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Diffstat (limited to 'FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves')
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diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/README.md b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/README.md index e69de29..b46936b 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/README.md +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/README.md @@ -0,0 +1,9 @@ +This repository contains the codes written by [Saarth Deshpande](https://github.com/saarthdeshpande) during the course of FOSSEE Summer Fellowship 2020 under the FLOSS: Mathematics using Python. + +__Sub-topics covered__: +* Equations of Planes and Lines +* General Parametric Curves +* Space Curves (an Intro to Coordinates in 3D) +* Velocity and Differentiability +* Finding Arc Length and Curvature +* TNB Frame and Serret-Frenet Formulae diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/README.md b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/README.md new file mode 100644 index 0000000..10786d6 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/README.md @@ -0,0 +1,11 @@ +**file1_simple_visualization.py** <br> + + +**file2_circle_curvature.py** <br> + + +**file3_curvature_interpretation.py** <br> + + +**file4_different_curvature_single_curve.py** <br> + diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.gif Binary files differnew file mode 100644 index 0000000..bbad112 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.py new file mode 100644 index 0000000..7c970e5 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.py @@ -0,0 +1,123 @@ +from manimlib.imports import * + + +class arcl(MovingCameraScene): + def construct(self): + # self.setup() + def curve_(x): + return 3 - (3653*x**2)/5292 + (2477*x**3)/31752 + (13*x**4)/784 - (17*x**5)/5292 + (17*x**6)/63504 + + curve = FunctionGraph(curve_, x_min=-2, x_max=6, stroke_width = 2, color = BLUE).scale(0.1).move_to(ORIGIN) + lines = [Line(length = 0.05, color = RED).scale(0.2).move_to(ORIGIN).shift(np.array([-4 + 0.1*i, curve_(-2.5 + 0.1*i), 0])).rotate(-25*DEGREES) for i in range(4)] + lines2 = [Line(length = 0.05, color = RED).scale(0.2).move_to(ORIGIN).shift(np.array([-4 + 0.125*i, curve_(-2.5 + 0.1*i), 0])).rotate(-25*DEGREES) for i in range(4, 9)] + # lines[0].rotate(-25*DEGREES).shift(np.array([-4,curve_(-2.5), 0])) + # lines[1].rotate(-25*DEGREES).shift(np.array([-3.78,curve_(-2.3), 0])) + # lines3 = [Line(length = 0.05, color = RED).scale(0.2).move_to(ORIGIN + 1.5*UP + 0.6*RIGHT).shift(np.array([-1 + 0.2*i, -1.5 - 0.2*i, 0])).rotate(30*DEGREES) for i in range(4)] + # lines2b = VGroup(*lines3).rotate(-8*DEGREES) + # lines4 = [Line(length = 0.05, color = RED).scale(0.2).move_to(ORIGIN + 1.6*UP + 0.5*RIGHT).shift(np.array([-1 + 0.18*i, -1.65 - 0.2*i, 0])).rotate(22*DEGREES) for i in range(4, 9)] + # lines5 = [Line(length = 0.05, color = RED).scale(0.2).move_to(ORIGIN + 7*RIGHT).shift(np.array([-4 + 0.1*i, curve_(-2.5 + 0.1*i), 0])).rotate(-25*DEGREES) for i in range(4)] + # lines6 = [Line(length = 0.05, color = RED).scale(0.2).move_to(ORIGIN +7.25*RIGHT).shift(np.array([-4 + 0.053*i, curve_(-2.5 + 0.1*i), 0])).rotate(-26*DEGREES) for i in range(4, 9)] + + # lc1 = [Line(length = 0.05, color = RED).scale(0.2).rotate((-25 + i*2) * DEGREES).shift(np.array([-1 + 0.125*i, curve_(-1.5 + 0.1*i), 0])) for i in range(2)] + # lc1b = VGroup(*lc1).shift(1.7*LEFT + 0.2*DOWN) + + text = TextMobject(r'$r(t) = \left\langle t, t^{3} - 2t, 0\right\rangle$ \\ $r\prime (t) = \left\langle 1, 3t^{2} - 2, 0\right\rangle$').scale(0.7).shift(3*UP + 4*RIGHT) + + # l = VGroup(*lines, *lines2, lines2b, *lines4, *lines5, *lines6, lc1b).shift(curve.get_center()) + l = VGroup(*lines, *lines2) + arc = Line(lines[3].get_center(), lines2[0].get_center() + np.array([0.005, 0 ,0]), color = GREEN_SCREEN).rotate(12*DEGREES) + arctext = TextMobject(r'$ds$', color = GREEN_SCREEN).scale(0.15).next_to(arc.get_center(), 0.001*DOWN + 0.01*RIGHT,buff = 0.01) + dy = Arrow(arc.get_start(), np.array([arc.get_start()[0], lines2[0].get_center()[1] + 0.01, 0]), color = YELLOW) + dx = Arrow(arc.get_start(), np.array([lines2[0].get_center()[0] - 0.01, arc.get_start()[1], 0]), color = BLUE) + dxt = DashedLine(dy.get_end(), dy.get_end() + np.array([0.13, 0 ,0])) + dyt = DashedLine(dx.get_end(), dx.get_end() + np.array([0, 0.3 ,0])) + dxtext = TextMobject(r'$dx$').scale(0.2).next_to(dx, RIGHT, buff = 0.01) + dytext = TextMobject(r'$dy$').scale(0.2).next_to(dy, LEFT, buff = 0.01) + formula = TextMobject(r"Consider a very small interval ", r'$ds$. \\', r"Using Pythagoras' theorem, \\", r'$ds$', r" = $\sqrt{(dx)^{2} + (dy)^{2}}$").scale(0.25).shift(5*LEFT + 0.5*UP) + formula.set_color_by_tex_to_color_map({ + "$ds$. \\": GREEN_SCREEN, + "$ds$": GREEN_SCREEN + }) + + formula2 = TextMobject(r'To compute the arc length \\ from $a$ to $b$, we need to \\ sum over all intervals ', r'$ds$').scale(0.25).shift(5.2*LEFT + 0.7*UP) + formula2.set_color_by_tex_to_color_map({ + "$ds$": GREEN_SCREEN + }) + + formula3 = TextMobject(r'$L = \int_{a}^{b} ds$ \\ $= \int_{a}^{b} \sqrt{(\frac{dx}{dt})^{2} + (\frac{dy}{dt})^{2} + (\frac{dz}{dt})^{2}}\quad dt$').scale(0.25).shift(5.2*LEFT + 0.1*UP) + + bl = DashedLine(lines2[4].get_center(), lines2[4].get_center() + np.array([1,0,0])) + blt = TextMobject(r'$b$').scale(0.5).next_to(bl.get_center(), DOWN, buff=0.1) + al = DashedLine(lines[0].get_center(), lines[0].get_center() + np.array([1,0,0])) + alt = TextMobject(r'$a$').scale(0.5).next_to(al.get_center(), UP, buff=0.1) + pts = VGroup(*[bl, blt, al, alt]) + + compute = TextMobject(r'To compute the arc length from \\ $t = -1.4$ to $t = -1.1$, \\ summation of small intervals $ds$ \\ is given by $L = \int_{-1.4}^{-1.1} ds$ \\').scale(0.7).shift(6.8*LEFT + 2.5*UP) + compute_ = TextMobject(r'L = $ \int_{-1.4}^{-1.1} \sqrt{(\frac{dx}{dt})^{2} + (\frac{dy}{dt})^{2} + (\frac{dz}{dt})^{2}}\quad dt$ \\ = $\int_{-1.4}^{-1.1} \sqrt{1^{2} + (3t^{2} - 2)^{2} + 0^{2}}\quad dt$').scale(0.7).shift(6.8*LEFT + -0.6*DOWN) + #compute = VGroup(*[compute, compute_]) + compute2 = TextMobject(r'$ = \int_{-1.4}^{-1.1} \sqrt{9t^{4} - 12t^{2} + 5}\quad dt$').scale(0.7).shift(6.8*LEFT + 0.7*DOWN) + compute3 = TextMobject(r'$L = 0.8693$').scale(0.7).shift(6.8*LEFT + 1.2*DOWN) + arclen = compute3.copy() + arclen = arclen.scale(0.8).next_to(arc.get_center(), RIGHT, buff = 0.1) + dsd = TextMobject(r'We can divide the curve \\ into multiple small arcs ', r'$ds$').scale(0.25).shift(5.2*LEFT + 0.2*UP) + dsd.set_color_by_tex_to_color_map({ + "$ds$": GREEN_SCREEN + }) + + # 13th sec, consider a v small interval ds, show Pythagoras + # reduce text size + # then show we can divide curve into small ds + # all red ds + # To compute arc length, we need to sum over all intervals ds + # a and b show and give dashes dy dx for first and last + # give dz in formula and show it's zero + # Zooom out, Remove red bars, draw yellow line + # Consider t = -1.4 to -1.1 + # at end show l = 0.693 near yellow line, smaller size + + ax1 = Vector((0,1,0), color = YELLOW) + ax1l = TextMobject(r'$y$').next_to(ax1, LEFT, buff = 0) + ax2 = Vector((1,0,0), color = BLUE) + ax2l = TextMobject(r'$x$').next_to(ax2, RIGHT, buff = 0) + ax = VGroup(*[ax1, ax1l, ax2, ax2l]).scale(0.6).shift(3*DOWN + 6*LEFT) + + self.play(FadeIn(curve), FadeIn(ax)) + self.play(ApplyMethod(curve.scale, 10), FadeIn(text)) + # self.play(FadeIn(l)) + self.wait(2) + self.play(FadeOut(text)) + self.play(self.camera_frame.set_width, 5, + self.camera_frame.move_to, 3.8*LEFT+0.4*DOWN, + ax.shift, UP, + ax.scale, 0.5, run_time = 4) + long = ArcBetweenPoints(lines[1].get_center() + 0.01, lines2[3].get_center(), color = YELLOW, angle = 10*DEGREES).rotate(180*DEGREES) + + + self.play(Write(formula),FadeIn(VGroup(*[arc, arctext, dy, dx, dxt, dyt, dxtext, dytext])), FadeIn(VGroup(*[lines[3], lines2[0]]))) + self.wait(2) + self.play(ReplacementTransform(formula, dsd), TransformFromCopy(VGroup(*[lines[3], lines2[0]]) , l)) + #Transform(l, VGroup(*[lines[3], lines2[0]])), ) + self.wait(2) + self.play(ReplacementTransform(dsd, formula2), FadeIn(pts)) + self.wait(3) + self.play(FadeIn(formula3)) + self.wait(2) + self.play(FadeOut(VGroup(*[formula3, l, pts, formula2, arc, arctext, dy, dx, dxt, dyt, dxtext, dytext]))) + self.play( + self.camera_frame.set_width, 15, + self.camera_frame.move_to, 3*LEFT, + ax.shift, DOWN + 3*LEFT, + ax.scale, 2.3, + run_time = 4) + text = text.shift(2*LEFT) + self.play(FadeIn(long), FadeIn(compute), FadeIn(text)) + self.wait(2) + self.play(FadeIn(compute_)) + self.wait(2) + self.play(FadeIn(compute2)) + self.wait(1) + self.play(FadeIn(compute3)) + self.wait(1) + self.play(TransformFromCopy(compute3, arclen)) + self.wait(2) + self.play(FadeOut(VGroup(*[ax, arclen, compute_, curve, text, compute, compute2, compute3, long]))) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.gif Binary files differnew file mode 100644 index 0000000..3f7ecd1 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.py new file mode 100644 index 0000000..05cad80 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.py @@ -0,0 +1,91 @@ +from manimlib.imports import * + +class a(GraphScene): + CONFIG = { + "x_min": -3, + "x_max": 6, + "y_min": -6, + "y_max": 10, + "graph_origin": ORIGIN + } + def construct(self): + intro = TextMobject('Consider the following curve.') + mid = TextMobject(r'Notice how the direction of the unit tangent vector\\changes with respect to the arc length.') + outro = TextMobject(r'The rate of change of unit tangent with \\ respect to the arc length $ds$ is called curvature.