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from manimlib.imports import *
class f(SpecialThreeDScene):
CONFIG = {
"axes_config": {
"x_min": -5,
"x_max": 5,
"y_min": -5,
"y_max": 5,
"z_min": -4,
"z_max": 4,
"x_axis_config": {
"tick_frequency": 100,
},
"y_axis_config": {
"tick_frequency": 100,
},
"z_axis_config": {
"tick_frequency": 100,
},
"num_axis_pieces": 1,
}
}
def construct(self):
axes = ThreeDAxes(**self.axes_config)
text = TextMobject(r'$r(t) = \left\langle\sinh{t}, \cosh{t}, 2t\right\rangle$').scale(0.7).shift(3*UP + 3*LEFT)
self.set_camera_orientation(phi = 75*DEGREES, theta=225*DEGREES)
figure = ParametricFunction(
lambda t: np.array([
np.sinh(t),
np.cosh(t),
2*t
]), t_min = -3, t_max = 3, color=ORANGE
).scale(0.5).move_to(ORIGIN)
dot = Dot(color=RED)
alpha = ValueTracker(0)
t = alpha.get_value
vector_x = self.get_binormal_vector(t()%1, figure,scale=2)
vector_y = self.get_normal_vector(t(),figure,scale=2)
vector_z = self.get_tangent_vector(t(), figure, scale=2)
vector_x.add_updater(
lambda m: m.become(
self.get_binormal_vector(t()%1,figure,scale=2)
)
)
vector_y.add_updater(
lambda m: m.become(
self.get_normal_vector(t(),figure,scale=2)
)
)
vector_z.add_updater(
lambda m: m.become(
self.get_tangent_vector(t(),figure,scale=2)
)
)
dot.add_updater(
lambda m: m.move_to(vector_x.get_start())
)
def curvature(t):
r = np.array([np.sinh(t), np.cosh(t), 2*t])
rp = np.array([np.cosh(t), np.sinh(t), 2])
rpp = np.array([np.sinh(t), np.cosh(t), 0])
cp = np.cross(rp, rpp)
k = cp / (np.dot(rp, rp)**1.5)
return abs(k[0])
def torsion(t):
r = np.array([np.sinh(t), np.cosh(t), 2*t])
rp = np.array([np.cosh(t), np.sinh(t), 2])
rpp = np.array([np.sinh(t), np.cosh(t), 0])
n = rpp / np.dot(rpp, rpp)
dbdt = np.array([2*np.sinh(t), 2*np.cosh(t), 0])
tor = np.dot(dbdt, n)
return tor
k = curvature(0.3)
k = "{:.2f}".format(k)
tor = torsion(0.3)
tor = "{:.2f}".format(tor)
kt1 = TextMobject(rf'At the given point, \\ $\kappa =$ {k} \\').scale(0.7).shift(3*UP + 4*RIGHT)
kt2 = TextMobject('$\implies \kappa$',r'$T$',r' is scaled as:').scale(0.7).next_to(kt1, DOWN, buff=0.1)
kt2.set_color_by_tex_to_color_map({
'$T$': YELLOW
})
tbt1 = TextMobject(rf'At the given point, \\ $\tau =$ {tor} \\').scale(0.7).shift(3*UP + 4*RIGHT)
tbt2 = TextMobject(r'$\implies \tau$',r'$B$',r' is scaled as:').scale(0.7).next_to(tbt1, DOWN, buff=0.1)
tbt2.set_color_by_tex_to_color_map({
'$B$': GREEN_E
})
ft = TextMobject(r'$\frac{dN}{ds}$',r'$ = -\kappa$',r'$T$', r'$ + \tau$',r'$B$ \\', r'and is given as:').scale(0.7).shift(3*UP + 4*RIGHT)
ft.set_color_by_tex_to_color_map({
r'$\frac{dN}{ds}$': GREEN_SCREEN,
'$T$': YELLOW,
r'$B$ \\': GREEN_E
})
self.add_fixed_in_frame_mobjects(text)
self.play(FadeIn(figure), FadeIn(axes), FadeIn(text))
# self.begin_ambient_camera_rotation(rate = 0.13)
self.wait(1)
self.add(vector_x, vector_y,vector_z,dot)
self.play(alpha.increment_value, 0.3, run_time=10, rate_func=rush_from)
self.wait(1)
# self.stop_ambient_camera_rotation()
# self.move_camera(phi = 75*DEGREES, theta=225*DEGREES)
square = Rectangle(width=3.2, fill_color=WHITE, fill_opacity=0.3, color=RED_C).rotate(40*DEGREES).shift(0.8*DOWN+1.2*RIGHT)
mat = [[0.7, 0.3], [1.0, -0.7]]
