diff options
Diffstat (limited to 'FSF-2020/calculus-of-several-variables')
5 files changed, 205 insertions, 46 deletions
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.gif Binary files differdeleted file mode 100644 index 6b4b438..0000000 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py index c719a1d..f3f5a9c 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py @@ -1,23 +1,91 @@ from manimlib.imports import * - -class fs1(ThreeDScene): +class fs1(GraphScene): + CONFIG = { + "x_min": -2, + "x_max": 2, + "y_min": -6, + "y_max": 6, + "graph_origin": ORIGIN + } def construct(self): + text = TextMobject(r'$\frac{dT}{ds} = \kappa N$ \\ $\frac{dT}{ds}$ gives the direction of N, \\ while $\kappa$ gives its magnitude.').scale(0.7).shift(3*UP + 3*LEFT) + + self.setup_axes() + def curve_(x): + return x**3 - 2*x + + def nm(x): + return abs(6 * x / ((9*(x**4) - 6*(x**2) + 5)**1.5)) + + figure = self.get_graph(curve_) + + + dot = Dot().rotate(PI/2) + alpha = ValueTracker(0) + t2_ = ValueTracker(-2) + t2 = t2_.get_value + t = alpha.get_value + vector_x = self.get_tangent_vector(t(),figure,scale=2) + vector_y = self.get_normal_vector(t(),figure,scale=2) - self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) - dot1 = Dot(np.array([np.cos(-np.pi/3), np.sin(-np.pi/3), -0.4*np.pi/3]) + np.array([0,0.2,0]), radius = 0.16, color=RED) - tgt1 = Arrow((0,0,0), (-2,-0.55,0), color = YELLOW).shift(dot1.get_center() + np.array([0.18,0.04,0])) - nm1 = Arrow((0,0,0), (0.4,-2,0), color = BLUE).shift(dot1.get_center() + np.array([0,0.26,0])).shift(np.array([0.8,4.76,0])).rotate(-15*DEGREES).scale(0.8) - bnm1 = Arrow((0,0,0), (0,2,0), color=GREEN_E).shift(2.1*RIGHT+2*DOWN) - plane1 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(dot1.get_center() + np.array([-0.4, -0.6, 0])).rotate(13*DEGREES).scale(1.2) - point1 = VGroup(*[dot1, tgt1, plane1]).scale(0.8).shift(np.array([1,4.86,0])).rotate(-15*DEGREES) - t = TextMobject(r'$T$', color = YELLOW).move_to(ORIGIN).shift(3.2*RIGHT + DOWN) - n = TextMobject(r'$N$', color = BLUE).shift(DOWN + RIGHT) - b = TextMobject(r'$B$', color = GREEN_E).next_to(bnm1, UP, buff = 0.1) - text = VGroup(*[t, n, b]) - self.add_fixed_in_frame_mobjects(bnm1, text) - self.play(FadeIn(point1), FadeIn(bnm1), FadeIn(text)) - self.wait() - self.play(TransformFromCopy(tgt1, nm1, run_time = 2)) + kappa = TextMobject(r'$\kappa = $').scale(0.7).shift(3*DOWN + 3*RIGHT) + + t_text = always_redraw( + lambda: DecimalNumber( + nm(t2()), + color=WHITE, + ).scale(0.7).next_to(kappa) + ).scale(0.6) + + self.play( + ShowCreation(figure), + GrowFromCenter(dot), + GrowArrow(vector_x), + GrowArrow(vector_y) + ) + vector_x.add_updater( + lambda m: m.become( + self.get_tangent_vector(t(),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_x.get_start())) + circle = Circle(radius = 2, color = GREEN_SCREEN). shift(2.63*RIGHT + 2.8*UP) + dot2 = Dot(np.array([2, curve_(2), 0]), color = WHITE).shift(2*DOWN + 2.5*RIGHT) + + self.add(vector_x, vector_y,dot, t_text, kappa, text) + self.play(t2_.set_value, 2, alpha.set_value, 1, run_time=18, rate_func=smooth) + self.play(FadeIn(dot2), FadeIn(circle)) self.wait(2) - self.play(FadeOut(VGroup(*[bnm1, text, point1, nm1]))) + self.play(FadeOut(VGroup(*[self.axes, dot2, figure, circle, text, kappa, t_text]))) + + + def