\\Mathematically, curvature $ = k = \left|{\frac{dT}{ds}}\right|$') + + XTD = self.x_axis_width/(self.x_max- self.x_min) + YTD = self.y_axis_height/(self.y_max- self.y_min) + + circle = Circle(radius = 0.95, color = GRAY, fill_opacity = 0.2, fill_color = RED) + circle.set_stroke(width = 0.1) + + tgt1 = Arrow((-2.2*XTD,-0.5*YTD,0),(-1*XTD,1,0)) + tgt2 = Arrow((-1.2*XTD, 1.93*YTD,0),(0*XTD,1.6,0)).scale(1.2) + tgt3 = Arrow((-0.3*XTD,3*YTD, 0), (1.5*XTD, 3*YTD,0)) + tgt4 = Arrow((1.4*XTD, 2*YTD,0),(2.4*XTD, 1*YTD,0)).scale(2.8) + tgt5 = Arrow((2.4*XTD, 0, 0), (3.8*XTD,-2*YTD, 0)).scale(1.2).shift(0.26*RIGHT) + tgt6 = Arrow((3.8*XTD,-1*YTD, 0), (4.8*XTD, -1*YTD, 0)).scale(2.8).shift(0.26*RIGHT) + tgt7 = Arrow((5.3*XTD, 0, 0),(6.3*XTD,1,0)).shift(0.35*LEFT+0.1*DOWN).scale(1.3) + + dot1 = Dot(tgt1.get_start(), color = RED) + dot2 = Dot(tgt2.get_start(), color = RED) + dot3 = Dot(tgt3.get_start(), color = RED) + dot4 = Dot(tgt4.get_start(), color = RED) + dot5 = Dot(tgt5.get_start(), color = RED) + dot6 = Dot(tgt6.get_start(), color = RED) + dot7 = Dot(tgt7.get_start(), color = RED) + + arc = ArcBetweenPoints(dot1.get_center(), dot2.get_center(), color = GREEN_SCREEN, angle = 10*DEGREES).rotate(180*DEGREES) + + dots = VGroup(*[dot1, dot2, dot3, dot4, dot5, dot6, dot7]) + + ds = CurvedArrow((-4, 2, 0), (tgt1.get_start() + tgt2.get_start()) / 2, color = YELLOW) + ds_text = TextMobject(r'$ds$').next_to(ds, UP, buff = 0.1).shift(1.3*LEFT) + + self.setup_axes(hideaxes=True) + + def curve(x): + return 3 - (3653*x**2)/5292 + (2477*x**3)/31752 + (13*x**4)/784 - (17*x**5)/5292 + (17*x**6)/63504 + + # parabola_x_out = FunctionGraph(curve, x_min=-2, x_max=6, stroke_width = 2, color = BLUE) + parabola_x_out = self.get_graph(curve) + + dot_x = Dot().rotate(PI/2).set_color(YELLOW_E) + alpha_x = ValueTracker(-2) + vector_x = self.get_tangent_vector(alpha_x.get_value(),parabola_x_out,scale=1.5) + dot_x.add_updater(lambda m: m.move_to(vector_x.get_center())) + vector_x.add_updater( + lambda m: m.become( + self.get_tangent_vector(alpha_x.get_value()%1,parabola_x_out,scale=1.5) + ) + ) + + self.play(FadeIn(intro)) + self.wait(2) + self.play(FadeOut(intro)) + self.setup_axes(hideaxes=False) + self.play(ShowCreation(parabola_x_out), FadeIn(dots), FadeIn(ds), FadeIn(ds_text), FadeIn(arc)) + self.wait(2) + self.play(FadeOut(self.axes), FadeOut(arc), FadeOut(parabola_x_out),FadeIn(mid), FadeOut(dots), FadeOut(ds), FadeOut(ds_text)) + self.wait(3) + self.play(FadeOut(mid)) + self.play(FadeIn(self.axes), FadeIn(parabola_x_out), FadeIn(dots)) + self.add(vector_x) + self.play(alpha_x.increment_value, 1, run_time=8, rate_func=linear) + self.remove(vector_x) + self.play(FadeOut(VGroup(*[self.axes, dots, parabola_x_out]))) + self.play(FadeIn(outro)) + self.wait(3) + self.play(FadeOut(outro)) + self.wait(1) + + + + + 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) + unit_vector = reference_line.get_unit_vector() * scale + vector = Arrow(coord_i , coord_i + unit_vector, color = YELLOW, buff=0) + return vector diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.gif Binary files differnew file mode 100644 index 0000000..989a3b7 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.py new file mode 100644 index 0000000..232ac41 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.py @@ -0,0 +1,27 @@ +from manimlib.imports import * + +class circleC(GraphScene): + CONFIG = { + "x_min": -6, + "x_max": 6, + "y_min": -6, + "y_max": 6, + "graph_origin": ORIGIN, + "x_axis_width": 12, + "y_axis_height": 12 + } + def construct(self): + epiphany = TextMobject(r'Driving a vehicle on which of \\ the two paths would be easier?').scale(0.6).shift(3.5*LEFT + 3*UP) + outro = TextMobject(r'The larger path, due to its \\ smaller curvature, since $k = \frac{1}{R}$.').scale(0.6).shift(3.7*LEFT + 3*UP) + XTD = self.x_axis_width/(self.x_max- self.x_min) + YTD = self.y_axis_height/(self.y_max- self.y_min) + + circle = Circle(radius = 2, color = BLUE) + circle2 = Circle(radius = 3, color = GREEN_E) + + self.setup_axes(hideaxes=True) + self.play(FadeIn(self.axes), Write(circle, run_time = 2), FadeIn(epiphany)) + self.play(Write(circle2, run_time = 3)) + self.play(ReplacementTransform(epiphany, outro)) + self.wait(2) + self.play(FadeOut(VGroup(*[self.axes, circle, circle2, epiphany, outro]))) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.gif Binary files differnew file mode 100644 index 0000000..22a450a --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.py new file mode 100644 index 0000000..f10fa26 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.py @@ -0,0 +1,114 @@ +from manimlib.imports import * + +class interpretation(ZoomedScene): + CONFIG = { + "zoomed_display_height": 3, + "zoomed_display_width": 3, + "zoom_factor": 0.15, + "zoomed_display_center": ORIGIN + 4*LEFT + DOWN, + } + def construct(self): + + tgt = Vector((1, 2, 0), color = YELLOW).shift(0.005*RIGHT + 0.007*DOWN) + dot = Dot(tgt.get_start(),color = RED) + curve = ParametricFunction( + lambda t: np.array([ + 2*(t**2), + 4*t, + 0 + ]), t_min = -5, t_max = 5 + ).scale(0.3).move_to(ORIGIN + 4*RIGHT).rotate(6*DEGREES) + + ds = ParametricFunction( + lambda t: np.array([ + 2*(t**2), + 4*t, + 0 + ]), t_min = 0, t_max = 0.05, color = GREEN_SCREEN + ).scale(0.9).shift(3.09*LEFT).rotate(-27.5*DEGREES).move_to(ORIGIN).shift(0.07*UP + 0.05*RIGHT).set_stroke(width=20) + + dsl = TextMobject(r'$ds$', color = GREEN_SCREEN).scale(0.2).next_to(ds, RIGHT, buff = 0) + + + tgtText = TextMobject(r'$r\prime (t) = \left\langle 1, 2, 0\right\rangle$').next_to(tgt, UP, buff = 0).scale(0.7) + tgt2 = DashedLine((0,0,0),(1, 2, 0), color = GRAY).shift(DOWN + 2*RIGHT) + circle = Circle(radius = 0.9, color = GREEN_SCREEN).shift(0.85*RIGHT + 0.38*DOWN) + circle.set_stroke(opacity = 1) + dl = DashedLine(circle.get_center(), dot.get_center()) + dltext = TextMobject(r'$R = 2.795$').scale(0.5).next_to(circle.get_center(), DOWN, buff = 0.1) + + main = TextMobject(r'r(t) = $\left\langle t^{2}, 2t, 0 \right\rangle\quad r\prime (t) = \left\langle 2t, 2, 0 \right\rangle\quad$ \\ $r\prime\prime (t) = \left\langle 2, 0, 0 \right\rangle$').scale(0.7).shift(3*UP + 3*LEFT) + main2 = TextMobject(r'Curvature at an arbitrary point \\ say r(t = 0.5) can be given as: \\ $\kappa = \frac{1}{R} = \frac{1}{2.795} = 0.357$').scale(0.7).shift(3.5*LEFT) + main3 = TextMobject(r'The ', 'tangent', r' and ', 'normal', r' vectors \\ can be represented as:').scale(0.7).shift(3.5*LEFT) + main3.set_color_by_tex_to_color_map({ + "tangent": YELLOW, + "normal": BLUE + }) + main4 = TextMobject(r'These vectors travel along \\ a small interval ', r'$ds$').scale(0.7).shift(1.5*UP + 3*LEFT) + main4.set_color_by_tex_to_color_map({ + "$ds$": GREEN_SCREEN + }) + + main5 = TextMobject(r'$\kappa = 0.357$').scale(0.7).shift(main.get_center() + np.array([2.4,-0.18,0])) + + nm = Vector((2, -1, 0), color = BLUE).shift(0.005*RIGHT + 0.007*DOWN) + nmText = TextMobject(r'$r\prime\prime (t) = \left\langle 2,0,0\right\rangle$').next_to(nm, DOWN+RIGHT, buff = 0).scale(0.7) + nm2 = DashedLine((0,0,0),(2, -1, 0), color = GRAY).shift(2*UP + RIGHT) + square = Square(fill_color = WHITE, fill_opacity = 0.2).rotate(63*DEGREES).shift(0.5*UP +1.5*RIGHT).scale(1.1) + square.set_stroke(width = 0.1) + square2 = Square(fill_color = PINK, fill_opacity = 0.2).scale(0.55).rotate(63*DEGREES).move_to((square.get_center() - dot.get_center()) / 2) + square2.set_stroke(width = 0.1) + arrow = CurvedArrow(square.get_center() + np.array([2,1,0]), square.get_center() + np.array([0.5,0,0])) + arrowText = TextMobject(r'$r\prime (t)\times r\prime\prime (t) = 4$').next_to(arrow.get_start(), DOWN+1*RIGHT, buff = 0).scale(0.7) + + text1 = TextMobject(r'$\left|\frac{dT}{ds}\right| = \frac{\left|\frac{dT}{dt}\right|}{\left|\frac{ds}{dt}\right|}$').shift(UP+3*LEFT).scale(0.7) + text2 = TextMobject(r'$\left|\frac{dT}{ds}\right| = \frac{\frac{r\prime\prime (t)}{\left| r\prime (t)\right|}\times\frac{r\prime (t)}{\left| r\prime (t)\right|}}{\left|r\prime (t)\right|}$').next_to(text1, DOWN, buff = 0.1).scale(0.7) + text3 = TextMobject(r'$= \frac{4}{(4t^{2} + 4)^{\frac{3}{2}}}$ \\ $= \frac{1}{2\sqrt{(1 + (0.5)^{2})^{3}}}$').next_to(text2, DOWN, buff = 0.1).scale(0.7) + text4 = TextMobject(r'$ = 0.357$').scale(0.7).next_to(text3, DOWN, buff = 0.2) + unit = VGroup(*[tgt, tgt2, nm, nm2]) + + tgt2text = TextMobject(r'$\frac{r\prime (t)}{\left| r\prime (t)\right|}$').shift(1.1*UP).scale(0.7).rotate(63*DEGREES ) + nm2text = TextMobject(r'$\frac{r\prime\prime (t)}{\left| r\prime (t)\right|}$').scale(0.7).shift(0.7*RIGHT+0.8*DOWN).rotate(-25*DEGREES) + unit2 = unit.copy().scale(0.5).shift(0.75*LEFT+0.25*DOWN) + + self.play(FadeIn(curve), FadeIn(main)) + self.wait(1) + self.play(ApplyMethod(curve.scale, 3), ApplyMethod(curve.shift, ORIGIN + 3.31*RIGHT)) + # self.wait(2) + self.play(FadeIn(main2), FadeIn(dot)) + self.play(FadeIn(circle), FadeIn(dl), FadeIn(dltext)) + self.wait() + self.play(ReplacementTransform(main2, main5), FadeIn(main3), FadeOut(circle), FadeOut(dl), FadeOut(dltext), FadeIn(VGroup(*[tgt, tgtText]))) + self.wait(1) + self.play(FadeIn(VGroup(*[nm, nmText]))) + self.wait(1) + self.remove(dot) + self.setup() + #self.camera_frame.set_width(4) + self.activate_zooming(animate = True) + self.play(FadeIn(ds), FadeIn(dsl), FadeOut(main3)) + self.wait(1) + self.play(FadeIn(main4)) + self.play(ApplyMethod(tgt.shift, 0.16*UP + 0.09*RIGHT), ApplyMethod(nm.shift, 0.16*UP + 0.09*RIGHT), run_time = 5) + self.wait(1) + self.play(FadeOut(ds), FadeOut(dsl), FadeOut(main4), FadeOut(self.zoomed_display, run_time = 1), FadeOut(self.zoomed_camera.frame, run_time = 1)) + # tgt = tgt.shift(0.16*DOWN + 0.08*LEFT) + # nm = nm.shift(0.16*DOWN + 0.08*LEFT) + self.play(ApplyMethod(tgt.shift, 0.16*DOWN + 0.09*LEFT, run_time = 1), ApplyMethod(nm.shift, 0.16*DOWN + 0.09*LEFT, run_time = 1)) + self.play(FadeIn(dot), FadeIn(VGroup(*[tgt2, nm2]))) + self.wait(1) + self.play(FadeIn(VGroup(*[square, arrow, arrowText]))) + self.wait(1) + self.play(FadeIn(unit2), FadeIn(square2)) + self.wait(1) + self.play(FadeIn(VGroup(*[tgt2text, nm2text]))) + self.wait(1) + self.play(FadeIn(text1)) + self.wait(1) + self.play(FadeIn(text2)) + self.wait(1) + self.play(FadeIn(text3)) + self.wait(1) + self.play(FadeIn(text4)) + self.wait(2) + self.play(FadeOut(VGroup(*[main, main5, square2, curve, dot, tgt2text, nm2text, text1, text2, text3, text4, tgt, tgtText,nm, nmText,tgt2, nm2,square, arrow, arrowText,unit2]))) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.gif Binary files differnew file mode 100644 index 0000000..3b78b5f --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.py new file mode 100644 index 0000000..0dc06bb --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.py @@ -0,0 +1,76 @@ +from manimlib.imports import * + +class GR(GraphScene): + CONFIG = { + "x_axis_label": "", + "y_axis_label": "", + "x_min": -4, + "x_max": 6, + "y_min": -6, + "y_max": 10, + "graph_origin": ORIGIN, + 'x_tick_frequency': 20, + 'y_tick_frequency': 20 + } + + def construct(self): + + self.setup_axes() + def curve(x): + return 3 - (3653*x**2)/5292 + (277*x**3)/31752 + (13*x**4)/784 - (17*x**5)/5292 + (170*x**6)/63504 + + graph = FunctionGraph(curve, x_min=-2, x_max=6, stroke_width = 2, color = BLUE) + + tracker = ValueTracker(-2) + + text = TextMobject(r'$\because R_{1} > R_{2}$, the curvature at \\ point $P_{1}$ is less than that \\ at point $P_{2}$ as $\kappa = \frac{1}{R}$').shift(3.2*LEFT+3*UP).scale(0.6) + + dot1 = Dot((0,3,0), color = YELLOW) + dot1label = TextMobject(r'$P_{1}$').next_to(dot1, UP+RIGHT, buff = 0.1) + dot2 = Dot((2.9,-0.47, 0), color = YELLOW) + dot2label = TextMobject(r'$P_{2}$').next_to(dot2, DOWN, buff = 0.1) + dots = VGroup(*[dot1, dot2, dot1label, dot2label]) + + def get_tangent_line(): + line = Line( + ORIGIN, 2 * RIGHT, + color=RED, + stroke_width=4, + ) + dx = 0.0001 + + x = tracker.get_value() + p0 = np.array([x-dx,curve(x-dx),0]) + p1 = np.array([x, curve(x), 0]) + p2 = np.array([x + dx, curve(x + dx), 0]) + + angle = angle_of_vector(p2 - p1) + line.rotate(angle) + line.move_to(p0) + return line + + circle1 = Circle(radius = 0.8, color = GREY, opacity = 0.2).shift(2.2*UP) + tgt1 = Line((-2,3,0), (2,3,0), color = GREY, opacity = 0.2).scale(0.4) + + r1 = Line(circle1.get_center(), circle1.get_center() + np.array([0,0.8,0]), color=GREEN_SCREEN) + r1label = TextMobject(r'$R_{1}$',color=WHITE).next_to(r1, RIGHT, buff = 0.1).scale(0.6) + + curvature1 = VGroup(*[circle1, tgt1, r1, r1label]) + + circle2 = Circle(radius = 0.2, color = GREY, opacity = 0.2).shift(0.3*DOWN + 2.9*RIGHT) + tgt2 = Line((4,-2,0), (6, -2, 0), color = GREY, opacity = 0.2).scale(0.5).shift(2.1*LEFT + 1.5*UP) + + r2 = Line(circle2.get_center(), circle2.get_center() - np.array([0,0.2,0]), color=GREEN_SCREEN) + r2label = TextMobject(r'$R_{2}$', color=WHITE).next_to(r2.get_start(), np.array([0,0,0]), buff = 0).scale(0.4) + + curvature2 = VGroup(*[circle2, tgt2, r2, r2label]) + + line = always_redraw(get_tangent_line) + + self.add(graph, line, dots, text) + self.wait(1.2) + self.play(tracker.set_value, 4, rate_func=smooth, run_time=10) + self.play(FadeIn(curvature1), FadeIn(curvature2)) + self.wait(2) + self.play(FadeOut(VGroup(*[curvature1, curvature2, graph, self.axes, line, dots, text]))) + self.wait() diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/README.md b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/README.md new file mode 100644 index 0000000..29d2f6a --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/README.md @@ -0,0 +1,14 @@ +**file1_line_eqn.py**<br> + + +**file2_point_normal_form_plane.py**<br> + + +**file3_intercept_form_plane.py**<br> + + +**file4_3d_plane.py**<br> + + +**file5_vector_form_line.py**<br> + diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file1_line_eqn.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file1_line_eqn.gif Binary files differnew file mode 100644 index 0000000..a8a301a --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file1_line_eqn.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file1_line_eqn.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file1_line_eqn.py new file mode 100644 index 0000000..402775b --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file1_line_eqn.py @@ -0,0 +1,26 @@ +from manimlib.imports import * + +class three(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + self.set_camera_orientation(phi=14.25* DEGREES,theta=0*DEGREES,distance=8) + self.play(FadeIn(axes)) + + plane = ParametricSurface( + lambda u,v: np.array([ + 6, + 8*v, + 3*u + ]), u_min = -0.8, u_max = 0.8, fill_opacity = 0.4).rotate(45*DEGREES).move_to(ORIGIN).shift(RIGHT+UP) + d2text = TextMobject(r'$\mathbb{R}^{2}: y = mx + c$').shift(3*LEFT + 2*UP).rotate(np.pi/2) + d3text = TextMobject(r'$\mathbb{R}^{3}: y = mx + c$').shift(4*RIGHT+3*UP) + self.play(FadeIn(plane), FadeIn(d2text)) + self.wait(3) + self.play(FadeOut(d2text)) + self.move_camera(phi = 60*DEGREES, theta=45*DEGREES,run_time=3) + self.begin_ambient_camera_rotation(rate=0.02) + self.add_fixed_in_frame_mobjects(d3text) + self.play(FadeIn(d3text)) + self.wait(3) + self.play(FadeOut(d3text), FadeOut(plane), FadeOut(axes)) + self.wait() diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file2_point_normal_form_plane.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file2_point_normal_form_plane.gif Binary files differnew file mode 100644 index 0000000..e651be0 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file2_point_normal_form_plane.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file2_point_normal_form_plane.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file2_point_normal_form_plane.py new file mode 100644 index 0000000..122a9ff --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file2_point_normal_form_plane.py @@ -0,0 +1,39 @@ +from manimlib.imports import * + +class pointnormal(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + + self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) + normal = Arrow((0,-0.15,-0.25), (-3,0,3), color = YELLOW) + plane1 = Polygon(np.array([1,0,2]),np.array([-1,2.5,1]),np.array([-3,2,1]),np.array([-1,-1,2]), color = GREEN_E, fill_color = WHITE, fill_opacity=0.5) + plane2 = Polygon(np.array([1,0,2]),np.array([-1,2.5,1]),np.array([-3,2,1]),np.array([-1,-1,2]), color = BLUE, fill_color = WHITE, fill_opacity=0.3) + normalLabel = TextMobject(r'$\overrightarrow{n}$').shift((2,2.5,0)) + pointLabel = TextMobject(r'$P$').shift((2,1.2,0)) + xlabel = TextMobject(r'$x$').shift(4.5*LEFT + 1.7*DOWN) + ylabel = TextMobject(r'$y$').shift(4.5*RIGHT + 1.8*DOWN) + zlabel = TextMobject(r'$z$').shift(3.3*UP+0.5*RIGHT) + + normaltext = TextMobject(r'Consider an arbitrary \\ normal vector $\overrightarrow{n}$').scale(0.6).shift(2*UP + 2.5*LEFT) + planetext = TextMobject(r'A single vector is normal \\ to infinitely many planes.').scale(0.6).shift(2*UP + 2.5*LEFT) + pointtext = TextMobject(r'Given a fixed point $P$, \\ a plane is obtained as:').scale(0.6).shift(2*UP + 2.5*LEFT) + + point = Dot(color = RED).shift((1.6,1.3,0)) + self.play(FadeIn(axes)) + self.add_fixed_in_frame_mobjects(xlabel, ylabel, zlabel) + self.wait(1) + self.play(FadeIn(normal)) + self.add_fixed_in_frame_mobjects(normalLabel, normaltext) + self.play(FadeIn(normaltext)) + self.wait(2) + self.add_fixed_in_frame_mobjects(planetext) + self.play(ReplacementTransform(normaltext, planetext), run_time=0.01) + self.play(MoveAlongPath(plane1, normal), run_time = 6) + self.add_fixed_in_frame_mobjects(pointtext) + self.play(ReplacementTransform(planetext, pointtext)) + self.add_fixed_in_frame_mobjects(point, pointLabel) + self.wait(1) + self.play(Transform(plane1, plane2)) + self.wait(2) + self.play(FadeOut(axes), FadeOut(plane2), FadeOut(plane1), FadeOut(point), FadeOut(pointLabel), FadeOut(normal), FadeOut(normalLabel), FadeOut(planetext), FadeOut(pointtext), FadeOut(normaltext), FadeOut(VGroup(*[xlabel, ylabel, zlabel]))) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file3_intercept_form_plane.