def sample_p(self, p, u1, u2): """Sample at point p.""" # Compute coordinate system for sphere sampling p_center = self.object_to_world(Point(0, 0, 0)) wc = normalize(p_center - p) wc_x, wc_y = coordinate_system(wc) # Sample uniformly on sphere if $\pt{}$ is inside it if (distance_squared(p, p_center) - self.radius * self.radius) < 1e-4: return self.sample(u1, u2) # Sample sphere uniformly inside subtended cone sin_theta_max2 = self.radius * self.radius / distance_squared( p, p_center) cos_theta_max = math.sqrt(max(0.0, 1.0 - sin_theta_max2)) raise Exception("next_line") # r = Ray(p, uniform_sample_cone(u1, u2, cos_theta_max, wcX, wcY, wc), 1e-3) r = Ray(p) intersect, t_hit, ray_epsilon, dg_sphere = self.intersect(r) if not intersect: t_hit = dot(p_center - p, normalize(r.d)) ps = r(t_hit) ns = Normal(normalize(ps - p_center)) if (self.reverse_orientation): ns *= -1.0 return ps, ns
def test_dot_w_zero(self): v1 = Vector(1.0, 3.0, 5.0) v2 = Vector(2.0, 3.0, 4.0) q1 = Quaternion(v1, 0.0) q2 = Quaternion(v2, 0.0) scalar = dot_quaternions(q1, q2) self.assertEqual(scalar, dot(v1, v2))
def sample_p(self, p, u1, u2): """Sample at point p.""" # Compute coordinate system for sphere sampling p_center = self.object_to_world(Point(0, 0, 0)) wc = normalize(p_center - p) wc_x, wc_y = coordinate_system(wc) # Sample uniformly on sphere if $\pt{}$ is inside it if (distance_squared(p, p_center) - self.radius * self.radius) < 1e-4: return self.sample(u1, u2) # Sample sphere uniformly inside subtended cone sin_theta_max2 = self.radius * self.radius / distance_squared(p, p_center) cos_theta_max = math.sqrt(max(0.0, 1.0 - sin_theta_max2)) raise Exception("next_line") # r = Ray(p, uniform_sample_cone(u1, u2, cos_theta_max, wcX, wcY, wc), 1e-3) r = Ray(p) intersect, t_hit, ray_epsilon, dg_sphere = self.intersect(r) if not intersect: t_hit = dot(p_center - p, normalize(r.d)) ps = r(t_hit) ns = Normal(normalize(ps - p_center)) if self.reverse_orientation: ns *= -1.0 return ps, ns
def intersect(self, r): """Intersect the ray with the shape.""" # Transform _Ray_ to object space ray = self.world_to_object(r) # Compute quadratic sphere coefficients A = ray.d.x * ray.d.x + ray.d.y * ray.d.y + ray.d.z * ray.d.z B = 2 * (ray.d.x * ray.o.x + ray.d.y * ray.o.y + ray.d.z * ray.o.z) C = ray.o.x*ray.o.x + ray.o.y*ray.o.y + \ ray.o.z*ray.o.z - self.radius*self.radius # Solve quadratic equation for _t_ values found, t0, t1 = quadratic(A, B, C) if not found: return False, float('inf'), 0.0, None # Compute intersection distance along ray if (t0 > ray.maxt or t1 < ray.mint): return False, float('inf'), 0.0, None t_hit = t0 if (t0 < ray.mint): t_hit = t1 if (t_hit > ray.maxt): return False, float('inf'), 0.0, None # Compute sphere hit position and $\phi$ phi_t = ray(t_hit) if (phi_t.x == 0.0 and phi_t.y == 0.0): phi_t.x = 1e-5 * self.radius phi = math.atan2(phi_t.y, phi_t.x) if (phi < 0.0): phi += 2.0 * math.pi # Test sphere intersection against clipping parameters if ((self.z_min > -self.radius and phi_t.z < self.z_min) or \ (self.z_max < self.radius and phi_t.z > self.z_max) or \ phi > self.phi_max): if (t_hit == t1): return False, float('inf'), 0.0, None if (t1 > ray.maxt): return False, float('inf'), 0.0, None t_hit = t1 # Compute sphere hit position and $\phi$ phi_t = ray(t_hit) if (phi_t.x == 0.0 and phi_t.y == 0.0): phi_t.x = 1e-5 * self.radius phi = math.atan2(phi_t.y, phi_t.x) if (phi < 0.0): phi += 2.0 * math.pi if ((self.z_min > -self.radius and phi_t.z < self.z_min) or \ (self.z_max < self.radius and phi_t.z > self.z_max) or \ phi > self.phi_max): return False, float('inf'), 0.0, None # Find parametric representation of sphere hit u = phi / self.phi_max theta = math.acos(clamp(phi_t.z / self.radius, -1.0, 1.0)) v = (theta - self.theta_min) / (self.theta_max - self.theta_min) # Compute sphere $\dpdu$ and $\dpdv$ zradius = math.sqrt(phi_t.x * phi_t.x + phi_t.y * phi_t.y) inv_z_radius = 1.0 / zradius cos_phi = phi_t.x * inv_z_radius sin_phi = phi_t.y * inv_z_radius dpdu = Vector(-self.phi_max * phi_t.y, self.phi_max * phi_t.x, 0) dpdv = (self.theta_max-self.theta_min) * \ Vector(phi_t.z * cos_phi, phi_t.z * sin_phi, -self.radius * math.sin(theta)) # Compute sphere $\dndu$ and $\dndv$ d2Pduu = -self.phi_max * self.phi_max * Vector(phi_t.x, phi_t.y, 0) d2Pduv = (self.theta_max - self.theta_min) * phi_t.z * self.phi_max * \ Vector(-sin_phi, cos_phi, 0.0) d2Pdvv = -(self.theta_max - self.theta_min) * \ (self.theta_max - self.theta_min) * \ Vector(phi_t.x, phi_t.y, phi_t.z) # Compute coefficients for fundamental forms E = dot(dpdu, dpdu) F = dot(dpdu, dpdv) G = dot(dpdv, dpdv) N = normalize(cross(dpdu, dpdv)) e = dot(N, d2Pduu) f = dot(N, d2Pduv) g = dot(N, d2Pdvv) # Compute $\dndu$ and $\dndv$ from fundamental form coefficients invEGF2 = 1.0 / (E * G - F * F) dndu = Normal.from_vector((f*F - e*G) * invEGF2 * dpdu + \ (e*F - f*E) * invEGF2 * dpdv) dndv = Normal.from_vector((g*F - f*G) * invEGF2 * dpdu + \ (f*F - g*E) * invEGF2 * dpdv) # Initialize _DifferentialGeometry_ from parametric information o2w = self.object_to_world dg = DifferentialGeometry.from_intersection(o2w(phi_t), o2w(dpdu), o2w(dpdv), o2w(dndu), o2w(dndv), u, v, self) # Compute _rayEpsilon_ for quadric intersection ray_epsilon = 5e-4 * t_hit return True, t_hit, ray_epsilon, dg
def dot_quaternions(q1, q2): """Dot product of two quaternions.""" return dot(q1.v, q2.v) + q1.w * q2.w
