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 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
