def find_tangent(LC_s, LC_t):
"""
Given a source and target lighthouse center with radius 1, find the tangent.
This is done by first finding the angle between LC_t, LC_s, tang and then calculating the tang itself via rotation.
"""
return rotate(LC_s, LC_t, np.arcsin(1 / dist_2d(LC_s, LC_t)))
def compute_darkness(N, print_res=True):
if N == 1:
DA = 0
elif N == 2:
DA = inf
else:
placement_center = (0.0, 0.0)
centers = find_lighthouse_centers(N, placement_center)
lighthouses = []
for c in centers:
left, mid, right = find_lighthouse_illum_points(
N, c, placement_center)
lighthouses.append({
"center": c,
"left": left,
"middle": mid,
"right": right
})
source, tang, _ = get_first_illumination_line(lighthouses)
# We can find the dark area
coeff = np.polyfit(
[tang[0], source[0]], [tang[1], source[1]],
1) # finds the coefficients of y = ax + b for points x, y.
target_cross_x = -coeff[1] / coeff[
0] # we look for 0 = ax' + b --> x = -b/a
x = dist_2d(
(target_cross_x, 0.0),
tang) # this is the nugget, we can find the dark area from this.
DA = N * (x - np.arctan(x)) # Theorem 4.3
if print_res:
print("D(" + str(N) + ") by Calculation:", DA)
return DA
def flat_locus(atlas, x0, y0, r, z):
for i, point in enumerate(atlas.points):
dz = 0
n = util.dist_2d(x0, y0, point[0], point[1])
if (n <= r):
dz = z
atlas.elevs[i] += dz
def bounded_quad(atlas, x0, y0, zmin, zmax, a, b, c):
for i, point in enumerate(atlas.points):
n = util.dist_2d(x0, y0, point[0], point[1])
dz = a * n**2 + b * n + c
dz = max(zmin, dz)
dz = min(zmax, dz)
atlas.elevs[i] += dz
def find_tangent(LL_s, LC_t):
"""
Given a source point and target lighthouse center with radius 1, find the tangent.
This is done by first finding the angle between LC_t, LL_s, tang and then calculating the tang itself via rotation. LL is "Lighthouse->Left".
It is possible that the angle between the supposed
"""
return rotate(LL_s, LC_t, np.arcsin(1 / dist_2d(LL_s, LC_t)))
def rand_bounded_quad(atlas, zmin, zmax, a, b, c):
x0 = randrange(0, atlas.width)
y0 = randrange(0, atlas.height)
for i, point in enumerate(atlas.points):
n = util.dist_2d(x0, y0, point[0], point[1])
dz = a * n**2 + b * n + c
dz = max(zmin, dz)
dz = min(zmax, dz)
atlas.elevs[i] += dz
def dist_2d(self, idx1, idx2):
p1 = self.points[idx1]
p2 = self.points[idx2]
return util.dist_2d(p1[0], p1[1], p2[0], p2[1])
def draw_all(N):
'''
Draw all lines until a match.
'''
placement_center = (0.0, 0.0)
centers = find_lighthouse_centers(N, placement_center)
lighthouses = []
for c in centers:
left, mid, right = find_lighthouse_illum_points(N, c, placement_center)
lighthouses.append({
"center": c,
"left": left,
"middle": mid,
"right": right
})
# plots
fig = plt.figure()
ax = plt.axes()
ax.scatter(placement_center[0], placement_center[1], color="red")
for l in lighthouses:
ax.add_patch(plt.Circle(l['center'], 1.0, color='black', fill=False))
ax.scatter(l['center'][0], l['center'][1], color="yellow")
ax.scatter(l['left'][0], l['left'][1], color="gray")
ax.scatter(l['right'][0], l['right'][1], color="gray")
ax.scatter(l['middle'][0], l['middle'][1], color="gray")
ax.add_line(
Line2D([l['center'][0], placement_center[0]],
[l['center'][1], placement_center[1]],
linestyle='--',
color='gray',
linewidth=0.4))
ax.add_line(
Line2D([l['center'][0], l['left'][0]],
[l['center'][1], l['left'][1]],
color='gray',
linewidth=0.5))
ax.add_line(
Line2D([l['center'][0], l['right'][0]],
[l['center'][1], l['right'][1]],
color='gray',
linewidth=0.5))
plt.xlim([-N - 1.5, N + 1.5])
plt.ylim([-N - 1.5, N + 1.5])
if N == 1:
DA = 0
else:
target = lighthouses[0]
for source in lighthouses[1:int(len(lighthouses) / 2) + 1]:
isValid, tang, line = get_illumination_line(
source['center'], target['center'], placement_center,
len(lighthouses))
ax.add_line(line)
if isValid:
ax.scatter(source['center'][0],
source['center'][1],
color="green")
ax.scatter(tang[0], tang[1], color="green")
break
else:
ax.scatter(source['center'][0],
source['center'][1],
color="red")
ax.scatter(tang[0], tang[1], color="red")
if source['center'][1] <= tang[1]:
# The dark area is infinite
DA = inf
else:
# We can find the dark area
coeff = np.polyfit(
[tang[0], source['center'][0]], [tang[1], source['center'][1]],
1) # finds the coefficients of y = ax + b for points x, y.
target_cross_x = -coeff[1] / coeff[
0] # we look for 0 = ax' + b --> x = -b/a
ax.scatter(target_cross_x, 0.0, color="orange")
ax.add_line(
Line2D([tang[0], target_cross_x], [tang[1], 0.0],
color='gray',
linewidth=0.5))
ax.add_line(
Line2D([lighthouses[0]['center'][0], target_cross_x],
[lighthouses[0]['center'][1], 0.0],
color='gray',
linewidth=0.5))
x = dist_2d(
(target_cross_x, 0.0), tang
) # this is the nugget, we can find the dark area from this.
DA = N * (x - np.arctan(x)) # Theorem 4.3
plt.xlim([-N - 1.5, target_cross_x + 1.5])
DA_theorem = theorem_4_3_formula(N)
print("D(" + str(N) + ") by Calculation:", DA)
print("D(" + str(N) + ") by Theorem:", DA_theorem)
ax.set_title(str(N) + " lighthouses")
plt.show()
return DA, DA_theorem
