bool Sphere::IntersectP(const Ray &r, bool testAlphaTexture) const { Float phi; Point3f pHit; // Transform _Ray_ to object space Vector3f oErr, dErr; Ray ray = (*WorldToObject)(r, &oErr, &dErr); // Compute quadratic sphere coefficients // Initialize _EFloat_ ray coordinate values EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z); EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z); EFloat a = dx * dx + dy * dy + dz * dz; EFloat b = 2 * (dx * ox + dy * oy + dz * oz); EFloat c = ox * ox + oy * oy + oz * oz - EFloat(radius) * EFloat(radius); // Solve quadratic equation for _t_ values EFloat t0, t1; if (!Quadratic(a, b, c, &t0, &t1)) return false; // Check quadric shape _t0_ and _t1_ for nearest intersection if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false; EFloat tShapeHit = t0; if (tShapeHit.LowerBound() <= 0) { tShapeHit = t1; if (tShapeHit.UpperBound() > ray.tMax) return false; } // Compute sphere hit position and $\phi$ pHit = ray((Float)tShapeHit); // Refine sphere intersection point pHit *= radius / Distance(pHit, Point3f(0, 0, 0)); if (pHit.x == 0 && pHit.y == 0) pHit.x = 1e-5f * radius; phi = std::atan2(pHit.y, pHit.x); if (phi < 0) phi += 2 * Pi; // Test sphere intersection against clipping parameters if ((zMin > -radius && pHit.z < zMin) || (zMax < radius && pHit.z > zMax) || phi > phiMax) { if (tShapeHit == t1) return false; if (t1.UpperBound() > ray.tMax) return false; tShapeHit = t1; // Compute sphere hit position and $\phi$ pHit = ray((Float)tShapeHit); // Refine sphere intersection point pHit *= radius / Distance(pHit, Point3f(0, 0, 0)); if (pHit.x == 0 && pHit.y == 0) pHit.x = 1e-5f * radius; phi = std::atan2(pHit.y, pHit.x); if (phi < 0) phi += 2 * Pi; if ((zMin > -radius && pHit.z < zMin) || (zMax < radius && pHit.z > zMax) || phi > phiMax) return false; } return true; }
bool Cylinder::IntersectP(const Ray &r, bool testAlphaTexture) const { Float phi; Point3f pHit; // Transform _Ray_ to object space Vector3f oErr, dErr; Ray ray = (*WorldToObject)(r, &oErr, &dErr); // Compute quadratic cylinder coefficients // Initialize _EFloat_ ray coordinate values EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z); EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z); EFloat a = dx * dx + dy * dy; EFloat b = 2 * (dx * ox + dy * oy); EFloat c = ox * ox + oy * oy - EFloat(radius) * EFloat(radius); // Solve quadratic equation for _t_ values EFloat t0, t1; if (!Quadratic(a, b, c, &t0, &t1)) return false; // Check quadric shape _t0_ and _t1_ for nearest intersection if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false; EFloat tShapeHit = t0; if (tShapeHit.LowerBound() <= 0) { tShapeHit = t1; if (tShapeHit.UpperBound() > ray.tMax) return false; } // Compute cylinder hit point and $\phi$ pHit = ray((Float)tShapeHit); // Refine cylinder intersection point Float hitRad = std::sqrt(pHit.x * pHit.x + pHit.y * pHit.y); pHit.x *= radius / hitRad; pHit.y *= radius / hitRad; phi = std::atan2(pHit.y, pHit.x); if (phi < 0) phi += 2 * Pi; // Test cylinder intersection against clipping parameters if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) { if (tShapeHit == t1) return false; tShapeHit = t1; if (t1.UpperBound() > ray.tMax) return false; // Compute cylinder hit point and $\phi$ pHit = ray((Float)tShapeHit); // Refine cylinder intersection point Float hitRad = std::sqrt(pHit.x * pHit.x + pHit.y * pHit.y); pHit.x *= radius / hitRad; pHit.y *= radius / hitRad; phi = std::atan2(pHit.y, pHit.x); if (phi < 0) phi += 2 * Pi; if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) return false; } return true; }
bool Paraboloid::IntersectP(const Ray &r) const { Float phi; Point3f pHit; // Transform _Ray_ to object space Vector3f oErr, dErr; Ray ray = (*WorldToObject)(r, &oErr, &dErr); // Compute quadratic paraboloid coefficients // Initialize _EFloat_ ray coordinate values EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z); EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z); EFloat k = EFloat(zMax) / (EFloat(radius) * EFloat(radius)); EFloat a = k * (dx * dx + dy * dy); EFloat b = 2.f * k * (dx * ox + dy * oy) - dz; EFloat c = k * (ox * ox + oy * oy) - oz; // Solve quadratic equation for _t_ values EFloat t0, t1; if (!Quadratic(a, b, c, &t0, &t1)) return false; // Check quadric shape _t0_ and _t1_ for nearest intersection if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false; EFloat tShapeHit = t0; if (tShapeHit.LowerBound() <= 0) { tShapeHit = t1; if (tShapeHit.UpperBound() > ray.tMax) return false; } // Compute paraboloid inverse mapping pHit = ray((Float)tShapeHit); phi = std::atan2(pHit.y, pHit.x); if (phi < 0.) phi += 2 * Pi; // Test paraboloid intersection against clipping parameters if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) { if (tShapeHit == t1) return false; tShapeHit = t1; if (t1.UpperBound() > ray.tMax) return false; // Compute paraboloid inverse mapping pHit = ray((Float)tShapeHit); phi = std::atan2(pHit.y, pHit.x); if (phi < 0.) phi += 2 * Pi; if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) return false; } return true; }
bool Cylinder::Intersect(const Ray &r, Float *tHit, SurfaceInteraction *isect, bool testAlphaTexture) const { Float phi; Point3f pHit; // Transform _Ray_ to object space Vector3f oErr, dErr; Ray ray = (*WorldToObject)(r, &oErr, &dErr); // Compute quadratic cylinder coefficients // Initialize _EFloat_ ray coordinate values EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z); EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z); EFloat a = dx * dx + dy * dy; EFloat b = 2 * (dx * ox + dy * oy); EFloat c = ox * ox + oy * oy - EFloat(radius) * EFloat(radius); // Solve quadratic equation for _t_ values EFloat t0, t1; if (!Quadratic(a, b, c, &t0, &t1)) return false; // Check quadric shape _t0_ and _t1_ for nearest intersection if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false; EFloat tShapeHit = t0; if (tShapeHit.LowerBound() <= 0) { tShapeHit = t1; if (tShapeHit.UpperBound() > ray.tMax) return false; } // Compute cylinder hit point and $\phi$ pHit = ray((Float)tShapeHit); // Refine cylinder intersection point Float hitRad = std::sqrt(pHit.x * pHit.x + pHit.y * pHit.y); pHit.x *= radius / hitRad; pHit.y *= radius / hitRad; phi = std::atan2(pHit.y, pHit.x); if (phi < 0) phi += 2 * Pi; // Test cylinder intersection against clipping parameters if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) { if (tShapeHit == t1) return false; tShapeHit = t1; if (t1.UpperBound() > ray.tMax) return false; // Compute cylinder hit point and $\phi$ pHit = ray((Float)tShapeHit); // Refine cylinder intersection point Float hitRad = std::sqrt(pHit.x * pHit.x + pHit.y * pHit.y); pHit.x *= radius / hitRad; pHit.y *= radius / hitRad; phi = std::atan2(pHit.y, pHit.x); if (phi < 0) phi += 2 * Pi; if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) return false; } // Find parametric representation of cylinder hit Float u = phi / phiMax; Float v = (pHit.z - zMin) / (zMax - zMin); // Compute cylinder $\dpdu$ and $\dpdv$ Vector3f dpdu(-phiMax * pHit.y, phiMax * pHit.x, 0); Vector3f dpdv(0, 0, zMax - zMin); // Compute cylinder $\dndu$ and $\dndv$ Vector3f d2Pduu = -phiMax * phiMax * Vector3f(pHit.x, pHit.y, 0); Vector3f d2Pduv(0, 0, 0), d2Pdvv(0, 0, 0); // Compute coefficients for fundamental forms Float E = Dot(dpdu, dpdu); Float F = Dot(dpdu, dpdv); Float G = Dot(dpdv, dpdv); Vector3f N = Normalize(Cross(dpdu, dpdv)); Float e = Dot(N, d2Pduu); Float f = Dot(N, d2Pduv); Float g = Dot(N, d2Pdvv); // Compute $\dndu$ and $\dndv$ from fundamental form coefficients Float invEGF2 = 1 / (E * G - F * F); Normal3f dndu = Normal3f((f * F - e * G) * invEGF2 * dpdu + (e * F - f * E) * invEGF2 * dpdv); Normal3f dndv = Normal3f((g * F - f * G) * invEGF2 * dpdu + (f * F - g * E) * invEGF2 * dpdv); // Compute error bounds for cylinder intersection Vector3f pError = gamma(3) * Abs(Vector3f(pHit.x, pHit.y, 0)); // Initialize _SurfaceInteraction_ from parametric information *isect = (*ObjectToWorld)(SurfaceInteraction(pHit, pError, Point2f(u, v), -ray.d, dpdu, dpdv, dndu, dndv, ray.time, this)); // Update _tHit_ for quadric intersection *tHit = (Float)tShapeHit; return true; }
bool Paraboloid::Intersect(const Ray &r, Float *tHit, SurfaceInteraction *isect) const { Float phi; Point3f pHit; // Transform _Ray_ to object space Vector3f oErr, dErr; Ray ray = (*WorldToObject)(r, &oErr, &dErr); // Compute quadratic paraboloid coefficients // Initialize _EFloat_ ray coordinate values EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z); EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z); EFloat k = EFloat(zMax) / (EFloat(radius) * EFloat(radius)); EFloat a = k * (dx * dx + dy * dy); EFloat b = 2.f * k * (dx * ox + dy * oy) - dz; EFloat c = k * (ox * ox + oy * oy) - oz; // Solve quadratic equation for _t_ values EFloat t0, t1; if (!Quadratic(a, b, c, &t0, &t1)) return false; // Check quadric shape _t0_ and _t1_ for nearest intersection if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false; EFloat tShapeHit = t0; if (tShapeHit.LowerBound() <= 0) { tShapeHit = t1; if (tShapeHit.UpperBound() > ray.tMax) return false; } // Compute paraboloid inverse mapping pHit = ray((Float)tShapeHit); phi = std::atan2(pHit.y, pHit.x); if (phi < 0.) phi += 2 * Pi; // Test paraboloid intersection against clipping parameters if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) { if (tShapeHit == t1) return false; tShapeHit = t1; if (t1.UpperBound() > ray.tMax) return false; // Compute paraboloid inverse mapping pHit = ray((Float)tShapeHit); phi = std::atan2(pHit.y, pHit.x); if (phi < 0.) phi += 2 * Pi; if (pHit.z < zMin || pHit.z > zMax || phi > phiMax) return false; } // Find parametric representation of paraboloid hit Float u = phi / phiMax; Float v = (pHit.z - zMin) / (zMax - zMin); // Compute paraboloid $\dpdu$ and $\dpdv$ Vector3f dpdu(-phiMax * pHit.y, phiMax * pHit.x, 0.); Vector3f dpdv = (zMax - zMin) * Vector3f(pHit.x / (2 * pHit.z), pHit.y / (2 * pHit.z), 1.); // Compute paraboloid $\dndu$ and $\dndv$ Vector3f d2Pduu = -phiMax * phiMax * Vector3f(pHit.x, pHit.y, 0); Vector3f d2Pduv = (zMax - zMin) * phiMax * Vector3f(-pHit.y / (2 * pHit.z), pHit.x / (2 * pHit.z), 0); Vector3f d2Pdvv = -(zMax - zMin) * (zMax - zMin) * Vector3f(pHit.x / (4 * pHit.z * pHit.z), pHit.y / (4 * pHit.z * pHit.z), 0.); // Compute coefficients for fundamental forms Float E = Dot(dpdu, dpdu); Float F = Dot(dpdu, dpdv); Float G = Dot(dpdv, dpdv); Vector3f N = Normalize(Cross(dpdu, dpdv)); Float e = Dot(N, d2Pduu); Float f = Dot(N, d2Pduv); Float g = Dot(N, d2Pdvv); // Compute $\dndu$ and $\dndv$ from fundamental form coefficients Float invEGF2 = 1 / (E * G - F * F); Normal3f dndu = Normal3f((f * F - e * G) * invEGF2 * dpdu + (e * F - f * E) * invEGF2 * dpdv); Normal3f dndv = Normal3f((g * F - f * G) * invEGF2 * dpdu + (f * F - g * E) * invEGF2 * dpdv); // Compute error bounds for paraboloid intersection // Compute error bounds for intersection computed with ray equation EFloat px = ox + tShapeHit * dx; EFloat py = oy + tShapeHit * dy; EFloat pz = oz + tShapeHit * dz; Vector3f pError = Vector3f(px.GetAbsoluteError(), py.GetAbsoluteError(), pz.GetAbsoluteError()); // Initialize _SurfaceInteraction_ from parametric information *isect = (*ObjectToWorld)(SurfaceInteraction(pHit, pError, Point2f(u, v), -ray.d, dpdu, dpdv, dndu, dndv, ray.time, this)); *tHit = (Float)tShapeHit; return true; }
