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billiards.py
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billiards.py
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import taichi as ti
import sys
import math
import numpy as np
import os
import matplotlib.pyplot as plt
real = ti.f32
ti.init(default_fp=real, flatten_if=True)
max_steps = 2048
vis_interval = 64
output_vis_interval = 16
steps = 1024
assert steps * 2 <= max_steps
vis_resolution = 1024
scalar = lambda: ti.field(dtype=real)
vec = lambda: ti.Vector.field(2, dtype=real)
loss = scalar()
init_x = vec()
init_v = vec()
x = vec()
x_inc = vec() # for TOI
v = vec()
impulse = vec()
billiard_layers = 4
n_balls = 1 + (1 + billiard_layers) * billiard_layers // 2
target_ball = n_balls - 1
# target_ball = 0
goal = [0.9, 0.75]
radius = 0.03
elasticity = 0.8
ti.root.dense(ti.i, max_steps).dense(ti.j, n_balls).place(x, v, x_inc, impulse)
ti.root.place(init_x, init_v)
ti.root.place(loss)
ti.root.lazy_grad()
dt = 0.003
alpha = 0.00000
learning_rate = 0.01
@ti.func
def collide_pair(t, i, j):
imp = ti.Vector([0.0, 0.0])
x_inc_contrib = ti.Vector([0.0, 0.0])
if i != j:
dist = (x[t, i] + dt * v[t, i]) - (x[t, j] + dt * v[t, j])
dist_norm = dist.norm()
rela_v = v[t, i] - v[t, j]
if dist_norm < 2 * radius:
dir = ti.Vector.normalized(dist, 1e-6)
projected_v = dir.dot(rela_v)
if projected_v < 0:
imp = -(1 + elasticity) * 0.5 * projected_v * dir
toi = (dist_norm - 2 * radius) / min(
-1e-3, projected_v) # Time of impact
x_inc_contrib = min(toi - dt, 0) * imp
x_inc[t + 1, i] += x_inc_contrib
impulse[t + 1, i] += imp
@ti.kernel
def collide(t: ti.i32):
for i in range(n_balls):
for j in range(i):
collide_pair(t, i, j)
for i in range(n_balls):
for j in range(i + 1, n_balls):
collide_pair(t, i, j)
@ti.kernel
def advance(t: ti.i32):
for i in range(n_balls):
v[t, i] = v[t - 1, i] + impulse[t, i]
x[t, i] = x[t - 1, i] + dt * v[t, i] + x_inc[t, i]
@ti.kernel
def compute_loss(t: ti.i32):
loss[None] = (x[t, target_ball][0] - goal[0])**2 + (x[t, target_ball][1] -
goal[1])**2
@ti.kernel
def initialize():
x[0, 0] = init_x[None]
v[0, 0] = init_v[None]
gui = ti.GUI("Billiards", (1024, 1024), background_color=0x3C733F)
def forward(visualize=False, output=None):
initialize()
interval = vis_interval
if output:
interval = output_vis_interval
os.makedirs('billiards/{}/'.format(output), exist_ok=True)
count = 0
for i in range(billiard_layers):
for j in range(i + 1):
count += 1
x[0, count] = [
i * 2 * radius + 0.5, j * 2 * radius + 0.5 - i * radius * 0.7
]
pixel_radius = int(radius * 1024) + 1
for t in range(1, steps):
collide(t - 1)
advance(t)
if (t + 1) % interval == 0 and visualize:
gui.clear()
gui.circle((goal[0], goal[1]), 0x00000, pixel_radius // 2)
for i in range(n_balls):
if i == 0:
color = 0xCCCCCC
elif i == n_balls - 1:
color = 0x3344cc
else:
color = 0xF20530
gui.circle((x[t, i][0], x[t, i][1]), color, pixel_radius)
if output:
gui.show('billiards/{}/{:04d}.png'.format(output, t))
else:
gui.show()
compute_loss(steps - 1)
@ti.kernel
def clear():
for t, i in ti.ndrange(max_steps, n_balls):
impulse[t, i] = ti.Vector([0.0, 0.0])
x_inc[t, i] = ti.Vector([0.0, 0.0])
def optimize():
init_x[None] = [0.1, 0.5]
init_v[None] = [0.3, 0.0]
clear()
# forward(visualize=True, output='initial')
for iter in range(200):
clear()
with ti.ad.Tape(loss):
if iter % 20 == 19:
output = 'iter{:04d}'.format(iter)
else:
output = None
forward(visualize=True, output=output)
print('Iter=', iter, 'Loss=', loss[None])
for d in range(2):
init_x[None][d] -= learning_rate * init_x.grad[None][d]
init_v[None][d] -= learning_rate * init_v.grad[None][d]
clear()
forward(visualize=True, output='final')
def scan(zoom):
N = 1000
angles = []
losses = []
forward(visualize=True, output='initial')
for i in range(N):
alpha = ((i + 0.5) / N - 0.5) * math.pi * zoom
init_x[None] = [0.1, 0.5]
init_v[None] = [0.3 * math.cos(alpha), 0.3 * math.sin(alpha)]
loss[None] = 0
clear()
forward(visualize=False)
print(loss[None])
losses.append(loss[None])
angles.append(math.degrees(alpha))
plt.plot(angles, losses)
fig = plt.gcf()
fig.set_size_inches(5, 3)
plt.title('Billiard Scene Objective')
plt.ylabel('Objective')
plt.xlabel('Angle of velocity')
plt.tight_layout()
plt.show()
if __name__ == '__main__':
if len(sys.argv) > 1:
scan(float(sys.argv[1]))
else:
optimize()