diff options
Diffstat (limited to 'FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file7_fs3.py')
| -rw-r--r-- | FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file7_fs3.py | 194 |
1 files changed, 194 insertions, 0 deletions
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 |