def intersect(self, r): """Intersect the ray with the shape.""" # Transform _Ray_ to object space ray = self.world_to_object(r) # Compute quadratic sphere coefficients A = ray.d.x * ray.d.x + ray.d.y * ray.d.y + ray.d.z * ray.d.z B = 2 * (ray.d.x * ray.o.x + ray.d.y * ray.o.y + ray.d.z * ray.o.z) C = ray.o.x * ray.o.x + ray.o.y * ray.o.y + ray.o.z * ray.o.z - self.radius * self.radius # Solve quadratic equation for _t_ values found, t0, t1 = quadratic(A, B, C) if not found: return False, float("inf"), 0.0, None # Compute intersection distance along ray if t0 > ray.maxt or t1 < ray.mint: return False, float("inf"), 0.0, None t_hit = t0 if t0 < ray.mint: t_hit = t1 if t_hit > ray.maxt: return False, float("inf"), 0.0, None # Compute sphere hit position and $\phi$ phi_t = ray(t_hit) if phi_t.x == 0.0 and phi_t.y == 0.0: phi_t.x = 1e-5 * self.radius phi = math.atan2(phi_t.y, phi_t.x) if phi < 0.0: phi += 2.0 * math.pi # Test sphere intersection against clipping parameters if ( (self.z_min > -self.radius and phi_t.z < self.z_min) or (self.z_max < self.radius and phi_t.z > self.z_max) or phi > self.phi_max ): if t_hit == t1: return False, float("inf"), 0.0, None if t1 > ray.maxt: return False, float("inf"), 0.0, None t_hit = t1 # Compute sphere hit position and $\phi$ phi_t = ray(t_hit) if phi_t.x == 0.0 and phi_t.y == 0.0: phi_t.x = 1e-5 * self.radius phi = math.atan2(phi_t.y, phi_t.x) if phi < 0.0: phi += 2.0 * math.pi if ( (self.z_min > -self.radius and phi_t.z < self.z_min) or (self.z_max < self.radius and phi_t.z > self.z_max) or phi > self.phi_max ): return False, float("inf"), 0.0, None # Find parametric representation of sphere hit u = phi / self.phi_max theta = math.acos(clamp(phi_t.z / self.radius, -1.0, 1.0)) v = (theta - self.theta_min) / (self.theta_max - self.theta_min) # Compute sphere $\dpdu$ and $\dpdv$ zradius = math.sqrt(phi_t.x * phi_t.x + phi_t.y * phi_t.y) inv_z_radius = 1.0 / zradius cos_phi = phi_t.x * inv_z_radius sin_phi = phi_t.y * inv_z_radius dpdu = Vector(-self.phi_max * phi_t.y, self.phi_max * phi_t.x, 0) dpdv = (self.theta_max - self.theta_min) * Vector( phi_t.z * cos_phi, phi_t.z * sin_phi, -self.radius * math.sin(theta) ) # Compute sphere $\dndu$ and $\dndv$ d2Pduu = -self.phi_max * self.phi_max * Vector(phi_t.x, phi_t.y, 0) d2Pduv = (self.theta_max - self.theta_min) * phi_t.z * self.phi_max * Vector(-sin_phi, cos_phi, 0.0) d2Pdvv = ( -(self.theta_max - self.theta_min) * (self.theta_max - self.theta_min) * Vector(phi_t.x, phi_t.y, phi_t.z) ) # Compute coefficients for fundamental forms E = dot(dpdu, dpdu) F = dot(dpdu, dpdv) G = dot(dpdv, dpdv) N = normalize(cross(dpdu, dpdv)) e = dot(N, d2Pduu) f = dot(N, d2Pduv) g = dot(N, d2Pdvv) # Compute $\dndu$ and $\dndv$ from fundamental form coefficients invEGF2 = 1.0 / (E * G - F * F) dndu = Normal.from_vector((f * F - e * G) * invEGF2 * dpdu + (e * F - f * E) * invEGF2 * dpdv) dndv = Normal.from_vector((g * F - f * G) * invEGF2 * dpdu + (f * F - g * E) * invEGF2 * dpdv) # Initialize _DifferentialGeometry_ from parametric information o2w = self.object_to_world dg = DifferentialGeometry.from_intersection(o2w(phi_t), o2w(dpdu), o2w(dpdv), o2w(dndu), o2w(dndv), u, v, self) # Compute _rayEpsilon_ for quadric intersection ray_epsilon = 5e-4 * t_hit return True, t_hit, ray_epsilon, dg