This is one of the 100 recipes of the IPython Cookbook, the definitive guide to high-performance scientific computing and data science in Python.
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.
import numpy as np import matplotlib.pyplot as plt
#%load_ext cythonmagic %load_ext Cython
%%cython import numpy as np cimport numpy as np w, h = 200, 200 # Size of the screen in pixels. def normalize(x): # This function normalizes a vector. x /= np.linalg.norm(x) return x def intersect_sphere(O, D, S, 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. a = np.dot(D, D) OS = O - S b = 2 * np.dot(D, OS) c = np.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(O, D): # 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 # Lambert shading (diffuse). col += diffuse * max(np.dot(N, toL), 0) * color # Blinn-Phong shading (specular). col += specular_c * color_light * \ max(np.dot(N, normalize(toL + toO)), 0) \ ** specular_k return col def run(): img = np.zeros((h, w, 3)) # 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, Q = x, y # Direction of the ray going through the optical center. D = normalize(Q - O) depth = 0 # Launch the ray and get the color of the pixel. col = trace_ray(O, D) if col is None: continue img[h - j - 1, i, :] = np.clip(col, 0, 1) return img # Sphere properties. position = np.array([0., 0., 1.]) radius = 1. color = np.array([0., 0., 1.]) diffuse = 1. specular_c = 1. specular_k = 50 # Light position and color. L = np.array([5., 5., -10.]) color_light = np.ones(3) ambient = .05 # Camera. O = np.array([0., 0., -1.]) # Position. Q = np.array([0., 0., 0.]) # Pointing to.
img = run() plt.imshow(img); plt.xticks(); plt.yticks();
%timeit -n1 -r1 run()