) = if(or(m-Dsummax,and(lt-t,Dsumlim-t)),0, |
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t+Dsum2(m+1,t,c*g/(m+1),c*g/(m+1)*Exp(-g-e/(m+1))/(m+1),e,g)); |
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Dsum(e,g) = Dsum2(1,0,g,g*Exp(-g-e),e,g); |
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D`r(kix,kiy,kiz) = sq(PI)/4/sq(lambda`r)*sq(tau) * |
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Dsum(sq(2*PI/lambda`r)/4*sq(tau)* |
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(sq(kix-Dx)+sq(kiy-Dy)+sq(kiz-Dz) - |
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sq(NxP*(kix-Dx)+NyP*(kiy-Dy)+NzP*(kiz-Dz))), |
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g`r(kix*NxP+kiy*NyP+kiz*NzP)); |
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D`g(kix,kiy,kiz) = sq(PI)/4/sq(lambda`g)*sq(tau) * |
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Dsum(sq(2*PI/lambda`g)/4*sq(tau)* |
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(sq(kix-Dx)+sq(kiy-Dy)+sq(kiz-Dz) - |
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sq(NxP*(kix-Dx)+NyP*(kiy-Dy)+NzP*(kiz-Dz))), |
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g`g(kix*NxP+kiy*NyP+kiz*NzP)); |
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D`b(kix,kiy,kiz) = sq(PI)/4/sq(lambda`b)*sq(tau) * |
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Dsum(sq(2*PI/lambda`b)/4*sq(tau)* |
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(sq(kix-Dx)+sq(kiy-Dy)+sq(kiz-Dz) - |
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sq(NxP*(kix-Dx)+NyP*(kiy-Dy)+NzP*(kiz-Dz))), |
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g`b(kix*NxP+kiy*NyP+kiz*NzP)); |
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{ rho_dd } |
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dd2(cti) = shadowf_r*shadowf(Acos(cti))/cti/RdotP; |
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dd`r(kix,kiy,kiz) = dd2(kix*NxP+kiy*NyP+kiz*NzP)*G(kix,kiy,kiz)* |
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fresnel2dd`r(kix,kiy,kiz)/PI*D`r(kix,kiy,kiz); |
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dd`g(kix,kiy,kiz) = dd2(kix*NxP+kiy*NyP+kiz*NzP)*G(kix,kiy,kiz)* |
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fresnel2dd`g(kix,kiy,kiz)/PI*D`g(kix,kiy,kiz); |
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dd`b(kix,kiy,kiz) = dd2(kix*NxP+kiy*NyP+kiz*NzP)*G(kix,kiy,kiz)* |
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fresnel2dd`b(kix,kiy,kiz)/PI*D`b(kix,kiy,kiz); |
