def local_intersect(self, ray):
if len(self.children) == 0:
return Intersections()
xs = reduce(lambda a, e: a + e.intersect(ray).sorted_intersections,
self.children, [])
return Intersections(*xs)
def local_intersect(self, ray):
def check_axis(origin, direction):
tmin_numerator = (-1 - origin)
tmax_numerator = (1 - origin)
if abs(direction) >= EPSILON:
tmin = tmin_numerator / direction
tmax = tmax_numerator / direction
else:
tmin = tmin_numerator * float('inf')
tmax = tmax_numerator * float('inf')
if tmin > tmax:
tmin, tmax = tmax, tmin
return (tmin, tmax)
xtmin, xtmax = check_axis(ray.origin.x, ray.direction.x)
ytmin, ytmax = check_axis(ray.origin.y, ray.direction.y)
ztmin, ztmax = check_axis(ray.origin.z, ray.direction.z)
tmin = max(xtmin, ytmin, ztmin)
tmax = min(xtmax, ytmax, ztmax)
if tmin > tmax:
return Intersections()
else:
return Intersections(Intersection(tmin, self),
Intersection(tmax, self))
def local_intersect(self, ray):
a = ray.direction.x**2 - ray.direction.y**2 + ray.direction.z**2
b = 2 * ray.origin.x * ray.direction.x \
- 2 * ray.origin.y * ray.direction.y \
+ 2 * ray.origin.z * ray.direction.z
c = ray.origin.x**2 - ray.origin.y**2 + ray.origin.z**2
if abs(a) < EPSILON:
if abs(b) < EPSILON:
return Intersections()
else:
xs = [Intersection(-c / (2 * b), self)] + \
self._intersect_caps(ray)
return Intersections(*xs)
disc = b**2 - 4 * a * c
if disc < 0:
return Intersections()
t0 = (-b - sqrt(disc)) / (2 * a)
t1 = (-b + sqrt(disc)) / (2 * a)
xs = []
y0 = ray.origin.y + t0 * ray.direction.y
if self.minimum < y0 and y0 < self.maximum:
xs.append(Intersection(t0, self))
y1 = ray.origin.y + t1 * ray.direction.y
if self.minimum < y1 and y1 < self.maximum:
xs.append(Intersection(t1, self))
xs = xs + self._intersect_caps(ray)
return Intersections(*xs)
def local_intersect(self, ray):
# the vector from the sphere's center, to the ray origin
# remember: the sphere is centered at the world origin
sphere_to_ray = ray.origin - Point(0, 0, 0)
a = ray.direction.dot(ray.direction)
b = 2 * ray.direction.dot(sphere_to_ray)
c = sphere_to_ray.dot(sphere_to_ray) - 1
discriminant = b**2 - 4 * a * c
if discriminant < 0:
return Intersections()
t1 = (-b - sqrt(discriminant)) / (2 * a)
t2 = (-b + sqrt(discriminant)) / (2 * a)
return Intersections(Intersection(t1, self), Intersection(t2, self))
def test_the_hit__when_all_intersections_have_positive_t():
# Given
s = Sphere()
i1 = Intersection(1, s)
i2 = Intersection(2, s)
xs = Intersections(i2, i1)
# When
i = xs.hit()
# Then
assert i == i1
def test_the_hit__when_some_intersections_have_negative_t():
# Given
s = Sphere()
i1 = Intersection(-1, s)
i2 = Intersection(1, s)
xs = Intersections(i2, i1)
# When
i = xs.hit()
# Then
assert i == i2
def test_the_hit__when_all_intersections_have_netative_t():
# Given
s = Sphere()
i1 = Intersection(-2, s)
i2 = Intersection(-1, s)
xs = Intersections(i2, i1)
# When
i = xs.hit()
# Then
assert i is None
def test_the_schlick_approximation_with_small_angle_and_n2_greater_than_n1():
# Given
shape = glass_sphere()
