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