def solve_energy(self):
loss_in_potential = TextMobject("Loss in potential: ")
loss_in_potential.shift(2 * UP)
potential = TexMobject("m g y".split())
potential.next_to(loss_in_potential)
kinetic = TexMobject(["\\dfrac{1}{2}", "m", "v", "^2", "="])
kinetic.next_to(potential, LEFT)
nudge = 0.1 * UP
kinetic.shift(nudge)
loss_in_potential.shift(nudge)
ms = Mobject(kinetic.split()[1], potential.split()[0])
two = TexMobject("2")
two.shift(ms.split()[1].get_center())
half = kinetic.split()[0]
sqrt = TexMobject("\\sqrt{\\phantom{2mg}}")
sqrt.shift(potential.get_center())
nudge = 0.2 * LEFT
sqrt.shift(nudge)
squared = kinetic.split()[3]
equals = kinetic.split()[-1]
new_eq = equals.copy().next_to(kinetic.split()[2])
self.play(
Transform(Point(loss_in_potential.get_left()), loss_in_potential),
*map(GrowFromCenter, potential.split()))
self.dither(2)
self.play(FadeOut(loss_in_potential), GrowFromCenter(kinetic))
self.dither(2)
self.play(ApplyMethod(ms.shift, 5 * UP))
self.dither()
self.play(Transform(half, two, path_func=counterclockwise_path()))
self.dither()
self.play(Transform(squared, sqrt, path_func=clockwise_path()),
Transform(equals, new_eq))
self.dither(2)
def solve_energy(self):
loss_in_potential = TextMobject("Loss in potential: ")
loss_in_potential.shift(2*UP)
potential = TexMobject("m g y".split())
potential.next_to(loss_in_potential)
kinetic = TexMobject([
"\\dfrac{1}{2}","m","v","^2","="
])
kinetic.next_to(potential, LEFT)
nudge = 0.1*UP
kinetic.shift(nudge)
loss_in_potential.shift(nudge)
ms = Mobject(kinetic.split()[1], potential.split()[0])
two = TexMobject("2")
two.shift(ms.split()[1].get_center())
half = kinetic.split()[0]
sqrt = TexMobject("\\sqrt{\\phantom{2mg}}")
sqrt.shift(potential.get_center())
nudge = 0.2*LEFT
sqrt.shift(nudge)
squared = kinetic.split()[3]
equals = kinetic.split()[-1]
new_eq = equals.copy().next_to(kinetic.split()[2])
self.play(
Transform(
Point(loss_in_potential.get_left()),
loss_in_potential
),
*map(GrowFromCenter, potential.split())
)
self.dither(2)
self.play(
FadeOut(loss_in_potential),
GrowFromCenter(kinetic)
)
self.dither(2)
self.play(ApplyMethod(ms.shift, 5*UP))
self.dither()
self.play(Transform(
half, two,
path_func = counterclockwise_path()
))
self.dither()
self.play(
Transform(
squared, sqrt,
path_func = clockwise_path()
),
Transform(equals, new_eq)
)
self.dither(2)
def construct(self):
n_terms = 4
def func((x, y, ignore)):
z = complex(x, y)
if (np.abs(x%1 - 0.5)<0.01 and y < 0.01) or np.abs(z)<0.01:
return ORIGIN
out_z = 1./(2*np.tan(np.pi*z)*(z**2))
return out_z.real*RIGHT - out_z.imag*UP
arrows = Mobject(*[
Arrow(ORIGIN, np.sqrt(2)*point)
for point in compass_directions(4, RIGHT+UP)
])
arrows.highlight(YELLOW)
arrows.ingest_submobjects()
all_arrows = Mobject(*[
arrows.copy().scale(0.3/(x)).shift(x*RIGHT)
for x in range(1, n_terms+2)
])
terms = TexMobject([
"\\dfrac{1}{%d^2} + "%(x+1)
for x in range(n_terms)
]+["\\cdots"])
terms.shift(2*UP)
plane = NumberPlane(color = BLUE_E)
axes = Mobject(NumberLine(), NumberLine().rotate(np.pi/2))
axes.highlight(WHITE)
for term in terms.split():
self.play(ShimmerIn(term, run_time = 0.5))
self.dither()
self.play(ShowCreation(plane), ShowCreation(axes))
self.play(*[
Transform(*pair)
for pair in zip(terms.split(), all_arrows.split())
])
self.play(PhaseFlow(
func, plane,
run_time = 5,
virtual_time = 8
))
class FluidFlow(Scene):
DEFAULT_CONFIG = {
"arrow_spacing" : 1,
"dot_spacing" : 0.5,
"dot_color" : BLUE_B,
"text_color" : WHITE,
"arrow_color" : GREEN_A,
"points_height" : SPACE_HEIGHT,
"points_width" : SPACE_WIDTH,
}
def use_function(self, function):
self.function = function
def get_points(self, spacing):
x_radius, y_radius = [
val-val%spacing
for val in self.points_width, self.points_height
]
return map(np.array, it.product(
np.arange(-x_radius, x_radius+spacing, spacing),
np.arange(-y_radius, y_radius+spacing, spacing),
[0]
))
def add_plane(self):
self.add(NumberPlane().fade())
def add_dots(self):
points = self.get_points(self.dot_spacing)
self.dots = Mobject(*map(Dot, points))
self.dots.highlight(self.dot_color)
self.play(ShowCreation(self.dots))
self.dither()
def add_arrows(self, true_length = False):
if not hasattr(self, "function"):
raise Exception("Must run use_function first")
points = self.get_points(self.arrow_spacing)
points = filter(
lambda p : np.linalg.norm(self.function(p)) > 0.01,
points
)
angles = map(angle_of_vector, map(self.function, points))
prototype = Arrow(
ORIGIN, RIGHT*self.arrow_spacing/2.,
color = self.arrow_color,
tip_length = 0.1,
buff = 0
)
arrows = []
for point in points:
arrow = prototype.copy()
output = self.function(point)
if true_length:
arrow.scale(np.linalg.norm(output))
arrow.rotate(angle_of_vector(output))
arrow.shift(point)
arrows.append(arrow)
self.arrows = Mobject(*arrows)
self.play(ShowCreation(self.arrows))
self.dither()
def add_paddle(self):
pass
def flow(self, **kwargs):
if not hasattr(self, "function"):
raise Exception("Must run use_function first")
self.play(ApplyToCenters(
PhaseFlow,
self.dots.split(),
function = self.function,
**kwargs
))
def label(self, text, time = 5):
mob = TextMobject(text)
mob.scale(1.5)
mob.to_edge(UP)
rectangle = region_from_polygon_vertices(*[
mob.get_corner(vect) + 0.3*vect
for vect in [
UP+RIGHT,
UP+LEFT,
DOWN+LEFT,
DOWN+RIGHT
]
])
mob.highlight(self.text_color)
rectangle = MobjectFromRegion(rectangle, "#111111")
rectangle.point_thickness = 3
self.add(rectangle, mob)
self.dither(time)
self.remove(mob, rectangle)
