This is one of the 100 recipes of the IPython Cookbook, the definitive guide to high-performance scientific computing and data science in Python.

# 5.5. Ray tracing: Cython array buffers¶

In this example, we will render a sphere with a diffuse and specular material. The principle is to model a scene with a light source and a camera, and use the physical properties of light propagation to calculate the light intensity and color of every pixel of the screen.

In [ ]:
import numpy as np
import matplotlib.pyplot as plt

In [ ]:
%matplotlib inline

In [ ]:
#%load_ext cythonmagic


## Take 1¶

In [ ]:
%%cython
import numpy as np
cimport numpy as np
from numpy import dot
from libc.math cimport sqrt

DBL = np.double
ctypedef np.double_t DBL_C
INT = np.int
ctypedef np.int_t INT_C
cdef INT_C w, h

w, h = 200, 200  # Size of the screen in pixels.

def normalize(np.ndarray[DBL_C, ndim=1] x):
# This function normalizes a vector.
x /= np.linalg.norm(x)
return x

def intersect_sphere(np.ndarray[DBL_C, ndim=1] O, np.ndarray[DBL_C, ndim=1] D,
np.ndarray[DBL_C, ndim=1] S, DBL_C R):
# Return the distance from O to the intersection
# of the ray (O, D) with the sphere (S, R), or
# +inf if there is no intersection.
# O and S are 3D points, D (direction) is a
# normalized vector, R is a scalar.

cdef DBL_C a, b, c, disc, distSqrt, q, t0, t1
cdef np.ndarray[DBL_C, ndim=1] OS

a = dot(D, D)
OS = O - S
b = 2 * dot(D, OS)
c = dot(OS, OS) - R*R
disc = b*b - 4*a*c
if disc > 0:
distSqrt = np.sqrt(disc)
q = (-b - distSqrt) / 2.0 if b < 0 \
else (-b + distSqrt) / 2.0
t0 = q / a
t1 = c / q
t0, t1 = min(t0, t1), max(t0, t1)
if t1 >= 0:
return t1 if t0 < 0 else t0
return np.inf

def trace_ray(np.ndarray[DBL_C, ndim=1] O, np.ndarray[DBL_C, ndim=1] D,
np.ndarray[DBL_C, ndim=1] position,
np.ndarray[DBL_C, ndim=1] color,
np.ndarray[DBL_C, ndim=1] L,
np.ndarray[DBL_C, ndim=1] color_light):

cdef DBL_C t
cdef np.ndarray[DBL_C, ndim=1] M, N, toL, toO, col

# Find first point of intersection with the scene.
t = intersect_sphere(O, D, position, radius)
# No intersection?
if t == np.inf:
return
# Find the point of intersection on the object.
M = O + D * t
N = normalize(M - position)
toL = normalize(L - M)
toO = normalize(O - M)
# Ambient light.
col = ambient * np.ones(3)
col += diffuse * max(dot(N, toL), 0) * color
col += specular_c * color_light * \
max(dot(N, normalize(toL + toO)), 0) \
** specular_k
return col

def run():
cdef np.ndarray[DBL_C, ndim=3] img
img = np.zeros((h, w, 3))
cdef INT_C i, j
cdef DBL_C x, y
cdef np.ndarray[DBL_C, ndim=1] O, Q, D, col, position, color, L, color_light

# Sphere properties.
position = np.array([0., 0., 1.])
color = np.array([0., 0., 1.])
L = np.array([5., 5., -10.])
color_light = np.ones(3)

# Camera.
O = np.array([0., 0., -1.])  # Position.
Q = np.array([0., 0., 0.])  # Pointing to.

# Loop through all pixels.
for i, x in enumerate(np.linspace(-1., 1., w)):
for j, y in enumerate(np.linspace(-1., 1., h)):
# Position of the pixel.
Q[0], Q[1] = x, y
# Direction of the ray going through the optical center.
D = normalize(Q - O)
# Launch the ray and get the color of the pixel.
col = trace_ray(O, D, position, color, L, color_light)
if col is None:
continue
img[h - j - 1, i, :] = np.clip(col, 0, 1)
return img

cdef DBL_C radius, ambient, diffuse, specular_k, specular_c

# Sphere and light properties.
diffuse = 1.
specular_c = 1.
specular_k = 50.
ambient = .05

In [ ]:
img = run()
plt.imshow(img);
plt.xticks([]); plt.yticks([]);

In [ ]:
%timeit -n1 -r1 run()


## Take 2¶

In this version, we rewrite normalize in pure C.