square = square.apply_matrix(mat).rotate(17*DEGREES).shift(2.1*DOWN+RIGHT)
tl, nl, bl = TextMobject(r'$T$', color=YELLOW).shift(2.8*RIGHT+0.5*DOWN), TextMobject(r'$N$', color=BLUE).shift(RIGHT), TextMobject(r'$B$', color=GREEN_E).shift(0.6*LEFT+0.5*DOWN)
self.add_fixed_in_frame_mobjects(tl, nl, bl)
self.play(FadeIn(VGroup(*[tl, nl, bl])))
self.wait(3)
self.add_fixed_in_frame_mobjects(square)
self.play(FadeIn(square), FadeOut(VGroup(*[tl, nl, bl])))
self.wait(2)
self.add_fixed_in_frame_mobjects(kt1)
self.play(FadeIn(kt1))
self.wait(2)
self.add_fixed_in_frame_mobjects(kt2)
self.play(FadeIn(kt2))
self.wait(2)
kt = self.get_tangent_vector(0.3, figure, scale = -4*float(k))
tb = self.get_binormal_vector(0.3, figure, scale = 2*float(tor))
self.play(
ReplacementTransform(vector_z, kt)
)
self.wait(3)
self.add_fixed_in_frame_mobjects(tbt1)
self.play(FadeOut(VGroup(*[kt1, kt2])), FadeIn(tbt1))
self.wait(2)
self.add_fixed_in_frame_mobjects(tbt2)
self.play(FadeIn(tbt2))
self.wait(2)
self.play(
ReplacementTransform(vector_x, tb)
)
self.wait(2)
self.add_fixed_in_frame_mobjects(ft)
self.play(FadeOut(VGroup(*[tbt1, tbt2])), FadeIn(ft))
self.wait(2)
dnds = Arrow(dot.get_center() + np.array([-0.1,-0.25,0]), np.array([-4,-1,2]), color=GREEN_SCREEN)
dndsl = TextMobject(r'$\frac{dN}{ds}$', color=GREEN_SCREEN).shift(2.5*LEFT + 1.2*UP)
self.add_fixed_in_frame_mobjects(dndsl)
self.play(FadeIn(dnds), FadeIn(dndsl))
self.wait(5)
self.play(FadeOut(VGroup(*[square, dot,vector_y, dnds, dndsl, text, ft, tb, kt])))
self.play(FadeOut(figure), FadeOut(axes))
def get_binormal_vector(self, proportion, curve, dx=0.001, scale=1):
t = proportion
coord_i = curve.point_from_proportion(proportion)
rprime = np.array([np.cosh(t), np.sinh(t), 2])
T = rprime / np.sqrt(np.dot(rprime, rprime))
rpp = np.array([np.sinh(t), np.cosh(t), 0])
n = rpp / np.dot(rpp, rpp)
# b = (np.cross(T, n)[0] - 0.5, np.cross(T, n)[1], coord_i[2] + 1)
b = np.cross(T, n)
# coord_f = curve.point_from_proportion(proportion + dx)
coord_f = b
reference_line = Line(coord_i,coord_f)
unit_vector = reference_line.get_unit_vector() * scale
vector = Arrow(coord_i , coord_i + unit_vector, color = GREEN_E, buff=0)
return vector
def get_normal_vector(self, proportion, curve, dx=0.001, scale=1):
coord_i = curve.point_from_proportion(proportion)
coord_f = curve.point_from_proportion(proportion + dx)
t = proportion.copy()/7
rpp = np.array([np.sinh(t), np.cosh(t), 0])
length = np.sqrt(np.dot(rpp, rpp))
length = 1/(1 + np.exp(-length))
reference_line = Line(coord_i,coord_f).rotate(PI/2).set_width(length).scale(2)
unit_vector = reference_line.get_unit_vector() * scale
vector = Arrow(coord_i, coord_i + unit_vector, color = BLUE, buff=0)
return vector
def get_tangent_vector(self, proportion, curve, dx=0.001, scale=1):
coord_i = curve.point_from_proportion(proportion)
coord_f = curve.point_from_proportion(proportion + dx)
reference_line = Line(coord_i,coord_f).scale(2)
if scale < 0:
reference_line = Line(coord_i,coord_f).scale(2).rotate(360*DEGREES)
unit_vector = reference_line.get_unit_vector() * scale
vector = Arrow(coord_i, coord_i + unit_vector, color = YELLOW, buff=0)
return vector
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