This gist is part of my article on Medium: https://medium.com/@omaraflak/ray-tracing-from-scratch-in-python-41670e6a96f9?source=friends_link&sk=4edf81600f5c0941aa58907bbfb2151d

medium_ray_tracing_code_13.py

import numpy as np
import matplotlib.pyplot as plt

def normalize(vector):
    return vector / np.linalg.norm(vector)

def reflected(vector, axis):
    return vector - 2 * np.dot(vector, axis) * axis

def sphere_intersect(center, radius, ray_origin, ray_direction):
    b = 2 * np.dot(ray_direction, ray_origin - center)
    c = np.linalg.norm(ray_origin - center) ** 2 - radius ** 2
    delta = b ** 2 - 4 * c
    if delta > 0:
        t1 = (-b + np.sqrt(delta)) / 2
        t2 = (-b - np.sqrt(delta)) / 2
        if t1 > 0 and t2 > 0:
            return min(t1, t2)
    return None

def nearest_intersected_object(objects, ray_origin, ray_direction):
    distances = [sphere_intersect(obj['center'], obj['radius'], ray_origin, ray_direction) for obj in objects]
    nearest_object = None
    min_distance = np.inf
    for index, distance in enumerate(distances):
        if distance and distance < min_distance:
            min_distance = distance
            nearest_object = objects[index]
    return nearest_object, min_distance

width = 300
height = 200

max_depth = 3

camera = np.array([0, 0, 1])
ratio = float(width) / height
screen = (-1, 1 / ratio, 1, -1 / ratio) # left, top, right, bottom

light = { 'position': np.array([5, 5, 5]), 'ambient': np.array([1, 1, 1]), 'diffuse': np.array([1, 1, 1]), 'specular': np.array([1, 1, 1]) }

objects = [
    { 'center': np.array([-0.2, 0, -1]), 'radius': 0.7, 'ambient': np.array([0.1, 0, 0]), 'diffuse': np.array([0.7, 0, 0]), 'specular': np.array([1, 1, 1]), 'shininess': 100, 'reflection': 0.5 },
    { 'center': np.array([0.1, -0.3, 0]), 'radius': 0.1, 'ambient': np.array([0.1, 0, 0.1]), 'diffuse': np.array([0.7, 0, 0.7]), 'specular': np.array([1, 1, 1]), 'shininess': 100, 'reflection': 0.5 },
    { 'center': np.array([-0.3, 0, 0]), 'radius': 0.15, 'ambient': np.array([0, 0.1, 0]), 'diffuse': np.array([0, 0.6, 0]), 'specular': np.array([1, 1, 1]), 'shininess': 100, 'reflection': 0.5 },
    { 'center': np.array([0, -9000, 0]), 'radius': 9000 - 0.7, 'ambient': np.array([0.1, 0.1, 0.1]), 'diffuse': np.array([0.6, 0.6, 0.6]), 'specular': np.array([1, 1, 1]), 'shininess': 100, 'reflection': 0.5 }
]

image = np.zeros((height, width, 3))
for i, y in enumerate(np.linspace(screen[1], screen[3], height)):
    for j, x in enumerate(np.linspace(screen[0], screen[2], width)):
        # screen is on origin
        pixel = np.array([x, y, 0])
        origin = camera
        direction = normalize(pixel - origin)

        color = np.zeros((3))
        reflection = 1

        for k in range(max_depth):
            # check for intersections
            nearest_object, min_distance = nearest_intersected_object(objects, origin, direction)
            if nearest_object is None:
                break

            intersection = origin + min_distance * direction
            normal_to_surface = normalize(intersection - nearest_object['center'])
            shifted_point = intersection + 1e-5 * normal_to_surface
            intersection_to_light = normalize(light['position'] - shifted_point)

            _, min_distance = nearest_intersected_object(objects, shifted_point, intersection_to_light)
            intersection_to_light_distance = np.linalg.norm(light['position'] - intersection)
            is_shadowed = min_distance < intersection_to_light_distance

            if is_shadowed:
                break

            illumination = np.zeros((3))

            # ambiant
            illumination += nearest_object['ambient'] * light['ambient']

            # diffuse
            illumination += nearest_object['diffuse'] * light['diffuse'] * np.dot(intersection_to_light, normal_to_surface)

            # specular
            intersection_to_camera = normalize(camera - intersection)
            H = normalize(intersection_to_light + intersection_to_camera)
            illumination += nearest_object['specular'] * light['specular'] * np.dot(normal_to_surface, H) ** (nearest_object['shininess'] / 4)

            # reflection
            color += reflection * illumination
            reflection *= nearest_object['reflection']

            origin = shifted_point
            direction = reflected(direction, normal_to_surface)

        image[i, j] = np.clip(color, 0, 1)
    print("%d/%d" % (i + 1, height))

plt.imsave('image.png', image)

One of my big circles is not rendering
can i talk to you on discord?

https://discord.gg/ms4BcMRUGf

Could you give me a detail about variable H in line 88?
I am so wondering how did you solved reflection equation.
Actually I modified the line with referencing this link
but my work seems worse than yours.

Could you give me a detail about variable H in line 88? I am so wondering how did you solved reflection equation. Actually I modified the line with referencing this link but my work seems worse than yours.

Have you read the article I mentioned above?
https://medium.com/@omaraflak/ray-tracing-from-scratch-in-python-41670e6a96f9?source=friends_link&sk=4edf81600f5c0941aa58907bbfb2151d

Look at equation 7 (they're all labeled). The specular component of the color is:

Ks * Is * [N • ((L+V) / ||L+V||)] ^ (alpha/4)

Here,
H = (L+V) / ||L+V||

Where,
L is a direction unit vector from the intersection point towards the light;
V is a direction unit vector from the intersection point towards the camera;

You'll have more context in the article.

Hello,
Where can I change the color of the spheres by RGB values?

Hello, Where can I change the color of the spheres by RGB values?

Play around with ambiant, diffuse, specular values