In [ ]:
%%cython
import numpy as np
cimport numpy as np
from numpy import dot
from libc.math cimport sqrt

DBL = np.double
ctypedef np.double_t DBL_C
INT = np.int
ctypedef np.int_t INT_C
cdef INT_C w, h

w, h = 200, 200  # Size of the screen in pixels.

# normalize is now a pure C function that does not make
# use NumPy for the computations
cdef normalize(np.ndarray[DBL_C, ndim=1] x):
cdef DBL_C n
n = sqrt(x[0] * x[0] + x[1] * x[1] + x[2] * x[2])
x[0] /= n
x[1] /= n
x[2] /= n
return x

def intersect_sphere(np.ndarray[DBL_C, ndim=1] O, np.ndarray[DBL_C, ndim=1] D,
np.ndarray[DBL_C, ndim=1] S, DBL_C R):
# Return the distance from O to the intersection
# of the ray (O, D) with the sphere (S, R), or
# +inf if there is no intersection.
# O and S are 3D points, D (direction) is a
# normalized vector, R is a scalar.

cdef DBL_C a, b, c, disc, distSqrt, q, t0, t1
cdef np.ndarray[DBL_C, ndim=1] OS

a = dot(D, D)
OS = O - S
b = 2 * dot(D, OS)
c = dot(OS, OS) - R*R
disc = b*b - 4*a*c
if disc > 0:
distSqrt = np.sqrt(disc)
q = (-b - distSqrt) / 2.0 if b < 0 \
else (-b + distSqrt) / 2.0
t0 = q / a
t1 = c / q
t0, t1 = min(t0, t1), max(t0, t1)
if t1 >= 0:
return t1 if t0 < 0 else t0
return np.inf

def trace_ray(np.ndarray[DBL_C, ndim=1] O, np.ndarray[DBL_C, ndim=1] D,
np.ndarray[DBL_C, ndim=1] position,
np.ndarray[DBL_C, ndim=1] color,
np.ndarray[DBL_C, ndim=1] L,
np.ndarray[DBL_C, ndim=1] color_light):

cdef DBL_C t
cdef np.ndarray[DBL_C, ndim=1] M, N, toL, toO, col

# Find first point of intersection with the scene.
t = intersect_sphere(O, D, position, radius)
# No intersection?
if t == np.inf:
return
# Find the point of intersection on the object.
M = O + D * t
N = normalize(M - position)
toL = normalize(L - M)
toO = normalize(O - M)
# Ambient light.
col = ambient * np.ones(3)
col += diffuse * max(dot(N, toL), 0) * color
col += specular_c * color_light * \
max(dot(N, normalize(toL + toO)), 0) \
** specular_k
return col

def run():
cdef np.ndarray[DBL_C, ndim=3] img
img = np.zeros((h, w, 3))
cdef INT_C i, j
cdef DBL_C x, y
cdef np.ndarray[DBL_C, ndim=1] O, Q, D, col, position, color, L, color_light

# Sphere properties.
position = np.array([0., 0., 1.])
color = np.array([0., 0., 1.])
L = np.array([5., 5., -10.])
color_light = np.ones(3)

# Camera.
O = np.array([0., 0., -1.])  # Position.
Q = np.array([0., 0., 0.])  # Pointing to.

# Loop through all pixels.
for i, x in enumerate(np.linspace(-1., 1., w)):
for j, y in enumerate(np.linspace(-1., 1., h)):
# Position of the pixel.
Q[0], Q[1] = x, y
# Direction of the ray going through the optical center.
D = normalize(Q - O)
# Launch the ray and get the color of the pixel.
col = trace_ray(O, D, position, color, L, color_light)
if col is None:
continue
img[h - j - 1, i, :] = np.clip(col, 0, 1)
return img

cdef DBL_C radius, ambient, diffuse, specular_k, specular_c

# Sphere and light properties.
diffuse = 1.
specular_c = 1.
specular_k = 50.
ambient = .05

In [ ]:
img = run()
plt.imshow(img);
plt.xticks([]); plt.yticks([]);

In [ ]:
%timeit -n1 -r1 run()


You'll find all the explanations, figures, references, and much more in the book (to be released later this summer).

IPython Cookbook, by Cyrille Rossant, Packt Publishing, 2014 (500 pages).