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{ |
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He-Torrance Reflectance Model (Siggraph 1991) |
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|
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This is the simplified version that doesn't account for |
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changes in reflection due to changes in wavelength. Also, |
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specular and directional-diffuse hightlights are left uncolored |
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because coloring them requires multiple evaluations of some |
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very expensive functions. |
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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 s s |
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0 0 0 |
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dd dd dd |
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He.cal |
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0 |
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13 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 n_imag |
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|
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For metals, the specular color may be modified like so: |
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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 dd dd |
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He.cal |
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0 |
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13 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 n_imag |
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|
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This doesn't work for the directional diffuse component, unfortunately. |
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A second set of functions dd_r, dd_g and dd_b may be used, but they |
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cost three times as much to compute! |
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} |
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|
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{ Constants } |
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lambda : .5; { 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 = arg(12); { real part of index of refraction } |
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n_imag = arg(13); { imaginary part of index of refraction } |
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{ Derived parameters } |
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n_k = n_imag/n_real; |
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{ Constant functions } |
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Exp(x) : if(-x-400, 0, exp(x)); { rayinit.cal version too timid for D() } |
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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(ct) = sq(n_real)*(1-sq(n_k)) - (1-sq(ct)); |
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fuvB(ct) = sqrt(sq(fuvA(ct)) + 4*sq(sq(n_real)*n_k)); |
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fu2(ct) = (fuvA(ct) + fuvB(ct))/2; |
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fv2(ct) = (-fuvA(ct) + fuvB(ct))/2; |
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fperp2(ct) = (sq(ct-sqrt(fu2(ct))) + fv2(ct)) / |
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(sq(ct+sqrt(fu2(ct))) + fv2(ct)); |
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fpara2(ct) = (sq(sq(n_real)*(1-sq(n_k))*ct - sqrt(fu2(ct))) + |
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sq(2*sq(n_real)*n_k*ct - Sqrt(fv2(ct)))) / |
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(sq(sq(n_real)*(1-sq(n_k))*ct + sqrt(fu2(ct))) + |
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sq(2*sq(n_real)*n_k*ct + Sqrt(fv2(ct)))); |
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fresnel2(ct) = (fperp2(ct) + fpara2(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 = fresnel2(RdotP)*Exp(-g(RdotP))*sq(shadowf_r); |
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s_r = s*arg(1)*CrP; |
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s_g = s*arg(2)*CgP; |
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s_b = s*arg(3)*CbP; |
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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(cti) = sq(2*PI/lambda*sigma(Acos(cti))*(cti+RdotP)); |
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{ |F|^2 } |
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fresnel2dd(kix,kiy,kiz) = fresnel2(sqrt(sq(kix-Dx) + sq(kiy-Dy) + |
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sq(kiz-Dz))/2); |
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{ G } |
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{ The bulk of G was found by Andrew Willmott to cancel. This is the original: |
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G2( kix,kiy,kiz, six,siy,siz ) = |
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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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sq(sq(Dy*kiz-Dz*kiy)+sq(Dz*kix-Dx*kiz)+sq(Dx*kiy-Dy*kix)) * |
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(sq(srx*kix+sry*kiy+srz*kiz) + |
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sq(prx*kix+pry*kiy+prz*kiz)) * |
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(sq(six*Dx+siy*Dy+siz*Dz) + |
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sq((siy*kiz-siz*kiy)*Dx+(siz*kix-six*kiz)*Dy+(six*kiy-siy*kix)*Dz)) / |
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srn2 / (sq(six)+sq(siy)+sq(siz)); |
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G(kix,kiy,kiz) = G2(kix,kiy,kiz, |
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kiy*NzP-kiz*NyP, kiz*NxP-kix*NzP, kix*NyP-kiy*NxP); |
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-- Newer version below is much simpler: } |
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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)),t, |
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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(kix,kiy,kiz) = sq(PI)/4/sq(lambda)*sq(tau) * |
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Dsum(sq(2*PI/lambda)/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(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(kix,kiy,kiz) = dd2(kix*NxP+kiy*NyP+kiz*NzP)*G(kix,kiy,kiz)* |
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fresnel2dd(kix,kiy,kiz)/PI*D(kix,kiy,kiz); |
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{ Color version 3x as slow! } |
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dd_r(kix,kiy,kiz) = dd(kix,kiy,kiz)*arg(1)*CrP; |
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dd_g(kix,kiy,kiz) = dd(kix,kiy,kiz)*arg(2)*CgP; |
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dd_b(kix,kiy,kiz) = dd(kix,kiy,kiz)*arg(3)*CbP; |
