mirror of
https://github.com/3b1b/manim.git
synced 2025-08-05 16:49:03 +00:00
Animations up to DefineDivergence of div_curl
This commit is contained in:
Grant Sanderson
parent
2eebaccde2
commit
edaaa26b7a
1 changed files with 292 additions and 60 deletions
|
|
@ -1,6 +1,6 @@
|
|||
from big_ol_pile_of_manim_imports import *
|
||||
|
||||
DEFAULT_SCALAR_FIELD_COLORS = [BLUE_E, WHITE, RED]
|
||||
DEFAULT_SCALAR_FIELD_COLORS = [BLUE_E, GREEN, YELLOW, RED]
|
||||
|
||||
# Quick note to anyone coming to this file with the
|
||||
# intent of recreating animations from the video. Some
|
||||
|
|
@ -44,6 +44,30 @@ def derivative(func, dt=1e-7):
|
|||
return lambda z: (func(z + dt) - func(z)) / dt
|
||||
|
||||
|
||||
def negative_gradient(potential_func, dt=1e-7):
|
||||
def result(p):
|
||||
output = potential_func(p)
|
||||
dx = dt * RIGHT
|
||||
dy = dt * UP
|
||||
dz = dt * OUT
|
||||
return -np.array([
|
||||
(potential_func(p + dx) - output) / dt,
|
||||
(potential_func(p + dy) - output) / dt,
|
||||
(potential_func(p + dz) - output) / dt,
|
||||
])
|
||||
return result
|
||||
|
||||
|
||||
def divergence(vector_func, dt=1e-1):
|
||||
def result(point):
|
||||
value = vector_func(point)
|
||||
return sum([
|
||||
(vector_func(point + dt * RIGHT) - value)[i] / dt
|
||||
for i, vect in enumerate([RIGHT, UP, OUT])
|
||||
])
|
||||
return result
|
||||
|
||||
|
||||
def cylinder_flow_vector_field(point, R=1, U=1):
|
||||
z = R3_to_complex(point)
|
||||
# return complex_to_R3(1.0 / derivative(joukowsky_map)(z))
|
||||
|
|
@ -64,8 +88,13 @@ def get_colored_background_image(scalar_field_func,
|
|||
fw = FRAME_WIDTH
|
||||
fh = FRAME_HEIGHT
|
||||
points_array = np.zeros((ph, pw, 3))
|
||||
x_array = np.linspace(-fw / 2, fw / 2, ph).repeat(pw).reshape((ph, pw))
|
||||
y_array = np.linspace(fh / 2, -fh / 2, pw).repeat(ph).reshape((pw, ph)).T
|
||||
x_array = np.linspace(-fw / 2, fw / 2, pw)
|
||||
x_array = x_array.reshape((1, len(x_array)))
|
||||
x_array = x_array.repeat(ph, axis=0)
|
||||
|
||||
y_array = np.linspace(fh / 2, -fh / 2, ph)
|
||||
y_array = y_array.reshape((len(y_array), 1))
|
||||
y_array.repeat(pw, axis=1)
|
||||
points_array[:, :, 0] = x_array
|
||||
points_array[:, :, 1] = y_array
|
||||
scalars = np.apply_along_axis(scalar_field_func, 2, points_array)
|
||||
|
|
@ -82,8 +111,8 @@ def get_rgb_gradient_function(min_value=0, max_value=1,
|
|||
def func(values):
|
||||
alphas = inverse_interpolate(min_value, max_value, values)
|
||||
alphas = np.clip(alphas, 0, 1)
|
||||
if flip_alphas:
|
||||
alphas = 1 - alphas
|
||||
# if flip_alphas:
|
||||
# alphas = 1 - alphas
|
||||
scaled_alphas = alphas * (len(rgbs) - 1)
|
||||
indices = scaled_alphas.astype(int)
|
||||
next_indices = np.clip(indices + 1, 0, len(rgbs) - 1)
|
||||
|
|
@ -135,7 +164,9 @@ def four_swirls_function(point):
|
|||
return result
|
||||
|
||||
|
||||
def get_force_field_func(*point_strength_pairs):
|
||||
def get_force_field_func(*point_strength_pairs, **kwargs):