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file3_intercept_form_plane.gif Binary files differnew file mode 100644 index 0000000..a8b7d75 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file3_intercept_form_plane.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file3_intercept_form_plane.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file3_intercept_form_plane.py new file mode 100644 index 0000000..258ac3c --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file3_intercept_form_plane.py @@ -0,0 +1,29 @@ +from manimlib.imports import * + +class pointnormal(ThreeDScene): + def construct(self): + axes = ThreeDAxes(x_min = 0, y_min = 0, z_min = 0) + self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) + + plane1 = Polygon(np.array([2,-3,2.5]),np.array([-1.45,2,2.5]),np.array([-0.5,4.5,-0.1]),np.array([3.5,-1,-0.2]), fill_color = WHITE, fill_opacity=0.3) + + xlabel = TextMobject(r'$x$').shift(5*LEFT + 1.5*DOWN) + ylabel = TextMobject(r'$y$').shift(5*RIGHT + 1.5*DOWN) + zlabel = TextMobject(r'$z$').shift(3.3*UP + 0.5*LEFT) + + zintercept = Dot().shift(2.5*UP) + zinterceptlabel = TextMobject(r'$(0,0,c\prime)$').shift(2.8*UP + RIGHT).scale(0.7) + + yintercept = Dot().shift(3.7*RIGHT + 0.925*DOWN) + yinterceptlabel = TextMobject(r'$(0,b\prime ,0)$').shift(3.7*RIGHT+1.5*DOWN).scale(0.7) + + xintercept = Dot().shift(2.9*LEFT + 0.75*DOWN) + xinterceptlabel = TextMobject(r'$(a\prime ,0,0)$').shift(3*LEFT+1.3*DOWN).scale(0.7) + + self.play(FadeIn(axes), FadeIn(plane1)) + self.add_fixed_in_frame_mobjects(xlabel, ylabel, zlabel, zintercept, zinterceptlabel, yintercept, yinterceptlabel, xintercept, xinterceptlabel) + self.wait(2) + self.remove(zintercept, zinterceptlabel, yintercept, yinterceptlabel, xintercept, xinterceptlabel, xlabel, ylabel, zlabel) + self.begin_ambient_camera_rotation(rate=0.5) + self.wait(5) + self.play(FadeOut(axes), FadeOut(plane1)) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file4_3d_plane.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file4_3d_plane.gif Binary files differnew file mode 100644 index 0000000..b4c259e --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file4_3d_plane.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file4_3d_plane.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file4_3d_plane.py new file mode 100644 index 0000000..26ad825 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file4_3d_plane.py @@ -0,0 +1,49 @@ +from manimlib.imports import * + +class pointnormal(ThreeDScene): + CONFIG = { + 'x_axis_label': '$x$', + 'y_axis_label': '$y$' + } + def construct(self): + axes = ThreeDAxes() + axes.add(axes.get_axis_labels()) + self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) + + plane = Polygon( + np.array([2,0,2.7]), + np.array([0,0,0.4]), + np.array([-3.2,0,0.55]), + np.array([-3,-2,2.5]), + fill_color = WHITE, fill_opacity = 0.25) + + normal = Arrow((0.25,2,0), (1.5,3.5,0)) + normalLabel = TextMobject(r'$\overrightarrow{n}$').shift((1.5,2.8,0)) + + point = Dot(color = RED).shift((1.6,1.3,0)) + pointLabel = TextMobject(r'$P_{0}$').shift((2,1.2,0)) + + point2 = Dot(color = RED).shift((-0.2,1.8,0)) + point2Label = TextMobject(r'$P$').shift((-0.3,2,0)) + + arrow1 = Arrow((0,-0.25,-0.2), (-2.55,0,1), color = YELLOW).set_stroke(width=3) + arrow2 = Arrow((0,0,-0.25), (0.3,0,2), color = YELLOW).set_stroke(width=3) + res = Arrow((1.8,1.23,0),(-0.35,1.85,0), color = BLUE).set_stroke(width=3) + + arrow1label = TextMobject(r'$\overrightarrow{r_{0}}$').next_to(arrow2, UP).shift(RIGHT + 0.16*DOWN).scale(0.7) + arrow2label = TextMobject(r'$\overrightarrow{r}$').next_to(arrow2, UP).shift(0.7*LEFT).scale(0.7) + reslabel = TextMobject(r'$\overrightarrow{r} - \overrightarrow{r_{0}}$').next_to(arrow2, UP).shift(0.7*RIGHT + 1.2*UP).scale(0.7) + + self.play(FadeIn(axes), FadeIn(plane)) + self.wait(1) + self.add_fixed_in_frame_mobjects(normal, normalLabel) + self.wait(1) + self.add_fixed_in_frame_mobjects(point, pointLabel) + self.add_fixed_in_frame_mobjects(point2, point2Label) + self.play(Write(arrow1), Write(arrow2)) + self.add_fixed_in_frame_mobjects(arrow2label, arrow1label) + self.wait(1) + self.add_fixed_in_frame_mobjects(res, reslabel) + self.play(Write(res), FadeIn(reslabel)) + self.wait(1) + self.play(FadeOut(axes), FadeOut(plane), FadeOut(point), FadeOut(pointLabel), FadeOut(normal), FadeOut(normalLabel), FadeOut(point2), FadeOut(point2Label), FadeOut(arrow1label), FadeOut(arrow2label), FadeOut(reslabel), FadeOut(arrow1), FadeOut(arrow2), FadeOut(res))
\ No newline at end of file diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file5_vector_form_line.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file5_vector_form_line.gif Binary files differnew file mode 100644 index 0000000..b6fdb51 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file5_vector_form_line.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file5_vector_form_line.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file5_vector_form_line.py new file mode 100644 index 0000000..e25c4eb --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/equations-of-planes-and-lines/file5_vector_form_line.py @@ -0,0 +1,47 @@ +from manimlib.imports import * + +class line_(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + xlabel = TextMobject(r'$x$').shift(4.5*LEFT + 1.7*DOWN) + ylabel = TextMobject(r'$y$').shift(4.5*RIGHT + 1.8*DOWN) + zlabel = TextMobject(r'$z$').shift(3.3*UP+0.5*RIGHT) + + self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) + pointLabel = TextMobject(r'$P$').shift((2.28,2.12,0)).scale(0.7) + point = Dot(color = RED).shift((1.95,1.9,0)) + + vlabel = TextMobject(r'$\overrightarrow{v}$').shift((0.5,1.3,0)).scale(0.7) + + inf_text = TextMobject(r'Infinitely many lines pass \\ through a single point.').scale(0.6).shift(2*UP + 2.5*LEFT) + pointtext = TextMobject(r'Given a direction vector $\overrightarrow{v}$, \\ a line is obtained as:').scale(0.6).shift(2*UP + 2.5*LEFT) + + + line = Line((0.7,0.7,0), (2,3,0)).shift(0.06*UP+0.6*RIGHT) + v = Vector((0.8,1,0), color = GREEN_E) + #finalLine = Line((-1.56,0,0.5),(-4,0,2.42), color = YELLOW) + finalLine = Line((1,0.8,0),(3,3,0), color = YELLOW).shift(0.05*LEFT) + self.play(FadeIn(axes)) + self.add_fixed_in_frame_mobjects(zlabel, ylabel, xlabel) + self.wait(1) + self.add_fixed_in_frame_mobjects(point, pointLabel) + self.wait(1) + self.add_fixed_in_frame_mobjects(inf_text) + self.wait(1) + self.add_fixed_in_frame_mobjects(line) + + for i in range(9): + self.play(ApplyMethod(line.rotate, -np.pi/12), run_time = 0.7) + if i == 8: + self.add_fixed_in_frame_mobjects(pointtext) + self.play(ReplacementTransform(inf_text, pointtext)) + self.add_fixed_in_frame_mobjects(v, vlabel) + # if i == 13: + # self.add_fixed_in_frame_mobjects(pointtext) + + self.add_fixed_in_frame_mobjects(finalLine) + self.play(FadeIn(finalLine)) + self.play(Transform(line, finalLine), run_time = 4) + #self.play(FadeOut(line), FadeIn(finalLine)) + self.wait(1.5) + self.play(FadeOut(VGroup(*[axes, xlabel, ylabel, zlabel, finalLine, v, vlabel, point, pointLabel, pointtext, line]))) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/README.md b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/README.md new file mode 100644 index 0000000..8a47a0e --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/README.md @@ -0,0 +1,11 @@ +**file1_parametric_circle..py** <br> + + +**file2_cycloid_manim.py** <br> + + +**file3_brachistochrone.py** <br> + + +**file4_helix_visualization.py** <br> + diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file1_parametric_circle.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file1_parametric_circle.gif Binary files differnew file mode 100644 index 0000000..732b6bb --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file1_parametric_circle.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file1_parametric_circle.