bool Sphere::Intersect(const Ray &r, Float *tHit, SurfaceInteraction *isect, bool testAlphaTexture) const { Float phi; Point3f pHit; // Transform _Ray_ to object space Vector3f oErr, dErr; Ray ray = (*WorldToObject)(r, &oErr, &dErr); // Compute quadratic sphere coefficients // Initialize _EFloat_ ray coordinate values EFloat ox(ray.o.x, oErr.x), oy(ray.o.y, oErr.y), oz(ray.o.z, oErr.z); EFloat dx(ray.d.x, dErr.x), dy(ray.d.y, dErr.y), dz(ray.d.z, dErr.z); EFloat a = dx * dx + dy * dy + dz * dz; EFloat b = 2 * (dx * ox + dy * oy + dz * oz); EFloat c = ox * ox + oy * oy + oz * oz - EFloat(radius) * EFloat(radius); // Solve quadratic equation for _t_ values EFloat t0, t1; if (!Quadratic(a, b, c, &t0, &t1)) return false; // Check quadric shape _t0_ and _t1_ for nearest intersection if (t0.UpperBound() > ray.tMax || t1.LowerBound() <= 0) return false; EFloat tShapeHit = t0; if (tShapeHit.LowerBound() <= 0) { tShapeHit = t1; if (tShapeHit.UpperBound() > ray.tMax) return false; } // Compute sphere hit position and $\phi$ pHit = ray((Float)tShapeHit); // Refine sphere intersection point pHit *= radius / Distance(pHit, Point3f(0, 0, 0)); if (pHit.x == 0 && pHit.y == 0) pHit.x = 1e-5f * radius; phi = std::atan2(pHit.y, pHit.x); if (phi < 0) phi += 2 * Pi; // Test sphere intersection against clipping parameters if ((zMin > -radius && pHit.z < zMin) || (zMax < radius && pHit.z > zMax) || phi > phiMax) { if (tShapeHit == t1) return false; if (t1.UpperBound() > ray.tMax) return false; tShapeHit = t1; // Compute sphere hit position and $\phi$ pHit = ray((Float)tShapeHit); // Refine sphere intersection point pHit *= radius / Distance(pHit, Point3f(0, 0, 0)); if (pHit.x == 0 && pHit.y == 0) pHit.x = 1e-5f * radius; phi = std::atan2(pHit.y, pHit.x); if (phi < 0) phi += 2 * Pi; if ((zMin > -radius && pHit.z < zMin) || (zMax < radius && pHit.z > zMax) || phi > phiMax) return false; } // Find parametric representation of sphere hit Float u = phi / phiMax; Float theta = std::acos(Clamp(pHit.z / radius, -1, 1)); Float v = (theta - thetaMin) / (thetaMax - thetaMin); // Compute sphere $\dpdu$ and $\dpdv$ Float zRadius = std::sqrt(pHit.x * pHit.x + pHit.y * pHit.y); Float invZRadius = 1 / zRadius; Float cosPhi = pHit.x * invZRadius; Float sinPhi = pHit.y * invZRadius; Vector3f dpdu(-phiMax * pHit.y, phiMax * pHit.x, 0); Vector3f dpdv = (thetaMax - thetaMin) * Vector3f(pHit.z * cosPhi, pHit.z * sinPhi, -radius * std::sin(theta)); // Compute sphere $\dndu$ and $\dndv$ Vector3f d2Pduu = -phiMax * phiMax * Vector3f(pHit.x, pHit.y, 0); Vector3f d2Pduv = (thetaMax - thetaMin) * pHit.z * phiMax * Vector3f(-sinPhi, cosPhi, 0.); Vector3f d2Pdvv = -(thetaMax - thetaMin) * (thetaMax - thetaMin) * Vector3f(pHit.x, pHit.y, pHit.z); // Compute coefficients for fundamental forms Float E = Dot(dpdu, dpdu); Float F = Dot(dpdu, dpdv); Float G = Dot(dpdv, dpdv); Vector3f N = Normalize(Cross(dpdu, dpdv)); Float e = Dot(N, d2Pduu); Float f = Dot(N, d2Pduv); Float g = Dot(N, d2Pdvv); // Compute $\dndu$ and $\dndv$ from fundamental form coefficients Float invEGF2 = 1 / (E * G - F * F); Normal3f dndu = Normal3f((f * F - e * G) * invEGF2 * dpdu + (e * F - f * E) * invEGF2 * dpdv); Normal3f dndv = Normal3f((g * F - f * G) * invEGF2 * dpdu + (f * F - g * E) * invEGF2 * dpdv); // Compute error bounds for sphere intersection Vector3f pError = gamma(5) * Abs((Vector3f)pHit); // Initialize _SurfaceInteraction_ from parametric information *isect = (*ObjectToWorld)(SurfaceInteraction(pHit, pError, Point2f(u, v), -ray.d, dpdu, dpdv, dndu, dndv, ray.time, this)); // Update _tHit_ for quadric intersection *tHit = (Float)tShapeHit; return true; }