|
||||
radius = kwargs.get("radius", 0.5)
|
||||
|
||||
def func(point):
|
||||
result = np.array(ORIGIN)
|
||||
for center, strength in point_strength_pairs:
|
||||
|
|
@ -143,8 +174,11 @@ def get_force_field_func(*point_strength_pairs):
|
|||
norm = np.linalg.norm(to_center)
|
||||
if norm == 0:
|
||||
continue
|
||||
elif norm < radius:
|
||||
to_center /= radius**3
|
||||
elif norm >= radius:
|
||||
to_center /= norm**3
|
||||
to_center *= strength
|
||||
to_center *= -strength
|
||||
result += to_center
|
||||
return result
|
||||
return func
|
||||
|
|
@ -183,7 +217,7 @@ class StreamLines(VGroup):
|
|||
"start_points_generator_config": {},
|
||||
"dt": 0.05,
|
||||
"virtual_time": 15,
|
||||
"n_anchors_per_line": 40,
|
||||
"n_anchors_per_line": 100,
|
||||
"stroke_width": 1,
|
||||
"stroke_color": WHITE,
|
||||
"color_lines_by_magnitude": True,
|
||||
|
|
@ -245,8 +279,7 @@ class VectorField(VGroup):
|
|||
def __init__(self, func, **kwargs):
|
||||
VGroup.__init__(self, **kwargs)
|
||||
self.func = func
|
||||
|
||||
rgb_gradient_function = get_rgb_gradient_function(
|
||||
self.rgb_gradient_function = get_rgb_gradient_function(
|
||||
self.min_magnitude,
|
||||
self.max_magnitude,
|
||||
self.colors,
|
||||
|
|
@ -255,23 +288,26 @@ class VectorField(VGroup):
|
|||
for x in np.arange(self.x_min, self.x_max, self.delta_x):
|
||||
for y in np.arange(self.y_min, self.y_max, self.delta_y):
|
||||
point = x * RIGHT + y * UP
|
||||
output = np.array(func(point))
|
||||
self.add(self.get_vector(point))
|
||||
|
||||
def get_vector(self, point):
|
||||
output = np.array(self.func(point))
|
||||
norm = np.linalg.norm(output)
|
||||
if norm == 0:
|
||||
output *= 0
|
||||
else:
|
||||
output *= self.length_func(norm) / norm
|
||||
# new_norm = np.linalg.norm(output)
|
||||
vect = Vector(output)
|
||||
vect.shift(point)
|
||||
vect.set_fill(rgb_to_color(
|
||||
rgb_gradient_function(np.array([norm]))[0]
|
||||
self.rgb_gradient_function(np.array([norm]))[0]
|
||||
))
|
||||
vect.set_stroke(
|
||||
self.stroke_color,
|
||||
self.stroke_width
|
||||
)
|
||||
self.add(vect)
|
||||
return vect
|
||||
|
||||
|
||||
# Continual animations
|
||||
|
||||
|
|
@ -338,7 +374,7 @@ class ShowPassingFlashWithThinningStrokeWidth(AnimationGroup):
|
|||
class StreamLineAnimation(ContinualAnimation):
|
||||
CONFIG = {
|
||||
"lag_range": 4,
|
||||
"line_anim_class": ShowPassingFlashWithThinningStrokeWidth,
|
||||
"line_anim_class": ShowPassingFlash,
|
||||
"line_anim_config": {
|
||||
"run_time": 4,
|
||||
"rate_func": None,
|
||||
|
|
@ -397,20 +433,23 @@ class TestVectorField(Scene):
|
|||
}
|
||||
|
||||
def construct(self):
|
||||
vector_field = VectorField(
|
||||
lambda p: rotate_vector(cylinder_flow_vector_field(p), TAU / 4)
|
||||
lines = StreamLines(
|
||||
four_swirls_function,
|
||||
virtual_time=3,
|
||||
min_magnitude=0,
|
||||
max_magnitude=2,
|
||||
)
|
||||
vector_field.remove(*filter(
|
||||