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file1_parametric_circle.py new file mode 100644 index 0000000..37d079e --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file1_parametric_circle.py @@ -0,0 +1,81 @@ +from manimlib.imports import * + +class parametricCircle(ThreeDScene, GraphScene): + def construct(self): + self.x_min = -5 + self.y_min = -5 + self.graph_origin = ORIGIN + self.x_max = 5 + self.y_max = 5 + self.x_axis_label = "" + self.y_axis_label = "" + self.x_axis_width = 10 + self.y_axis_height = 10 + self.y_tick_frequency = 1.9 + self.x_tick_frequency = 1.4 + + axes = [] + + # self.setup_axes() + ax = Axes(y_tick_frequency = 1, x_axis_width = 10, y_axis_height = 10, y_min = -5, x_max = 5, y_max = 5, x_tick_frequency = 1, x_axis_label = "", y_axis_label = "", x_min = -5, ) + ax.scale(0.5).shift(3*LEFT) + axes.append(ax) + self.setup_axes() + self.axes.scale(0.3).shift(3*RIGHT + 2*UP) + axes.append(self.axes) + self.setup_axes() + self.axes.scale(0.3).shift(3*RIGHT + 2*DOWN) + axes.append(self.axes) + + axes = VGroup(*axes) + t_value = ValueTracker(-3.14) + t_tex = DecimalNumber(t_value.get_value()).add_updater(lambda v: v.set_value(t_value.get_value())) + t_label = TexMobject("t = ") + group = VGroup(t_tex,t_label).shift(3*DOWN) + t_label.next_to(t_tex,LEFT, buff=0.2,aligned_edge=t_label.get_bottom()) + + asint_text = TextMobject(r'$x = a\sin{t}$').scale(0.7).shift(4*RIGHT + 3*UP) + xlabel1 = TextMobject(r'$x$').shift(3.3*RIGHT + 3.7*UP).scale(0.7) + tlabel1 = TextMobject(r'$t$').shift(4.8*RIGHT + 2*UP).scale(0.7) + up_text = VGroup(*[asint_text, xlabel1, tlabel1]) + asint = ParametricFunction( + lambda t: np.array([ + t, + 2*np.sin(t), + 0 + ]), t_min = -np.pi, t_max = np.pi, color = GREEN_E + ).shift(3*RIGHT + 2*UP).scale(0.4) + + acost_text = TextMobject(r'$y = a\cos{t}$').scale(0.7).shift(4*RIGHT + DOWN) + ylabel1 = TextMobject(r'$y$').shift(3.3*RIGHT+0.3*DOWN).scale(0.7) + tlabel2 = TextMobject(r'$t$').shift(4.8*RIGHT + 2*DOWN).scale(0.7) + down_text = VGroup(*[acost_text, ylabel1, tlabel2]) + acost = ParametricFunction( + lambda t: np.array([ + t, + 2*np.cos(t), + 0 + ]), t_min = -np.pi, t_max = np.pi, color = BLUE + ).shift(3*RIGHT + 2*DOWN).scale(0.4) + + up_dot = Dot(color = RED) + down_dot = Dot(color = RED) + circle_dot = Dot(color = RED) + + ylabel2 = TextMobject(r'$y$').scale(0.7).shift(3*UP + 3*LEFT) + xlabel2 = TextMobject(r'$x$').scale(0.7) + ellipse_text = TextMobject(r'$x = a\sin{t}$ \\ $y = a\cos{t}$').scale(0.7).shift(2*UP + 1.3*LEFT) + main_text = VGroup(*[xlabel2, ylabel2, ellipse_text]) + circle = ParametricFunction( + lambda t: np.array([ + np.cos(t), + np.sin(t), + 0 + ]), t_min = -np.pi, t_max = np.pi, color = WHITE + ).shift(3*LEFT) + self.play(FadeIn(axes), FadeIn(asint), FadeIn(acost), FadeIn(circle), FadeIn(up_text), FadeIn(down_text), FadeIn(main_text), FadeIn(group)) + self.wait(1) + self.play(MoveAlongPath(up_dot, asint, run_time = 7), MoveAlongPath(down_dot, acost, run_time = 7), MoveAlongPath(circle_dot, circle, run_time = 7), t_value.set_value,3.14, rate_func=linear, run_time=7) + self.wait(1) + self.play(FadeOut(VGroup(*[axes, asint, acost, circle, up_text, down_text, main_text, up_dot, down_dot, circle_dot, group]))) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file2_cycloid_manim.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file2_cycloid_manim.gif Binary files differnew file mode 100644 index 0000000..e68b841 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file2_cycloid_manim.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file2_cycloid_manim.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file2_cycloid_manim.py new file mode 100644 index 0000000..7b6c0d1 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file2_cycloid_manim.py @@ -0,0 +1,46 @@ +from manimlib.imports import * + +t_offset = 0 +c_t = 0 + +class cycloid(Scene): + def construct(self): + + cycl = ParametricFunction( + lambda t: np.array([ + t - np.sin(t), + 1 - np.cos(t), + 0 + ]), t_min = -2.75*np.pi, t_max = 3*np.pi, color = BLUE + ).shift(0.73*RIGHT) + wheel_radius = 1 + wheel_function_path = lambda x : 0 + wheel_radius + + line = FunctionGraph(lambda x : 0, color = BLACK) + wheel_path = FunctionGraph(wheel_function_path) + + velocity_factor = 0.25 + frame_rate = self.camera.frame_rate + self.dt = 1 / frame_rate + + wheel = Circle(color = BLACK, radius = 1) + dot = Dot(radius = 0.16, color = RED) + #dot.move_to(wheel.get_arc_center() + np.array([0,2,0])) + + def update_dot(mob,dt): + global t_offset,c_t + if dt == 0 and c_t == 0: + rate= - velocity_factor * self.dt + c_t += 1 + else: + rate = - dt*velocity_factor + if dt > 0: + c_t = 0 + mob.move_to(wheel.point_from_proportion(((t_offset + rate))%1)) + t_offset += rate + #self.add(mob.copy()) + + #dot.move_to(wheel.get_arc_center() + np.array([0,2,0])) + dot.add_updater(update_dot) + self.add(wheel,dot, line, cycl) + self.play(MoveAlongPath(wheel, wheel_path, run_time = 9, rate_func = linear)) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file3_brachistochrone.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file3_brachistochrone.gif Binary files differnew file mode 100644 index 0000000..8daf4c0 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file3_brachistochrone.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file3_brachistochrone.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file3_brachistochrone.py new file mode 100644 index 0000000..633e500 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file3_brachistochrone.py @@ -0,0 +1,13 @@ +from manimlib.imports import * + +class brachistochrone(Scene): + def construct(self): + curve = ParametricFunction( + lambda t: np.array([ + 0.5*(t - np.sin(t)), + 0.5*(1 - np.cos(t)), + 0 + ]), t_max = np.pi + ).scale(5).rotate(540*DEGREES) + dot = Dot(color = RED, radius = 0.2) + self.play(FadeIn(curve), MoveAlongPath(dot, curve, run_time = 2)) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file4_helix_visualization.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file4_helix_visualization.gif Binary files differnew file mode 100644 index 0000000..16d2509 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file4_helix_visualization.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file4_helix_visualization.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file4_helix_visualization.py new file mode 100644 index 0000000..eddd3fe --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/general-parametric-curves/file4_helix_visualization.py @@ -0,0 +1,31 @@ +from manimlib.imports import * + +class helix_(ThreeDScene): + CONFIG = { + "x_min": -6, + "x_max": 6, + "y_min": -6, + "y_max": 6, + "graph_origin": ORIGIN + } + def construct(self): + axes = ThreeDAxes() + helix = ParametricFunction( + lambda t: np.array([ + 1.5*np.cos(TAU*t), + 1.5*np.sin(TAU*t), + 2*t + ]), t_min = -1, t_max = 2, color = RED + ) + cylinder = ParametricSurface( + lambda u, v: np.array([ + 1.5*np.cos(TAU*v), + 1.5*np.sin(TAU*v), + 2*u + ]), u_min = -1, u_max = 2, fill_opacity = -.4, fill_color = WHITE, color = WHITE + ) + self.set_camera_orientation(phi=60* DEGREES,theta=45*DEGREES) + self.play(FadeIn(axes), FadeIn(cylinder), ShowCreation(helix, run_time = 4)) + self.begin_ambient_camera_rotation(rate=0.5) + self.wait(5) + self.play(FadeOut(axes),FadeOut(helix), FadeOut(cylinder)) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/README.md b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/README.md new file mode 100644 index 0000000..42f5df1 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/README.md @@ -0,0 +1,11 @@ +**file1_parametric_ellipse.py** <br> + + +**file2_parametric_helix.py** <br> + + +**file3_circletosphere.py** <br> + + +**file4_cone.py** <br> + diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file1_parametric_ellipse.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file1_parametric_ellipse.gif Binary files differnew file mode 100644 index 0000000..90c0349 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file1_parametric_ellipse.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file1_parametric_ellipse.