bool Sphere::Sample(const Interaction &ref, const Point2f &sample,
Interaction *it) const {
// Compute coordinate system for sphere sampling
Point3f pCenter = (*ObjectToWorld)(Point3f(0, 0, 0));
Point3f pOrigin =
OffsetRayOrigin(ref.p, ref.pError, ref.n, pCenter - ref.p);
Vector3f wc = Normalize(pCenter - ref.p);
Vector3f wcX, wcY;
CoordinateSystem(wc, &wcX, &wcY);
// Sample uniformly on sphere if $\pt{}$ is inside it
if (DistanceSquared(pOrigin, pCenter) <= 1.0001f * radius * radius)
return Sample(sample, it);
// Sample sphere uniformly inside subtended cone
Float sinThetaMax2 = radius * radius / DistanceSquared(ref.p, pCenter);
Float cosThetaMax =
std::sqrt(std::max((Float)0., (Float)1. - sinThetaMax2));
SurfaceInteraction isectSphere;
Float tHit;
Ray r = ref.SpawnRay(UniformSampleCone(sample, cosThetaMax, wcX, wcY, wc));
if (!Intersect(r, &tHit, &isectSphere)) return false;
*it = isectSphere;
if (ReverseOrientation) it->n *= -1.f;
return true;
}
Float Sphere::Pdf(const Interaction &ref, const Vector3f &wi) const {
Point3f pCenter = (*ObjectToWorld)(Point3f(0, 0, 0));
// Return uniform PDF if point is inside sphere
Point3f pOrigin =
OffsetRayOrigin(ref.p, ref.pError, ref.n, pCenter - ref.p);
if (DistanceSquared(pOrigin, pCenter) <= radius * radius)
return Shape::Pdf(ref, wi);
// Compute general sphere PDF
Float sinThetaMax2 = radius * radius / DistanceSquared(ref.p, pCenter);
Float cosThetaMax = std::sqrt(std::max((Float)0, 1 - sinThetaMax2));
return UniformConePdf(cosThetaMax);
}
Interaction Sphere::Sample(const Interaction &ref, const Point2f &u) const {
// Compute coordinate system for sphere sampling
Point3f pCenter = (*ObjectToWorld)(Point3f(0, 0, 0));
Vector3f wc = Normalize(pCenter - ref.p);
Vector3f wcX, wcY;
CoordinateSystem(wc, &wcX, &wcY);
// Sample uniformly on sphere if $\pt{}$ is inside it
Point3f pOrigin =
OffsetRayOrigin(ref.p, ref.pError, ref.n, pCenter - ref.p);
if (DistanceSquared(pOrigin, pCenter) <= radius * radius) return Sample(u);
// Sample sphere uniformly inside subtended cone
// Compute $\theta$ and $\phi$ values for sample in cone
Float sinThetaMax2 = radius * radius / DistanceSquared(ref.p, pCenter);
Float cosThetaMax = std::sqrt(std::max((Float)0, 1 - sinThetaMax2));
Float cosTheta = (1 - u[0]) + u[0] * cosThetaMax;
Float sinTheta = std::sqrt(1 - cosTheta * cosTheta);
Float phi = u[1] * 2 * Pi;
// Compute angle $\alpha$ from center of sphere to sampled point on surface
Float dc = Distance(ref.p, pCenter);
Float ds = dc * cosTheta -
std::sqrt(std::max(
(Float)0, radius * radius - dc * dc * sinTheta * sinTheta));
Float cosAlpha = (dc * dc + radius * radius - ds * ds) / (2 * dc * radius);
Float sinAlpha = std::sqrt(std::max((Float)0, 1 - cosAlpha * cosAlpha));
// Compute surface normal and sampled point on sphere
Vector3f nObj =
SphericalDirection(sinAlpha, cosAlpha, phi, -wcX, -wcY, -wc);
Point3f pObj = radius * Point3f(nObj.x, nObj.y, nObj.z);
// Return _Interaction_ for sampled point on sphere
Interaction it;
// Reproject _pObj_ to sphere surface and compute _pObjError_
pObj *= radius / Distance(pObj, Point3f(0, 0, 0));
Vector3f pObjError = gamma(5) * Abs((Vector3f)pObj);
it.p = (*ObjectToWorld)(pObj, pObjError, &it.pError);
it.n = (*ObjectToWorld)(Normal3f(nObj));
if (reverseOrientation) it.n *= -1.f;
return it;
}