lambda v: np.linalg.norm(v.get_start()) <= 1,
|
||||
vector_field
|
||||
self.add(StreamLineAnimation(
|
||||
lines,
|
||||
line_anim_class=ShowPassingFlash
|
||||
))
|
||||
|
||||
self.add(vector_field)
|
||||
self.wait(10)
|
||||
|
||||
|
||||
class Introduction(Scene):
|
||||
CONFIG = {
|
||||
"production_quality_flow": True,
|
||||
"vector_field_func": cylinder_flow_vector_field,
|
||||
}
|
||||
|
||||
def construct(self):
|
||||
|
|
@ -631,7 +670,7 @@ class Introduction(Scene):
|
|||
return result
|
||||
|
||||
def get_stream_lines(self):
|
||||
func = cylinder_flow_vector_field
|
||||
func = self.vector_field_func
|
||||
if self.production_quality_flow:
|
||||
delta_x = 0.5
|
||||
delta_y = 0.1
|
||||
|
|
@ -962,7 +1001,7 @@ class IntroduceVectorField(Scene):
|
|||
|
||||
self.add(stream_line_animation)
|
||||
self.play(
|
||||
vector_field.set_fill, {"opacity": 0.3}
|
||||
vector_field.set_fill, {"opacity": 0.5}
|
||||
)
|
||||
self.wait(7)
|
||||
self.play(
|
||||
|
|
@ -979,7 +1018,7 @@ class IntroduceVectorField(Scene):
|
|||
earth.move_to(earth_center)
|
||||
moon.move_to(moon_center)
|
||||
|
||||
gravity_func = get_force_field_func((earth_center, 6), (moon_center, 1))
|
||||
gravity_func = get_force_field_func((earth_center, -6), (moon_center, -1))
|
||||
gravity_field = VectorField(
|
||||
gravity_func,
|
||||
**self.vector_field_config
|
||||
|
|
@ -997,7 +1036,7 @@ class IntroduceVectorField(Scene):
|
|||
|
||||
def show_magnetic_force(self):
|
||||
magnetic_func = get_force_field_func(
|
||||
(3 * LEFT, 1), (3 * RIGHT, - 1)
|
||||
(3 * LEFT, -1), (3 * RIGHT, +1)
|
||||
)
|
||||
magnetic_field = VectorField(
|
||||
magnetic_func,
|
||||
|
|
@ -1086,24 +1125,7 @@ class ChangingElectricField(Scene):
|
|||
}
|
||||
|
||||
def construct(self):
|
||||
particles = self.particles = VGroup()
|
||||
|
||||
for n in range(9):
|
||||
if n % 2 == 0:
|
||||
particle = get_proton(radius=0.2)
|
||||
particle.charge = +1
|
||||
else:
|
||||
particle = get_electron(radius=0.2)
|
||||
particle.charge = -1
|
||||
particle.velocity = np.array(ORIGIN)
|
||||
particles.add(particle)
|
||||
particle.shift(
|
||||
0.2 * random.random() * RIGHT +
|
||||
0.2 * random.random() * UP
|
||||
)
|
||||
|
||||
particles.arrange_submobjects_in_grid(buff=LARGE_BUFF)
|
||||
|
||||
particles = self.get_particles()
|
||||
vector_field = self.get_vector_field()
|
||||
|
||||
def update_vector_field(vector_field):
|
||||
|
|
@ -1124,9 +1146,219 @@ class ChangingElectricField(Scene):
|
|||
)
|
||||
self.wait(20)
|
||||
|
||||
def get_particles(self):
|
||||
particles = self.particles = VGroup()
|
||||
for n in range(9):
|
||||
if n % 2 == 0:
|
||||
particle = get_proton(radius=0.2)
|
||||
particle.charge = +1
|
||||
else:
|
||||
particle = get_electron(radius=0.2)
|
||||
particle.charge = -1
|
||||
particle.velocity = np.random.normal(0, 0.1, 3)
|
||||
particles.add(particle)
|
||||
particle.shift(np.random.normal(0, 0.2, 3))
|
||||
|
||||