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file1_parametric_ellipse.py new file mode 100644 index 0000000..1ce29d7 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file1_parametric_ellipse.py @@ -0,0 +1,78 @@ +from manimlib.imports import * + +class parametricEllipse(ThreeDScene, GraphScene): + def construct(self): + self.x_min = -5 + self.y_min = -5 + self.graph_origin = ORIGIN + self.x_max = 5 + self.y_max = 5 + self.x_axis_label = "" + self.y_axis_label = "" + self.x_axis_width = 10 + self.y_axis_height = 10 + + axes = [] + + self.setup_axes() + self.axes.scale(0.5).shift(3*LEFT) + axes.append(self.axes) + self.setup_axes() + self.axes.scale(0.3).shift(3*RIGHT + 2*UP) + axes.append(self.axes) + self.setup_axes() + self.axes.scale(0.3).shift(3*RIGHT + 2*DOWN) + axes.append(self.axes) + + axes = VGroup(*axes) + t_value = ValueTracker(-3.14) + t_tex = DecimalNumber(t_value.get_value()).add_updater(lambda v: v.set_value(t_value.get_value())) + t_label = TexMobject("t = ") + group = VGroup(t_tex,t_label).shift(3*DOWN) + t_label.next_to(t_tex,LEFT, buff=0.2,aligned_edge=t_label.get_bottom()) + + asint_text = TextMobject(r'$x = a\sin{t}$').scale(0.7).shift(4*RIGHT + 3*UP) + xlabel1 = TextMobject(r'$x$').shift(3.3*RIGHT + 3.7*UP).scale(0.7) + tlabel1 = TextMobject(r'$t$').shift(4.8*RIGHT + 2*UP).scale(0.7) + up_text = VGroup(*[asint_text, xlabel1, tlabel1]) + asint = ParametricFunction( + lambda t: np.array([ + t, + np.sin(t), + 0 + ]), t_min = -np.pi, t_max = np.pi, color = GREEN_E + ).shift(3*RIGHT + 2*UP).scale(0.4) + + bcost_text = TextMobject(r'$y = b\cos{t}$').scale(0.7).shift(4*RIGHT + DOWN) + ylabel1 = TextMobject(r'$y$').shift(3.3*RIGHT+0.3*DOWN).scale(0.7) + tlabel2 = TextMobject(r'$t$').shift(4.8*RIGHT + 2*DOWN).scale(0.7) + down_text = VGroup(*[bcost_text, ylabel1, tlabel2]) + bcost = ParametricFunction( + lambda t: np.array([ + t, + 1.5*np.cos(t), + 0 + ]), t_min = -np.pi, t_max = np.pi, color = BLUE + ).shift(3*RIGHT + 2*DOWN).scale(0.4) + + up_dot = Dot(color = RED) + down_dot = Dot(color = RED) + ellipse_dot = Dot(color = RED) + + ylabel2 = TextMobject(r'$y$').scale(0.7).shift(3*UP + 3*LEFT) + xlabel2 = TextMobject(r'$x$').scale(0.7) + ellipse_text = TextMobject(r'$x = a\sin{t}$ \\ $y = b\cos{t}$').scale(0.7).shift(2*UP + 1.3*LEFT) + main_text = VGroup(*[xlabel2, ylabel2, ellipse_text]) + ellipse = ParametricFunction( + lambda t: np.array([ + 1.5*np.cos(t), + np.sin(t), + 0 + ]), t_min = -np.pi, t_max = np.pi, color = WHITE + ).shift(3*LEFT) + self.play(FadeIn(axes), FadeIn(asint), FadeIn(bcost), FadeIn(ellipse), FadeIn(up_text), FadeIn(down_text), FadeIn(main_text), FadeIn(group)) + self.wait(1) + self.play(MoveAlongPath(up_dot, asint, run_time = 7), MoveAlongPath(down_dot, bcost, run_time = 7), MoveAlongPath(ellipse_dot, ellipse, run_time = 7), t_value.set_value,3.14, rate_func=linear, run_time=7) + self.wait(1) + self.play(FadeOut(VGroup(*[axes, asint, bcost, ellipse, up_text, down_text, main_text, up_dot, down_dot, ellipse_dot, group]))) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file2_parametric_helix.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file2_parametric_helix.gif Binary files differnew file mode 100644 index 0000000..4f349b1 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file2_parametric_helix.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file2_parametric_helix.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file2_parametric_helix.py new file mode 100644 index 0000000..3791752 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file2_parametric_helix.py @@ -0,0 +1,91 @@ +from manimlib.imports import * + +class parametricHelix(ThreeDScene, GraphScene): + def construct(self): + self.x_min = -5 + self.y_min = -4 + self.graph_origin = ORIGIN + self.x_max = 5 + self.y_max = 4 + self.x_axis_label = "" + self.y_axis_label = "" + self.x_axis_width = 10 + self.y_axis_height = 7.5 + ax1 = ThreeDAxes().scale(0.65).shift(2.6*RIGHT+DOWN+np.array([0,0,0.5])) + axes_group = [] + + self.setup_axes() + self.axes.shift(3*RIGHT + 2*UP).scale(0.3) + axes_group.append(self.axes) + + self.setup_axes() + self.axes.shift(3*RIGHT + 2*DOWN).scale(0.3) + axes_group.append(self.axes) + + axes_group = VGroup(*axes_group) + + asint_text = TextMobject(r'$x = a\sin{t}$').scale(0.7).shift(4*RIGHT + 3*UP) + xlabel1 = TextMobject(r'$x$').shift(3.3*RIGHT + 3.7*UP).scale(0.7) + tlabel1 = TextMobject(r'$t$').shift(5*RIGHT + 2*UP).scale(0.7) + up_text = VGroup(*[asint_text, xlabel1, tlabel1]) + asint = ParametricFunction( + lambda t: np.array([ + t, + np.sin(t), + 0 + ]), t_min = -4*np.pi, t_max = 4*np.pi, color = GREEN_E + ).shift(3*RIGHT + 2*UP).scale(0.12) + + acost_text = TextMobject(r'$y = a\cos{t}$').scale(0.7).shift(4*RIGHT + DOWN) + ylabel1 = TextMobject(r'$y$').shift(3.3*RIGHT+0.3*DOWN).scale(0.7) + tlabel2 = TextMobject(r'$t$').shift(5*RIGHT + 2*DOWN).scale(0.7) + down_text = VGroup(*[acost_text, ylabel1, tlabel2]) + acost = ParametricFunction( + lambda t: np.array([ + t, + np.cos(t), + 0 + ]), t_min = -4*np.pi, t_max = 4*np.pi, color = BLUE + ).shift(3*RIGHT + 2*DOWN).scale(0.12) + + up_dot = Dot(color = RED).scale(0.6) + down_dot = Dot(color = RED).scale(0.6) + helix_dot = Dot(radius = 0.16, color = RED) + + zlabel = TextMobject(r'$z$').scale(0.7).shift(3*UP + 2.8*LEFT) + ylabel2 = TextMobject(r'$y$').scale(0.7).shift(0.3*DOWN+0.15*RIGHT) + xlabel2 = TextMobject(r'$x$').scale(0.7).shift(0.5*DOWN + 6.4*LEFT) + helix_text = TextMobject(r'$x = a\sin{t}$ \\ $y = a\cos{t}$ \\ $z = ct$').scale(0.7).shift(2.3*UP + 1.3*LEFT) + main_text = VGroup(*[xlabel2, ylabel2, zlabel, helix_text]) + helix = ParametricFunction( + lambda t: np.array([ + np.cos(TAU*t), + np.sin(TAU*t), + 0.4*t + ]), t_min = -2*np.pi/3, t_max = 1.8*np.pi/3, color = WHITE + ).shift(ax1.get_center()) + + self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) + + t_tracker = ValueTracker(-12.56) + t=t_tracker.get_value + + t_label = TexMobject( + "t = ",color=WHITE + ).next_to(helix_text,DOWN, buff=0.2).scale(0.6) + + t_text = always_redraw( + lambda: DecimalNumber( + t(), + color=WHITE, + ).next_to(t_label, RIGHT, buff=0.2) + ).scale(0.6) + + group = VGroup(t_text,t_label).scale(1.5).move_to(ORIGIN).shift(2*DOWN) + self.add_fixed_in_frame_mobjects(axes_group, main_text, up_text, down_text, acost, asint) + self.play(FadeIn(ax1), FadeIn(axes_group), FadeIn(asint), FadeIn(acost), FadeIn(helix), FadeIn(up_text), FadeIn(down_text), FadeIn(main_text)) + #self.begin_ambient_camera_rotation(rate = 0.06) + self.add_fixed_in_frame_mobjects(up_dot, down_dot, group) + self.play(MoveAlongPath(up_dot, asint, run_time = 8), MoveAlongPath(down_dot, acost, run_time = 8), MoveAlongPath(helix_dot, helix, run_time = 8), t_tracker.set_value,12.56, rate_func=linear, run_time=8) + self.play(FadeOut(VGroup(*[ax1, axes_group, asint, acost, helix, up_text, down_text, main_text, up_dot, down_dot, helix_dot, group]))) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file3_circletosphere.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file3_circletosphere.gif Binary files differnew file mode 100644 index 0000000..d6a8afc --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file3_circletosphere.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file3_circletosphere.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file3_circletosphere.py new file mode 100644 index 0000000..6c0e810 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file3_circletosphere.py @@ -0,0 +1,45 @@ +from manimlib.imports import * + +class sphere(GraphScene, ThreeDScene): + CONFIG = { + 'x_min': -10, + 'x_max': 10, + 'y_min': -10, + 'y_max': 10, + 'graph_origin': ORIGIN, + "x_axis_width": 10, + "y_axis_height": 10, + } + def construct(self): + XTD = self.x_axis_width/(self.x_max- self.x_min) + YTD = self.y_axis_height/(self.y_max- self.y_min) + circleeqn = TextMobject(r'Hence, $x^{2} + y^{2} = 2(r^{2} - u^{2})$') + plottext = TextMobject(r'$x = \sqrt{r^{2} - u^{2}}cos\theta$ \\ $y = \sqrt{r^{2} - u^{2}}sin\theta$').shift(2*UP + 3*RIGHT) + + + self.setup_axes() + self.play(FadeIn(self.axes), FadeIn(plottext)) + + dots = [] + for t in range(19): + dot = Dot().shift((3*XTD*np.cos(t), 3*YTD*np.sin(t),0)) + dots.append(dot) + self.play(FadeIn(dot), run_time = 0.2) + dots = VGroup(*dots) + circle = Circle(radius = 3*XTD).set_color(WHITE).set_stroke(width = 10) + self.play(FadeIn(circle), FadeOut(dots), FadeOut(plottext)) + self.wait(2) + + + axes = ThreeDAxes(**self.CONFIG) + sph = Sphere(radius = 3).scale(0.5) + text2 = TextMobject(r'Setting $u = 3$,\\$z = u$').shift(4*YTD*UP + 5*XTD*RIGHT) + + self.play(Transform(self.axes,axes), ReplacementTransform(circle, sph)) + self.add(text2) + self.wait(2) + self.remove(text2) + self.move_camera(phi = 60*DEGREES, theta=45*DEGREES,run_time=5) + self.begin_ambient_camera_rotation(rate=0.03) + self.play(FadeOut(axes), FadeOut(sph), FadeOut(self.axes)) + self.wait(1) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file4_cone.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file4_cone.gif Binary files differnew file mode 100644 index 0000000..b126d20 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file4_cone.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file4_cone.