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from manimlib.imports import *
class t(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):
text = TextMobject(r'Torsion can be intuitively \\ thought of as the measure \\ of "twisting" of a curve.').scale(0.7).shift(2.5*UP + 4.2*LEFT)
dot = Dot().rotate(PI/2)
f1 = ParametricFunction(
lambda t: np.array([
2*np.sin(TAU*t),
2*np.cos(TAU*t),
2*t
]), t_min = -2, t_max = 2, color = BLUE
).scale(0.5)
d1 = Dot(color = RED).next_to(f1.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.2*UP + 2.4*RIGHT)
t1 = self.get_torsion(2, 0.174)
t1 = "{:.2f}".format(t1)
t1 = TextMobject(fr'At the given point, $\tau = {t1}$').shift(3.5*DOWN).scale(0.7)
f2 = ParametricFunction(
lambda t: np.array([
3*np.sin(TAU*t),
3*np.cos(TAU*t),
2*t
]), t_min = -2, t_max = 2, color = BLUE
).scale(0.5)
d2 = Dot(color = RED).next_to(f2.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.2*UP + 2.95*RIGHT)
t2 = self.get_torsion(3, 0.1765)
t2 = "{:.2f}".format(t2)
t2 = TextMobject(fr'At the given point, $\tau = {t2}$').shift(3.5*DOWN).scale(0.7)
f3 = ParametricFunction(
lambda t: np.array([
4*np.sin(TAU*t),
4*np.cos(TAU*t),
2*t
]), t_min = -2, t_max = 2, color = BLUE
).scale(0.5)
d3 = Dot(color = RED).next_to(f3.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.2*UP + 3.45*RIGHT)
t3 = self.get_torsion(4, 0.179)
t3 = "{:.2f}".format(t3)
t3 = TextMobject(fr'At the given point, $\tau = {t3}$').shift(3.5*DOWN).scale(0.7)
f4 = ParametricFunction(
lambda t: np.array([
1.5*np.sin(TAU*t),
1.5*np.cos(TAU*t),
2*t
]), t_min = -2, t_max = 2, color = BLUE
).scale(0.5)
d4 = Dot(color = RED).next_to(f4.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.215*UP + 2.128*RIGHT)
t4 = self.get_torsion(1.5, 0.173)
t4 = "{:.2f}".format(t4)
t4 = TextMobject(fr'At the given point, $\tau = {t4}$').shift(3.5*DOWN).scale(0.7)
f5 = ParametricFunction(
lambda t: np.array([
np.sin(TAU*t),
np.cos(TAU*t),
2*t
]), t_min = -2, t_max = 2, color = BLUE
).scale(0.5)
d5 = Dot(color = RED).next_to(f5.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.3*UP + 1.858*RIGHT)
t5 = self.get_torsion(1, 0.17)
t5 = "{:.2f}".format(t5)
t5 = TextMobject(fr'At the given point, $\tau = {t5}$').shift(3.5*DOWN).scale(0.7)
axes = ThreeDAxes(**self.axes_config)
self.set_camera_orientation(phi = 60*DEGREES, theta=45*DEGREES)
self.add_fixed_in_frame_mobjects(t1, text)
self.play(FadeIn(VGroup(*[f1, d1, t1, axes, text])))
self.wait(2)
self.add_fixed_in_frame_mobjects(t2)
self.play(ReplacementTransform(d1, d2), ReplacementTransform(f1, f2), ReplacementTransform(t1, t2))
self.wait(2)
self.add_fixed_in_frame_mobjects(t3)
self.play(ReplacementTransform(d2, d3), ReplacementTransform(f2, f3), ReplacementTransform(t2, t3))
self.wait(2)
self.add_fixed_in_frame_mobjects(t4)
self.play(ReplacementTransform(d3, d4), ReplacementTransform(f3, f4), ReplacementTransform(t3, t4))
self.wait(2)
self.add_fixed_in_frame_mobjects(t5)
self.play(ReplacementTransform(d4, d5), ReplacementTransform(f4, f5), ReplacementTransform(t4, t5))
self.wait(2)
self.play(FadeOut(VGroup(*[d5, f5, t5, text, axes])))
def get_torsion(self, a, t):
rprime = np.array([a*np.cos(t), -a*np.sin(t), 2])
T = rprime / np.sqrt(np.dot(rprime, rprime))
rpp = np.array([-a*np.sin(t), -a*np.cos(t), 0])
n = rpp / np.dot(rpp, rpp)
b = np.cross(T, n)
dbdt = np.array([-2*np.sin(t), -2*np.cos(t), 0])
tor = np.dot(dbdt, n)
return tor
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