particles.arrange_submobjects_in_grid(buff=LARGE_BUFF)
|
||||
return particles
|
||||
|
||||
def get_vector_field(self):
|
||||
func = get_force_field_func(*zip(
|
||||
map(Mobject.get_center, self.particles),
|
||||
[p.charge for p in self.particles]
|
||||
))
|
||||
return VectorField(func, **self.vector_field_config)
|
||||
self.vector_field = VectorField(func, **self.vector_field_config)
|
||||
return self.vector_field
|
||||
|
||||
|
||||
class InsertAirfoildTODO(TODOStub):
|
||||
CONFIG = {"message": "Insert airfoil flow animation"}
|
||||
|
||||
|
||||
class ThreeDVectorField(ExternallyAnimatedScene):
|
||||
pass
|
||||
|
||||
|
||||
class GravityFluidFlow(IntroduceVectorField):
|
||||
def construct(self):
|
||||
self.vector_field = VectorField(
|
||||
lambda p: np.array(ORIGIN),
|
||||
**self.vector_field_config
|
||||
)
|
||||
self.show_gravitational_force()
|
||||
self.show_fluid_flow()
|
||||
|
||||
|
||||
class TotallyToScale(Scene):
|
||||
def construct(self):
|
||||
words = TextMobject(
|
||||
"Totally drawn to scale. \\\\ Don't even worry about it."
|
||||
)
|
||||
words.scale_to_fit_width(FRAME_WIDTH - 1)
|
||||
words.add_background_rectangle()
|
||||
self.add(words)
|
||||
self.wait()
|
||||
|
||||
|
||||
# TODO: Revisit this
|
||||
class FluidFlowAsHillGradient(Introduction, ThreeDScene):
|
||||
CONFIG = {
|
||||
"production_quality_flow": False,
|
||||
}
|
||||
|
||||
def construct(self):
|
||||
def potential(point):
|
||||
x, y = point[:2]
|
||||
result = 2 - 0.01 * op.mul(
|
||||
((x - 4)**2 + y**2),
|
||||
((x + 4)**2 + y**2)
|
||||
)
|
||||
return max(-10, result)
|
||||
|
||||
vector_field_func = negative_gradient(potential)
|
||||
|
||||
stream_lines = StreamLines(
|
||||
vector_field_func,
|
||||
virtual_time=3,
|
||||
color_lines_by_magnitude=False,
|
||||
start_points_generator_config={
|
||||
"delta_x": 0.2,
|
||||
"delta_y": 0.2,
|
||||
}
|
||||
)
|
||||
for line in stream_lines:
|
||||
line.points[:, 2] = np.apply_along_axis(
|
||||
potential, 1, line.points
|
||||
)
|
||||
stream_lines_animation = self.get_stream_lines_animation(
|
||||
stream_lines
|
||||
)
|
||||
|
||||
plane = NumberPlane()
|
||||
|
||||
self.add(plane)
|
||||
self.add(stream_lines_animation)
|
||||
self.wait(3)
|
||||
self.begin_ambient_camera_rotation(rate=0.1)
|
||||
self.move_camera(
|
||||
phi=70 * DEGREES,
|
||||
run_time=2
|
||||
)
|
||||
self.wait(5)
|
||||
|
||||
|
||||
class DefineDivergence(ChangingElectricField):
|
||||
CONFIG = {
|
||||
"vector_field_config": {
|
||||
"length_func": lambda norm: 0.3,
|
||||
},
|
||||
"stream_line_config": {
|
||||
"start_points_generator_config": {
|
||||
"delta_x": 0.125,
|
||||
"delta_y": 0.125,
|
||||
},
|
||||
"virtual_time": 2,
|
||||
"n_anchors_per_line": 10,
|
||||
"min_magnitude": 0,
|
||||
"max_magnitude": 6,
|
||||
},
|
||||
"stream_line_animation_config": {
|
||||
"line_anim_class": ShowPassingFlash,
|
||||
},
|
||||
"flow_time": 10,
|
||||
}
|
||||
|
||||