r = Ray(Point(0, 0.99, -2), Vector(0, 0, 1))
xs = Intersections(Intersection(1.8589, shape))
# When
comps = xs[0].prepare_computations(r, xs)
reflectance = comps.schlick()
# Then
assert equal(reflectance, 0.48873)
def test_the_schlick_approximation_with_a_perpendicular_viewing_angle():
# Given
shape = glass_sphere()
r = Ray(Point(0, 0, 0), Vector(0, 1, 0))
xs = Intersections(Intersection(-1, shape), Intersection(1, shape))
# When
comps = xs[1].prepare_computations(r, xs)
reflectance = comps.schlick()
# Then
assert equal(reflectance, 0.04)
def test_the_refracted_color_with_an_opaque_surface():
# Given
w = default_world()
shape = w.objects[0]
r = Ray(Point(0, 0, -5), Vector(0, 0, 1))
xs = Intersections(Intersection(4, shape), Intersection(6, shape))
# When
comps = xs[0].prepare_computations(r, xs)
c = w.refracted_color(comps, 5)
# Then
assert c == Color(0, 0, 0)
def test_the_schlick_approximation_under_total_internal_reflection():
# Given
shape = glass_sphere()
r = Ray(Point(0, 0, sqrt(2) / 2), Vector(0, 1, 0))
xs = Intersections(Intersection(-sqrt(2) / 2, shape),
Intersection(sqrt(2) / 2, shape))
# When
comps = xs[1].prepare_computations(r, xs)
reflectance = comps.schlick()
# Then
assert reflectance == 1.0
def test_aggregating_intersections():
# Given
s = Sphere()
i1 = Intersection(1, s)
i2 = Intersection(2, s)
# When
xs = Intersections(i1, i2)
# Then
assert len(xs) == 2
assert xs[0].t == 1
assert xs[1].t == 2
def test_the_hit_is_always_the_lowest_nonnegative_intersection():
# Given
s = Sphere()
i1 = Intersection(5, s)
i2 = Intersection(7, s)
i3 = Intersection(-3, s)
i4 = Intersection(2, s)
xs = Intersections(i1, i2, i3, i4)
# When
i = xs.hit()
# Then
assert i == i4
def test_the_under_point_is_offset_below_the_surface():
# Given
r = Ray(Point(0, 0, -5), Vector(0, 0, 1))
shape = glass_sphere()
shape.transform = translation(0, 0, 1)
i = Intersection(5, shape)
xs = Intersections(i)
# When
comps = i.prepare_computations(r, xs)
# Then
assert comps.under_point.z > EPSILON / 2
assert comps.point.z < comps.under_point.z
def test_the_refracted_color_at_the_maximum_recursive_depth():
# Given
w = default_world()
shape = w.objects[0]
shape.material.transparency = 1.0
shape.material.refractive_index = 1.5
r = Ray(Point(0, 0, -5), Vector(0, 0, 1))
xs = Intersections(Intersection(4, shape), Intersection(6, shape))
# When
comps = xs[0].prepare_computations(r, xs)
c = w.refracted_color(comps, 0)
# Then
assert c == Color(0, 0, 0)
def test_finding_n1_and_n2_at_various_intersections():
# Given
A = glass_sphere()
A.transform = scaling(2, 2, 2)
A.material.refractive_index = 1.5
B = glass_sphere()
B.transform = translation(0, 0, -0.25)
B.material.refractive_index = 2.0
C = glass_sphere()
C.transform = translation(0, 0, 0.25)
C.material.refractive_index = 2.5
r = Ray(Point(0, 0, -4), Vector(0, 0, 1))
xs = Intersections(Intersection(2, A), Intersection(2.75, B),
Intersection(3.25, C), Intersection(4.75, B),
Intersection(5.25, C), Intersection(6, A))
EXAMPLES = [
{
'n1': 1.0,
'n2': 1.5
