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import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from math import pi, sin, cos, degrees, gcd #, lcm (python3.9+ only)
# Python 3.8 or lower
def lcm(a, b):
return a*b // gcd(a,b)
###############################################################################
# This part is generic for any size
###############################################################################
def is_corner(n, sizes, point):
for i in range(n):
if point[i] % sizes[i] != 0:
return False
return True
def is_bouncing(n, sizes, point):
for i in range(n):
if point[i] % sizes[i] == 0:
return True
return False
def lcm_list(L):
ret = 1
for e in L:
ret = lcm(ret, e)
return ret
# Generic n-dimensional path.
def get_path(n, sizes, r):
if n <= 1 or len(sizes) != n or len(r) != n:
print("Invalid sizes or starting point:", n, sizes, r)
return
path = [r]
p = r
d = [1] * n
steps = 0
while steps < 2*lcm_list(sizes):
p = [p[i] + d[i] for i in range(n)]
path.append(p)
steps = steps+1
d = [d[i] if p[i] % sizes[i] != 0 else -d[i] for i in range(n)]
return path
###############################################################################
# This part is specific for 2d drawings
###############################################################################
class Transformation:
def __init__(self, angle=0.0, mirror=False, shift=(0,0)):
self.angle = angle
self.mirror = mirror
self.shift = shift
def rotate(self, point):
c, s = cos(self.angle), sin(self.angle)
return [c*point[0] - s*point[1], s*point[0] + c*point[1]]
def reflect(self, point):
return [point[0], -point[1] if self.mirror else point[1]]
def translate(self, point):
return [point[i] + self.shift[i] for i in range(2)]
def apply_to(self, point):
return self.translate(self.rotate(self.reflect(point)))
def draw_line_2d(p1, p2, c="black", w=1):
plt.plot([p1[0], p2[0]], [p1[1], p2[1]], color=c, linewidth=w)
def draw_grid_and_rectangle(sizes, t):
for i in range(0, sizes[0]+1):
p1 = t.apply_to([i,0])
p2 = t.apply_to([i,sizes[1]])
color, width = ("red", 1) if i % sizes[0] == 0 else ("grey", 0.5)
draw_line_2d(p1, p2, c=color, w=width)
for i in range(0, sizes[1]+1):
p1 = t.apply_to([0,i])
p2 = t.apply_to([sizes[0],i])
color, width = ("red", 1) if i % sizes[1] == 0 else ("grey", 0.5)
draw_line_2d(p1, p2, c=color, w=width)
def draw_path_2d(sizes, r, transformation):
if len(sizes) != 2:
print("Cannot draw non-2d path")
return
plt.axis("off")
plt.axis("equal")
draw_grid_and_rectangle(sizes, transformation)
path = [transformation.apply_to(p) for p in get_path(2, sizes, r)]
plt.plot([p[0] for p in path], [p[1] for p in path], color="blue")
###############################################################################
# This part is specific for drawing the 2d projections of 3d billiards
###############################################################################
class Face:
def __init__(self, fixed_coordinate, value, transformation):
self.f = fixed_coordinate
self.v = value
self.t = transformation
def contains(self, point):
return point[self.f] == self.v
def proj(self, point):
return [point[i] for i in range(3) if i != self.f]
def draw_bouncing_points_3d2d(sizes, r, face):
path_3d = get_path(3, sizes, r)
bp = [face.t.apply_to(face.proj(p))
for p in path_3d if is_bouncing(3, sizes, p) and face.contains(p)]
plt.scatter([p[0] for p in bp], [p[1] for p in bp], color="green")
def draw_3d_projections(sizes, r):
a, b, c = sizes
bottom = Face(2, 0, Transformation())
top = Face(2, c, Transformation(shift=(a+c, -(b+c))))
front = Face(1, 0, Transformation(mirror=True))
back = Face(1, b, Transformation(shift=(0,b)))
left = Face(0, 0, Transformation(angle=pi/2))
right = Face(0, a, Transformation(angle=pi/2, mirror=True, shift=(a,0)))
for face in [bottom, top, front, back, left, right]:
draw_path_2d(face.proj(sizes), face.proj(r), face.t)
draw_bouncing_points_3d2d(sizes, r, face)
plt.show()
###############################################################################
# This part is for drawing 3d pictures
###############################################################################
def draw_line_3d(ax, p1, p2, c="black", w=1):
ax.plot([p1[0], p2[0]], [p1[1], p2[1]], [p1[2], p2[2]], color=c, lw=w)
def draw_box_3d(ax, s):
color, width = ("red", 1)
draw_line_3d(ax, ( 0, 0, 0), (s[0], 0, 0), color, width)
draw_line_3d(ax, ( 0, s[1], 0), (s[0], s[1], 0), color, width)
draw_line_3d(ax, ( 0, 0, s[2]), (s[0], 0, s[2]), color, width)
draw_line_3d(ax, ( 0, s[1], s[2]), (s[0], s[1], s[2]), color, width)
draw_line_3d(ax, ( 0, 0, 0), ( 0, s[1], 0), color, width)
draw_line_3d(ax, (s[0], 0, 0), (s[0], s[1], 0), color, width)
draw_line_3d(ax, ( 0, 0, s[2]), ( 0, s[1], s[2]), color, width)
draw_line_3d(ax, (s[0], 0, s[2]), (s[0], s[1], s[2]), color, width)
draw_line_3d(ax, ( 0, 0, 0), ( 0, 0, s[2]), color, width)
draw_line_3d(ax, (s[0], 0, 0), (s[0], 0, s[2]), color, width)
draw_line_3d(ax, ( 0, s[1], 0), ( 0, s[1], s[2]), color, width)
draw_line_3d(ax, (s[0], s[1], 0), (s[0], s[1], s[2]), color, width)
def draw_3d_picture(sizes, r):
ax = plt.figure().add_subplot(111, projection="3d")
plt.axis("off")
ax.set_xlim(0, max(sizes))
ax.set_ylim(0, max(sizes))
ax.set_zlim(0, max(sizes))
draw_box_3d(ax, sizes)
path = get_path(3, sizes, r)
plt.plot([p[0] for p in path], [p[1] for p in path], [p[2] for p in path],
color="blue")
plt.show()
###############################################################################
# Billiard data / user input
###############################################################################
def user_input():
a = int(input("Value for a: "))
b = int(input("Value for b: "))
c = int(input("Value for c: "))
ra = int(input("a-coordinate of r: "))
rb = int(input("b-coordinate of r: "))
rc = int(input("c-coordinate of r: "))
pic = input("Choose p for projections or 3 for 3d (empty = both): ")
return [a,b,c], [ra,rb,rc], pic
# You can choose to write your input here or to get it interactively
#sizes, r, pic = [15, 9, 7], [2, 0, 0], "p"
#sizes, r, pic = [2, 3, 4], [0, 0, 0], "3"
sizes, r, pic = user_input()
if pic == "p":
draw_3d_projections(sizes, r)
elif pic == "3":
draw_3d_picture(sizes, r)
else:
draw_3d_projections(sizes, r)
draw_3d_picture(sizes, r)
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