def construct(self):
|
||||
self.draw_vector_field()
|
||||
# self.show_flow()
|
||||
self.point_out_sources_and_sinks()
|
||||
self.show_divergence_values()
|
||||
|
||||
def draw_vector_field(self):
|
||||
particles = self.get_particles()
|
||||
random.shuffle(particles.submobjects)
|
||||
particles.remove(particles[0])
|
||||
particles.arrange_submobjects_in_grid(
|
||||
n_cols=4, buff=3
|
||||
)
|
||||
for particle in particles:
|
||||
particle.shift(
|
||||
np.random.normal(0, 1, 3)
|
||||
)
|
||||
particle.shift(particle.get_center()[2] * IN)
|
||||
vector_field = self.get_vector_field()
|
||||
|
||||
self.play(
|
||||
LaggedStart(GrowArrow, vector_field),
|
||||
LaggedStart(GrowFromCenter, particles),
|
||||
run_time=4
|
||||
)
|
||||
self.wait()
|
||||
self.play(LaggedStart(FadeOut, particles))
|
||||
|
||||
def show_flow(self):
|
||||
stream_lines = StreamLines(
|
||||
self.vector_field.func,
|
||||
**self.stream_line_config
|
||||
)
|
||||
stream_line_animation = StreamLineAnimation(
|
||||
stream_lines,
|
||||
**self.stream_line_animation_config
|
||||
)
|
||||
self.add(stream_line_animation)
|
||||
self.wait(self.flow_time)
|
||||
|
||||
def point_out_sources_and_sinks(self):
|
||||
particles = self.particles
|
||||
self.positive_points, self.negative_points = [
|
||||
[
|
||||
particle.get_center()
|
||||
for particle in particles
|
||||
if particle.charge == charge
|
||||
]
|
||||
for charge in +1, -1
|
||||
]
|
||||
pair_of_vector_circle_groups = VGroup()
|
||||
for point_set in self.positive_points, self.negative_points:
|
||||
vector_circle_groups = VGroup()
|
||||
for point in point_set:
|
||||
vector_circle_group = VGroup()
|
||||
for angle in np.linspace(0, TAU, 12, endpoint=False):
|
||||
step = 0.5 * rotate_vector(RIGHT, angle)
|
||||
vect = self.vector_field.get_vector(point + step)
|
||||
vect.set_color(WHITE)
|
||||
vect.set_stroke(width=2)
|
||||
vector_circle_group.add(vect)
|
||||
vector_circle_groups.add(vector_circle_group)
|
||||
pair_of_vector_circle_groups.add(vector_circle_groups)
|
||||
|
||||
self.play(
|
||||
LaggedStart(
|
||||
LaggedStart, vector_circle_groups,
|
||||
lambda vcg: (GrowArrow, vcg),
|
||||
),
|
||||
self.vector_field.set_fill, {"opacity": 0.5}
|
||||
)
|
||||
self.wait()
|
||||
self.play(FadeOut(vector_circle_groups))
|
||||
|
||||
self.positive_vector_circle_groups = pair_of_vector_circle_groups[0]
|
||||
self.negative_vector_circle_groups = pair_of_vector_circle_groups[1]
|
||||
self.wait()
|
||||
|
||||
def show_divergence_values(self):
|
||||
div_func = divergence(self.vector_field.func)
|
||||
|
||||
|
||||
|
||||
class DefineDivergenceJustFlow(DefineDivergence):
|
||||
CONFIG = {
|
||||
"flow_time": 10,
|
||||
}
|
||||
|
||||
def construct(self):
|
||||
self.force_skipping()
|
||||
self.draw_vector_field()
|
||||
self.revert_to_original_skipping_status()
|
||||
self.clear()
|
||||
self.show_flow()
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue