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-rw-r--r--FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_torsion_intuition.py (renamed from FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_torsion_intuition.py)0
-rw-r--r--FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_fs2.py90
-rw-r--r--FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file7_fs3.py194
3 files changed, 284 insertions, 0 deletions
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_torsion_intuition.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_torsion_intuition.py
index 31b9a85..31b9a85 100644
--- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_torsion_intuition.py
+++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_torsion_intuition.py
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_fs2.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_fs2.py
new file mode 100644
index 0000000..0c74685
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_fs2.py
@@ -0,0 +1,90 @@
+from manimlib.imports import *
+
+class fs2(SpecialThreeDScene):
+ CONFIG = {
+ "x_min": -2,
+ "x_max": 2,
+ "y_min": -6,
+ "y_max": 6,
+ "graph_origin": ORIGIN
+ }
+ def construct(self):
+ axes = ThreeDAxes()
+ # text = TextMobject(r'$\frac{dB}{ds} = -\tau N$ \\ $\frac{dB}{ds}$ gives the direction of N, \\ while $\tau$ gives its magnitude.').scale(0.7).shift(3*UP + 3*LEFT)
+ self.set_camera_orientation(phi = 75*DEGREES, theta=135*DEGREES)
+ # self.move_camera(distance=0)
+
+ # rprime = np.array([2*np.cos(t), -np.sin(t) - (2*np.sin(2*t)), 0])
+ # t = rprime / np.sqrt(np.dot(rprime, rprime))
+ # rpp = np.array([-2*np.sin(t), -np.cos(t) - (4*np.cos(2*t)), 0])
+ # n = rpp / np.dot(rpp, rpp)
+ # b = np.cross(rprime, rpp)
+ text = TextMobject(r'$\frac{dB}{ds}$', r'$= -\tau$', r'$N$').shift(2*UP + 4*LEFT)
+ text.set_color_by_tex_to_color_map({
+ r'$\frac{dB}{ds}$': YELLOW,
+ r'$N$': RED_C
+ })
+
+ dot = Dot().rotate(PI/2)
+ alpha = ValueTracker(0)
+ t = alpha.get_value
+ figure = ParametricFunction(
+ lambda t: np.array([
+ np.sinh(t),
+ np.cosh(t),
+ 2*t
+ ]), t_min = -3, t_max = 3, color=BLUE
+ ).scale(0.5).move_to(ORIGIN)
+ vector_x = self.get_tangent_vector(t()%1, figure,scale=2)
+ vector_y = self.get_normal_vector(t(),figure,scale=2)
+ vector_x.add_updater(
+ lambda m: m.become(
+ self.get_tangent_vector(t()%1,figure,scale=2)
+ )
+ )
+ vector_y.add_updater(
+ lambda m: m.become(
+ self.get_normal_vector(t(),figure,scale=2)
+ )
+ )
+ dot.add_updater(lambda m: m.move_to(vector_y.get_start()))
+
+
+
+ self.add_fixed_in_frame_mobjects(text)
+ self.play(FadeIn(figure), FadeIn(axes), FadeIn(text))
+ self.begin_ambient_camera_rotation(rate = 0.1)
+ self.wait(1)
+ self.add(vector_x, vector_y,dot)
+ self.play(alpha.increment_value, 0.999, run_time=20, rate_func=rush_from)
+ self.wait(1)
+ self.remove(figure, vector_x, vector_y,dot)
+ self.play(FadeOut(figure), FadeOut(axes), FadeOut(text))
+
+ def get_tangent_vector(self, proportion, curve, dx=0.001, scale=1):
+ t = proportion.copy()
+ 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() * 1
+ vector = Arrow(coord_i , coord_i + unit_vector, color = YELLOW, 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_vector() * 0.7
+ vector = Arrow(coord_i, coord_i + unit_vector, color = RED_C, buff=0)
+ return vector
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file7_fs3.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file7_fs3.py
new file mode 100644
index 0000000..698ca74
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file7_fs3.py
@@ -0,0 +1,194 @@
+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
generated by cgit (git 2.34.1) at 2025-01-09 19:44:53 +0000