I developed a ray tracer that use standard phong/blinn phong lighting model. Now I'm modifying it to support physically based rendering, so I'm implementing various BRDF models. At the moment I'm focused on Oren-Nayar and Torrance-Sparrow model. Each one of these is based on spherical coordinates used to express incident wi and outgoing wo light direction.

My question is: which way is the right one convert wi and wo from cartesian coordinate to spherical coordinate?

I'm applying the standard formula reported here https://en.wikipedia.org/wiki/Spherical_coordinate_system#Coordinate_system_conversions but I'm not sure I'm doing the right thing, because my vector are not with tail at the origin of the cartesian coordinate system, but are centered on the intersection point of the ray with the object.

Here you can find my current implementation:

• https://github.com/chicio/Multispectral-Ray-tracing/tree/brdf/RayTracing/RayTracer/Objects/BRDF

• https://github.com/chicio/Multispectral-Ray-tracing/blob/brdf/RayTracing/RayTracer/Math/Vector3D.cpp

Can anyone help me giving an explanation of the correct way to convert the wi and wo vector from cartesian to spherical coordinate?

UPDATE

I copy here the relevant part of code:

spherical coordinate calculation

float Vector3D::sphericalTheta() const {

    float sphericalTheta = acosf(Utils::clamp(y, -1.f, 1.f));

    return sphericalTheta;
}

float Vector3D::sphericalPhi() const {
    
    float phi = atan2f(z, x);
    
    return (phi < 0.f) ? phi + 2.f * M_PI : phi;
}

Oren Nayar

OrenNayar::OrenNayar(Spectrum<constant::spectrumSamples> reflectanceSpectrum, float degree) : reflectanceSpectrum{reflectanceSpectrum} {

    float sigma = Utils::degreeToRadian(degree);
    float sigmaPowerTwo = sigma * sigma;
    
    A = 1.0f - (sigmaPowerTwo / 2.0f * (sigmaPowerTwo + 0.33f));
    B = 0.45f * sigmaPowerTwo / (sigmaPowerTwo + 0.09f);
};

Spectrum<constant::spectrumSamples> OrenNayar::f(const Vector3D& wi, const Vector3D& wo, const Intersection* intersection) const {

    float thetaI = wi.sphericalTheta();
    float phiI = wi.sphericalPhi();
    
    float thetaO = wo.sphericalTheta();
    float phiO = wo.sphericalPhi();
    
    float alpha = std::fmaxf(thetaI, thetaO);
    float beta = std::fminf(thetaI, thetaO);
    
    Spectrum<constant::spectrumSamples> orenNayar = reflectanceSpectrum * constant::inversePi * (A + B * std::fmaxf(0, cosf(phiI - phiO) * sinf(alpha) * tanf(beta)));
    
    return orenNayar;
}

Torrance-Sparrow

float TorranceSparrow::G(const Vector3D& wi, const Vector3D& wo, const Vector3D& wh, const Intersection* intersection) const {
    
    Vector3D normal = intersection->normal;
    normal.normalize();
    
    float normalDotWh = fabsf(normal.dot(wh));
    float normalDotWo = fabsf(normal.dot(wo));
    float normalDotWi = fabsf(normal.dot(wi));
    float woDotWh = fabsf(wo.dot(wh));
    
    float G = fminf(1.0f, std::fminf((2.0f * normalDotWh * normalDotWo)/woDotWh, (2.0f * normalDotWh * normalDotWi)/woDotWh));
    
    return G;
}

float TorranceSparrow::D(const Vector3D& wh, const Intersection* intersection) const {
    
    Vector3D normal = intersection->normal;
    normal.normalize();

    float cosThetaH = fabsf(wh.dot(normal));
    
    float Dd = (exponent + 2) * constant::inverseTwoPi * powf(cosThetaH, exponent);
    
    return Dd;
}

Spectrum<constant::spectrumSamples> TorranceSparrow::f(const Vector3D& wi, const Vector3D& wo, const Intersection* intersection) const {

    Vector3D normal = intersection->normal;
    normal.normalize();
    
    float thetaI = wi.sphericalTheta();
    float thetaO = wo.sphericalTheta();
    
    float cosThetaO = fabsf(cosf(thetaO));
    float cosThetaI = fabsf(cosf(thetaI));
    
    if(cosThetaI == 0 || cosThetaO == 0) {
        
        return reflectanceSpectrum * 0.0f;
    }
    
    Vector3D wh = (wi + wo);
    wh.normalize();
    
    float cosThetaH = wi.dot(wh);
    
    float F = Fresnel::dieletricFresnel(cosThetaH, refractiveIndex);
    float g = G(wi, wo, wh, intersection);
    float d = D(wh, intersection);

    printf("f %f g %f d %f \n", F, g, d);
    printf("result %f \n", ((d * g * F) / (4.0f * cosThetaI * cosThetaO)));
    
    Spectrum<constant::spectrumSamples> torranceSparrow = reflectanceSpectrum * ((d * g * F) / (4.0f * cosThetaI * cosThetaO));
    
    return torranceSparrow;
}

UPDATE 2

After some search I found this implementation of Oren-Nayar BRDF.

http://content.gpwiki.org/index.php/D3DBook:(Lighting)_Oren-Nayar

In the implementation above theta for wi and wo is obtained simply doing arccos(wo.dotProduct(Normal)) and arccos(wi.dotProduct(Normal)). This seems reasonable to me, as we can use the normal of the intersection point as the zenith direction for our spherical coordinate system and do the calculation. The calculation of gamma = cos(phi_wi - phi_wo) do some sort of projection of wi and wo on what it calls "tangent space". Assuming everything is correct in this implementation, can i just use the formulas |View - Normal x (View.dotProduct(Normal))| and |Light - Normal x (Light.dotProduct(Normal))| to obtain the phi coordinate (instead of using arctan("something"))?