get_tangent_vector(self, proportion, curve, dx=0.001, scale=0.5): + coord_i = curve.point_from_proportion(proportion) + coord_f = curve.point_from_proportion(proportion + dx) + reference_line = Line(coord_i,coord_f) + unit_vector = reference_line.get_unit_vector() * 0.7 + 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): + t = proportion.copy()/6 + coord_i = curve.point_from_proportion(proportion) + coord_f = curve.point_from_proportion(proportion + dx) + length = 6 * t / ((9*(t**4) - 6*(t**2) + 5)**1.5) + if coord_i[0] <= 0 and coord_i[0] > -0.5: + reference_line = Line(coord_i,coord_f).rotate(PI/2).set_width(0).scale(2) + elif coord_i[0] > 0 and (coord_i[0] < 0.5 or coord_i[0] > 2.7): + reference_line = Line(coord_i,coord_f).rotate(PI/2).set_width(0).scale(2) + elif coord_i[0] > 0: + reference_line = Line(coord_i,coord_f).rotate(PI/2).set_width(length).scale(2) + else: + reference_line = Line(coord_i,coord_f).rotate(-PI/2).set_width(length).scale(2) + unit_vector = reference_line.get_vector() * scale + 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/file5_fs2.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.gif Binary files differdeleted file mode 100644 index ce367b6..0000000 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.gif +++ /dev/null diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.py deleted file mode 100644 index 0261fed..0000000 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.py +++ /dev/null @@ -1,28 +0,0 @@ -from manimlib.imports import * - -class fs1(ThreeDScene): - def construct(self): - - - self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) - dot1 = Dot(np.array([np.cos(-np.pi/3), np.sin(-np.pi/3), -0.4*np.pi/3]) + np.array([0,0.2,0]), radius = 0.16, color=RED) - tgt1 = Arrow((0,0,0), (-2,-0.55,0), color = YELLOW).shift(dot1.get_center() + np.array([0.18,0.04,0])) - nm1 = Arrow((0,0,0), (0.4,-2,0), color = BLUE).shift(dot1.get_center() + np.array([0,0.26,0])).shift(np.array([0.8,4.76,0])).rotate(-15*DEGREES).scale(0.8) - bnm1 = Arrow((0,0,0), (0,2,0), color=GREEN_E).shift(2.1*RIGHT+2*DOWN) - - bnms = Line((0,0,0), (0,0,1.6), color = GREEN_E).shift(np.array([3.1,5.2,0])).scale(0.6) - bnmsa = ArrowTip(color = GREEN_E).next_to(bnms, np.array([0,0,1]), buff = 0).rotate(45*DEGREES) - bns = VGroup(*[bnms, bnmsa]) - - plane1 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(dot1.get_center() + np.array([-0.4, -0.6, 0])).rotate(13*DEGREES).scale(1.2) - point1 = VGroup(*[dot1, tgt1, plane1]).scale(0.8).shift(np.array([1,4.86,0])).rotate(-15*DEGREES) - t = TextMobject(r'$T$', color = YELLOW).move_to(ORIGIN).shift(3.2*RIGHT + DOWN) - n = TextMobject(r'$N$', color = BLUE).shift(DOWN + RIGHT) - b = TextMobject(r'$B$', color = GREEN_E).next_to(bnm1, UP, buff = 0.1) - text = VGroup(*[t, n, b]) - self.add_fixed_in_frame_mobjects(bnm1, text) - self.play(FadeIn(point1), FadeIn(text), FadeIn(bnm1)) - self.wait() - self.play(TransformFromCopy(bnms, nm1, run_time = 3)) - self.wait(2) - self.play(FadeOut(VGroup(*[bnms, text, point1, nm1, bnm1]))) 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/file6_torsion_intuition.py new file mode 100644 index 0000000..31b9a85 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_torsion_intuition.py @@ -0,0 +1,119 @@ +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 |