}, # index: 0
{
'n1': 1.5,
'n2': 2.0
}, # index: 1
{
'n1': 2.0,
'n2': 2.5
}, # index: 2
{
'n1': 2.5,
'n2': 2.5
}, # index: 3
{
'n1': 2.5,
'n2': 1.5
}, # index: 4
{
'n1': 1.5,
'n2': 1.0
}, # index: 5
]
for index in range(len(EXAMPLES)):
# When
comps = xs[index].prepare_computations(r, xs=xs)
# Then
assert comps.n1 == EXAMPLES[index]['n1']
assert comps.n2 == EXAMPLES[index]['n2']
def local_intersect(self, ray):
a = ray.direction.x ** 2 + ray.direction.z ** 2
# ray is parallel to the y axis
if abs(a) < EPSILON:
xs = self._intersect_caps(ray)
return Intersections(*xs)
b = 2 * ray.origin.x * ray.direction.x + \
2 * ray.origin.z * ray.direction.z
c = ray.origin.x ** 2 + ray.origin.z ** 2 - 1
disc = b ** 2 - 4 * a * c
# ray does not intersect the cylinder
if disc < 0:
return Intersections()
t0 = (-b - sqrt(disc)) / (2 * a)
t1 = (-b + sqrt(disc)) / (2 * a)
# if t0 > t1:
# t0, t1 = t1, t0
xs = []
y0 = ray.origin.y + t0 * ray.direction.y
if self.minimum < y0 and y0 < self.maximum:
xs.append(Intersection(t0, self))
y1 = ray.origin.y + t1 * ray.direction.y
if self.minimum < y1 and y1 < self.maximum:
xs.append(Intersection(t1, self))
xs = xs + self._intersect_caps(ray)
return Intersections(*xs)
def test_the_refracted_color_under_total_internal_reflection():
# Given
w = default_world()
shape = w.objects[0]
shape.material.transparency = 1.0
shape.material.refractive_index = 1.5
r = Ray(Point(0, 0, sqrt(2) / 2), Vector(0, 1, 0))
xs = Intersections(Intersection(-sqrt(2) / 2, shape),
Intersection(sqrt(2) / 2, shape))
# When
# NOTE: this time you're inside the sphere, so you need
# to look at the second intersection, xs[1], not xs[0]
comps = xs[1].prepare_computations(r, xs)
c = w.refracted_color(comps, 5)
# Then
assert c == Color(0, 0, 0)
def test_refracted_color_with_a_refracted_ray():
# Then
w = default_world()
A = w.objects[0]
A.material.ambient = 1.0
A.material.pattern = MockPattern()
B = w.objects[1]
B.material.transparency = 1.0
B.material.refractive_index = 1.5
r = Ray(Point(0, 0, 0.1), Vector(0, 1, 0))
xs = Intersections(Intersection(-0.9899, A), Intersection(-0.4899, B),
Intersection(0.4899, B), Intersection(0.9899, A))
# When
comps = xs[2].prepare_computations(r, xs)
c = w.refracted_color(comps, 5)
# Then
assert c == Color(0, 0.99887, 0.04721)
def test__shade_hit__with_a_transparent_material():
# Given
w = default_world()
floor = Plane()
floor.transform = translation(0, -1, 0)
floor.material.transparency = 0.5
floor.material.refractive_index = 1.5
w.objects.append(floor)
ball = Sphere()
ball.material.color = Color(1, 0, 0)
ball.material.ambient = 0.5
ball.transform = translation(0, -3.5, -0.5)
w.objects.append(ball)
r = Ray(Point(0, 0, -3), Vector(0, -sqrt(2) / 2, sqrt(2) / 2))
xs = Intersections(Intersection(sqrt(2), floor))
# When
comps = xs[0].prepare_computations(r, xs)
color = w.shade_hit(comps, 5)
# Then
assert color == Color(0.93642, 0.68642, 0.68642)
def intersect_world(self, ray):
xs = [x for object in self.objects for x in object.intersect(ray)]
return Intersections(*xs)