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file4_cone.py new file mode 100644 index 0000000..e6ae1c6 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/space-curves/file4_cone.py @@ -0,0 +1,33 @@ +from manimlib.imports import * + +class cone(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + eqn = TextMobject(r'$z^{2} = x^{2} + y^{2}$') + + conecurve = ParametricFunction( + lambda t: np.array([ + t*np.cos(TAU*t), + t*np.sin(TAU*t), + t + ]), t_min = -2.6, t_max = 2.6 + ).scale(0.85) + + conesurface = ParametricSurface( + lambda u,v: np.array([ + 3*np.sin(u)*np.cos(TAU*v), + 3*np.sin(u)*np.sin(TAU*v), + 2.7*u + ]), u_min = -1 + ).scale(0.85) + + self.play(FadeIn(eqn)) + self.wait(2) + self.play(FadeOut(eqn)) + self.set_camera_orientation(phi = 75*DEGREES, theta=50*DEGREES) + self.play(FadeIn(axes), ShowCreation(conecurve, run_time = 3)) + self.play(FadeOut(conecurve), FadeIn(conesurface)) + self.begin_ambient_camera_rotation(rate=0.03) + self.wait(2) + self.play(FadeOut(axes), FadeOut(conesurface)) + self.wait(2) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/README.md b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/README.md new file mode 100644 index 0000000..7874f43 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/README.md @@ -0,0 +1,15 @@ +**file1_tnb_creation.py**<br> + + + +**file2_tnb_basic.py** <br> + + +**file3_tnb_frame_manim.py** <br> + + +**file4_fs1.py** <br> + + +**file5_fs2.py** <br> + diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file1_tnb_creation.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file1_tnb_creation.gif Binary files differnew file mode 100644 index 0000000..eae8686 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file1_tnb_creation.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file1_tnb_creation.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file1_tnb_creation.py new file mode 100644 index 0000000..80372ee --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file1_tnb_creation.py @@ -0,0 +1,66 @@ +from manimlib.imports import * + +class tnb(ThreeDScene): + def construct(self): + self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) + + helix1 = ParametricFunction( + lambda t: np.array([ + np.cos(TAU*t), + np.sin(TAU*t), + 0.4*t + ]), t_min = -2*np.pi/3, t_max = -1.638*np.pi/3, color = WHITE + ) + + helix2 = ParametricFunction( + lambda t: np.array([ + np.cos(TAU*t), + np.sin(TAU*t), + 0.4*t + ]), t_min = -1.638*np.pi/3, t_max = -1.33*np.pi/3, color = WHITE + ) + + pointText = TextMobject(r'Consider an arbitrary point \\ on the given curve.').scale(0.8).shift(1.5*UP) + tgtText = TextMobject(r'Unit', ' tangent ', r'vector at \\ this point is given as:').scale(0.8).shift(1.5*UP) + tgtText.set_color_by_tex_to_color_map({ + "tangent": YELLOW + }) + normalText = TextMobject(r'Unit', ' normal ', r'vector at \\ this point is given as:').scale(0.8).shift(1.5*UP) + normalText.set_color_by_tex_to_color_map({ + "normal": BLUE + }) + planeText = TextMobject(r'$\overrightarrow{T}$ and $\overrightarrow{N}$ \\ prescribe a plane.').scale(0.8).shift(1.5*UP) + bnmText = TextMobject(r'The vector normal to this plane \\ is called the', ' binormal ', 'vector.').scale(0.8).shift(1.5*UP) + bnmText.set_color_by_tex_to_color_map({ + "binormal": GREEN_E + }) + + 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])) + 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, nm1, plane1]).scale(0.8).shift(np.array([1,4.86,0])).rotate(-15*DEGREES) + + + + helix = VGroup(*[helix1, helix2]) + self.play(FadeIn(helix)) + self.play(ApplyMethod(helix.scale, 4)) + self.add_fixed_in_frame_mobjects(pointText) + self.play(FadeIn(dot1), FadeIn(pointText)) + self.wait(2) + self.add_fixed_in_frame_mobjects(tgtText) + self.play(Write(tgt1), ReplacementTransform(pointText, tgtText)) + self.wait(2) + self.add_fixed_in_frame_mobjects(normalText) + self.play(Write(nm1), ReplacementTransform(tgtText, normalText)) + self.wait(2) + self.add_fixed_in_frame_mobjects(planeText) + self.play(FadeIn(plane1), ReplacementTransform(normalText, planeText)) + self.wait(2) + self.add_fixed_in_frame_mobjects(bnmText) + self.add_fixed_in_frame_mobjects(bnm1) + self.play(ReplacementTransform(planeText, bnmText), Write(bnm1)) + self.wait(2) + self.play(FadeOut(VGroup(*[helix, bnm1, bnmText, dot1, tgt1, nm1, plane1]))) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file2_tnb_basic.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file2_tnb_basic.gif Binary files differnew file mode 100644 index 0000000..67aaea2 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file2_tnb_basic.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file2_tnb_basic.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file2_tnb_basic.py new file mode 100644 index 0000000..c870210 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file2_tnb_basic.py @@ -0,0 +1,36 @@ +from manimlib.imports import * + +class tnb(ThreeDScene): + def construct(self): + t = TextMobject(r'T', color = YELLOW) + n = TextMobject(r'N', color = BLUE).next_to(t, RIGHT, buff=0) + b = TextMobject(r'B', color = GREEN_E).next_to(n, RIGHT, buff=0) + frame = TextMobject(r'Frame').next_to(b, RIGHT, buff=0.2) + f1 = TextMobject(r'$\overrightarrow{B}$ ', color = GREEN_E) + f2 = TextMobject(r' = $\overrightarrow{T}$', color = YELLOW).next_to(f1, RIGHT, buff=0.2) + f3 = TextMobject(r'$\times\overrightarrow{N}$', color = BLUE).next_to(f2, RIGHT, buff=0.1) + formula = VGroup(*[f1, f2, f3]).move_to(ORIGIN).shift(3*UP) + + # text = VGroup(*[t,n,b,frame]).move_to(ORIGIN).shift(3*UP) + curve = ParametricFunction( + lambda t: np.array([ + np.sin(TAU*t), + np.cos(TAU*t), + 0 + ]) + ).scale(2.5) + dot = Dot(color = RED).scale(1.5).shift(1.05*LEFT) + tgt = Arrow(dot.get_center(), (-2, 2, 0), color = YELLOW).shift(0.3*DOWN + 0.09*RIGHT) + normal = Arrow(tgt.get_start(), (1, 1, 0), color = BLUE).shift(0.2*LEFT + 0.05*DOWN) + binormal = Arrow(dot.get_center() - np.array([0,0,0.3]), (tgt.get_start()[0], tgt.get_start()[1],2), color = GREEN) + square = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).move_to(tgt.get_start()).rotate(27*DEGREES).shift(UP+0.4*RIGHT).scale(1.2) + group = VGroup(*[dot, tgt, normal, square, binormal]).shift(np.array([-1.24,-1,0])) + + self.add_fixed_in_frame_mobjects(formula) + self.add(curve, group) + self.wait(1) + self.move_camera(phi = 75*DEGREES, theta=45*DEGREES, run_time = 2) + self.add_fixed_in_frame_mobjects(formula) + self.begin_ambient_camera_rotation(rate = 0.5) + self.wait(5) + self.play(FadeOut(VGroup(*[formula, curve, dot, tgt, normal, square, binormal]))) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file3_tnb_frame_manim.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file3_tnb_frame_manim.gif Binary files differnew file mode 100644 index 0000000..78e3aa3 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file3_tnb_frame_manim.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file3_tnb_frame_manim.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file3_tnb_frame_manim.py new file mode 100644 index 0000000..091c1e2 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file3_tnb_frame_manim.py @@ -0,0 +1,218 @@ +from manimlib.imports import * + +class tnb(ThreeDScene): + def construct(self): + self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES) + + t = TextMobject(r'T', color = YELLOW) + n = TextMobject(r'N', color = BLUE).next_to(t, RIGHT, buff=0) + b = TextMobject(r'B', color = GREEN_E).next_to(n, RIGHT, buff=0) + frame = TextMobject(r'Frame').next_to(b, RIGHT, buff=0.2) + + text = VGroup(*[t,n,b,frame]).move_to(ORIGIN).shift(3*UP) + + c1 = TextMobject(r'$r(t) = \left\langle\cos{t}, \sin{t}, 0.4t\right\rangle\quad r\prime (t) =\left\langle -\sin{t}, \cos{t}, 0.4\right\rangle$').next_to(text, DOWN, buff = 0.1).scale(0.7) + + + helix1 = ParametricFunction( + lambda t: np.array([ + np.cos(TAU*t), + np.sin(TAU*t), + 0.4*t + ]), t_min = -2*np.pi/3, t_max = -1.638*np.pi/3, color = WHITE + ) + + helix2 = ParametricFunction( + lambda t: np.array([ + np.cos(TAU*t), + np.sin(TAU*t), + 0.4*t + ]), t_min = -1.638*np.pi/3, t_max = -1.33*np.pi/3, color = WHITE + ) + + helix3 = ParametricFunction( + lambda t: np.array([ + np.cos(TAU*t), + np.sin(TAU*t), + 0.4*t + ]), t_min = -1.33*np.pi/3, t_max = -np.pi/3, color = WHITE + ) + + helix4 = ParametricFunction( + lambda t: np.array([ + np.cos(TAU*t), + np.sin(TAU*t), + 0.4*t + ]), t_min = -np.pi/3, t_max = -1.3*np.pi/6, color = WHITE + ) + + helix5 = ParametricFunction( + lambda t: np.array([ + np.cos(TAU*t), + np.sin(TAU*t), + 0.4*t + ]), t_min = -1.3*np.pi/6, t_max = 0, color = WHITE + ) + + helix_dot = Dot(radius = 0.16, color = RED) + + t_tracker = ValueTracker(-2*np.pi/3) + t=t_tracker.get_value + + # t_label = TexMobject( + # "t = ",color=WHITE + # ).next_to(helix1,DOWN, buff=0.2).scale(0.6) + + cval1 = TextMobject(r'r(').next_to(c1, DOWN+16.5*LEFT, buff = 0.1).scale(0.7) + + t_text = always_redraw( + lambda: DecimalNumber( + t(), + color=WHITE, + ).next_to(cval1, RIGHT, buff=0.05).scale(0.7) + ).scale(0.6) + + + cval2 = always_redraw( + lambda: TextMobject(r') = $\left\langle$').scale(0.7).next_to(t_text, RIGHT, buff = 0.05) + ) + + cos = always_redraw( + lambda: DecimalNumber( + np.cos(t()), + color=WHITE, + ).next_to(cval2, RIGHT, buff=0.1).scale(0.7) + ).scale(0.6) + + sin = always_redraw( + lambda: DecimalNumber( + np.sin(t()), + color=WHITE, + ).next_to(cos, RIGHT, buff=0.1).scale(0.7) + ).scale(0.6) + + zpart = always_redraw( + lambda: DecimalNumber( + 0.4* t(), + color=WHITE, + ).next_to(sin, RIGHT, buff=0.1).scale(0.7) + ).scale(0.6) + + cvalend = always_redraw( + lambda: TextMobject(r' $\right\rangle$').next_to(zpart, RIGHT, buff = 0.2).scale(0.7) + ).scale(0.6) + + + valgroup = VGroup(*[cval1, cval2,cos,sin,zpart, cvalend]) + + rp1 = always_redraw( + lambda: TextMobject(r'$r\prime ($').scale(0.7).next_to(cvalend, RIGHT, buff = 0.6) + ) + + t_text2 = always_redraw( + lambda: DecimalNumber( + t(), + color=WHITE, + ).next_to(rp1, RIGHT, buff=0.05).scale(0.7) + ).scale(0.6) + + rp2 = always_redraw( + lambda: TextMobject(r') = $\left\langle$').scale(0.7).next_to(t_text2, RIGHT, buff = 0.05) + ) + + rps = always_redraw( + lambda: DecimalNumber( + -np.sin(t()), + color=WHITE, + ).next_to(rp2, RIGHT, buff=0.1).scale(0.7) + ).scale(0.6) + + + rpc = always_redraw( + lambda: DecimalNumber( + np.cos(t()), + color=WHITE, + ).next_to(rps, RIGHT, buff=0.1).scale(0.7) + ).scale(0.6) + + + const = always_redraw( + lambda: TextMobject(r'0.4 $\right\rangle$').next_to(rpc, RIGHT, buff = 0.2).scale(0.7) + ).scale(0.6).shift(0.1*DOWN) + + val2group = VGroup(*[rp1, rp2, rps, rpc, const]) + + #group = VGroup(t_text, t_text2).scale(1.5).move_to(ORIGIN).shift(3.7*DOWN) + + + dot0 = Dot(np.array([np.cos(-2*np.pi/3), np.sin(-2*np.pi/3), -0.8*np.pi/3]), radius = 0.16, color=RED).shift(np.array([4.65,0,-0.8])) + tgt0 = Arrow((0,0,0), (1,2,0), color = YELLOW).shift(dot0.get_center() - np.array([0.04,0.2,0])) + nm0 = Arrow((0,0,0), (-2,1,0), color = BLUE).shift(dot0.get_center() + np.array([0.3,0,0])) + bnm0 = Arrow((0,0,0), (0,2,0), color = GREEN_E).shift(6.1*LEFT + 3*DOWN) + plane0 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(dot0.get_center() + np.array([-0.35, 0.85, 0])).scale(1.2).rotate(65*DEGREES) + point0 = VGroup(*[dot0, tgt0, nm0, bnm0, plane0]).scale(0.8).shift(np.array([1,0,0])) + + 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])) + bnm1 = Arrow((0,0,0), (0,2,0), color=GREEN_E).shift(3.68*RIGHT+2.48*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, nm1, plane1]).scale(0.8).shift(np.array([1,6.25,0])) + + dot2 = Dot(np.array([np.cos(-np.pi/6), np.sin(-np.pi/6), -0.2*np.pi/3]) - np.array([1.9,0,0]), radius=0.16,color=RED) + tgt2 = Arrow((0,0,0), (1,-2,0), color = YELLOW).shift(dot2.get_center() + np.array([-0.2,0.2,0])) + nm2 = Arrow((0,0,0), (2,1,0), color = BLUE).shift(dot2.get_center() + np.array([-0.2,-0.06,0])) + bnm2 = Arrow((0,0,0), (0,2,0), color=GREEN_E).shift(0.4*RIGHT + 0.16*DOWN) + plane2 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(dot2.get_center() + np.array([0.92, -0.5, 0])).rotate(23*DEGREES).scale(1.2) + point2 = VGroup(*[dot2, tgt2, nm2, bnm2, plane2]) + + helix = VGroup(*[helix1, helix2, helix3, helix4, helix5]) + self.add_fixed_in_frame_mobjects(text, c1) + self.play(FadeIn(helix), FadeIn(text), FadeIn(c1)) + self.play(ApplyMethod(helix.scale, 4)) + self.add_fixed_in_frame_mobjects(bnm0, valgroup, val2group, t_text, t_text2) + self.play(FadeIn(point0), FadeIn(t_text), FadeIn(t_text2), FadeIn(valgroup), FadeIn(val2group)) + self.play(ApplyMethod(point0.set_color, GRAY, opacity = 0.1, run_time = 0.5), MoveAlongPath(helix_dot, helix1, run_time=5), t_tracker.set_value,-1.638*np.pi/3, rate_func=linear, run_time=5) + + self.add_fixed_in_frame_mobjects(bnm1) + self.play(FadeIn(point1)) + self.play(ApplyMethod(point1.set_color, GRAY, opacity = 0.1, run_time = 0.5), ApplyMethod(bnm1.set_color, GRAY, opacity = 0.1, run_time = 0.5), MoveAlongPath(helix_dot, helix2, run_time = 5), t_tracker.set_value,-1.33*np.pi/3, rate_func=linear, run_time=5) + + self.add_fixed_in_frame_mobjects(bnm2) + self.play(FadeIn(point2)) + self.play(ApplyMethod(point2.set_color, GRAY, opacity = 0.1, run_time = 0.5), MoveAlongPath(helix_dot, helix3, run_time=5), t_tracker.set_value,-np.pi/3, rate_func=linear, run_time=5) + + dot3 = Dot(np.array([np.cos(-np.pi/3), np.sin(-np.pi/3), -0.4*np.pi/3]) + np.array([3.3,-0.25,0]), radius = 0.16, color=RED) + tgt3 = Arrow((0,0,0), (0,2,0), color = YELLOW).shift(helix_dot.get_center() - np.array([-0.05,0.2,0])) + nm3 = Arrow((0,0,0), (-2,0,0), color = BLUE).shift(helix_dot.get_center() + np.array([0.25,0,0])) + bnm3 = Arrow((0,0,0), (0,2,0), color = GREEN_E).shift(3.87*LEFT + 1.24*DOWN) + plane3 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(helix_dot.get_center() + np.array([-0.5, 0.62, 0])) + point3 = VGroup(*[dot3, tgt3, nm3, bnm3, plane3]).shift(np.array([0,0,0])) + + dot4 = Dot(np.array([np.cos(-np.pi/12), np.sin(-np.pi/12), -0.1*np.pi/3]) + np.array([-3.4,3.4,0]), radius = 0.16, color=RED) + tgt4 = Arrow((0,0,0), (-2,-0.85,0), color = YELLOW).shift(dot4.get_center() - np.array([-0.05,0,0])) + nm4 = Arrow((0,0,0), (0.8,-2,0), color = BLUE).shift(dot4.get_center() + np.array([-0.1,0.25,0])) + bnm4 = Arrow((0,0,0), (0,2,0), color = GREEN_E).shift(4.03*RIGHT + 0.5*DOWN) + plane4 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(dot4.get_center() + np.array([-0.4,-1,0])).rotate(22*DEGREES).scale(1.2) + point4 = VGroup(*[dot4, tgt4, nm4, bnm4, plane4]) + + dot5 = Dot((1,0,0) + np.array([2.3,-1,1])) + tgt5 = Arrow((0,0,0), (0,2,0), color = YELLOW).shift(dot5.get_center() - np.array([-0.05,0.2,0])) + nm5 = Arrow((0,0,0), (-2,0,0), color = BLUE).shift(dot5.get_center() + np.array([0.25,0,0])) + bnm5 = Arrow((0,0,0), (0,2,0), color = GREEN_E).shift(3.34*LEFT+0.3*UP) + plane5 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(dot5.get_center() + np.array([-0.5,0.5,0])) + point5 = VGroup(*[tgt5, nm5, bnm5, plane5]) + + self.add_fixed_in_frame_mobjects(bnm3) + self.play(FadeIn(point3)) + self.play(ApplyMethod(point3.set_color, GRAY, opacity = 0.1, run_time = 0.5), MoveAlongPath(helix_dot, helix4, run_time=5), t_tracker.set_value,-1.3*np.pi/6, rate_func=linear, run_time=5) + + self.add_fixed_in_frame_mobjects(bnm4) + self.play(FadeIn(point4)) + self.play(ApplyMethod(point4.set_color, GRAY, opacity = 0.1, run_time = 0.5), MoveAlongPath(helix_dot, helix5, run_time=5), t_tracker.set_value,0, rate_func=linear, run_time=5) + + self.add_fixed_in_frame_mobjects(bnm5) + self.play(FadeIn(point5)) + self.wait(2) + + self.play(FadeOut(VGroup(*[valgroup, val2group, t_text, t_text2, c1, text, helix, bnm1, point0, point1, point2, point3, point4, point5, helix_dot]))) 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 new file mode 100644 index 0000000..f3f5a9c --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py @@ -0,0 +1,91 @@ +from manimlib.imports import * +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) + + 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(*[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_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 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/file5_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 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 diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/README.md b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/README.md new file mode 100644 index 0000000..02678fd --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/README.md @@ -0,0 +1,2 @@ +**file3_tangent_space_curve.py** <br> + diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file1_smooth_curves.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file1_smooth_curves.gif Binary files differnew file mode 100644 index 0000000..5801796 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file1_smooth_curves.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file2_non_differentiable.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file2_non_differentiable.py new file mode 100644 index 0000000..a91da6b --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file2_non_differentiable.py @@ -0,0 +1,36 @@ +from manimlib.imports import * + +class nd(Scene): + def construct(self): + ld1 = Line().rotate(20*DEGREES) + pd1 = Dot(ld1.get_end(), fill_opacity = 0) + pd1.set_stroke(width = 0.5) + ld2 = Line().rotate(40*DEGREES).shift(1.4*UP + 1.7*RIGHT) + pd2 = Dot(ld2.get_start(), fill_opacity = 1, color = PURPLE) + t1 = TextMobject('A discontinuous function.').scale(0.7).shift(UP + 2*RIGHT) + + obj1 = VGroup(*[ld1, pd1, ld2, pd2]).shift(4*LEFT) + self.play(FadeIn(obj1), FadeIn(t1)) + self.wait(2) + + ld3 = ld2.copy().rotate(-60*DEGREES).shift(1.4*DOWN + 0.2*RIGHT) + pd3 = Dot(ld1.get_end(), fill_opacity = 1, color = PURPLE) + t2 = TextMobject('Graph containing a sharp corner.').scale(0.7).shift( 2*RIGHT) + + obj2 = VGroup(*[ld3, pd3]) + + self.play(Transform(VGroup(*[ld2, pd2]), obj2), ReplacementTransform(t1, t2)) + + self.wait(2) + + ld4 = Line().rotate(90*DEGREES) + pd4 = Dot(ld4.get_center(), color = PURPLE) + a1 = Arc(start_angle = -180*DEGREES, angle = 90*DEGREES).move_to(ld4.get_end()).rotate(-90*DEGREES).shift(0.5*(UP+RIGHT)) + a2 = Arc(start_angle = -180*DEGREES, angle = 90*DEGREES).move_to(ld4.get_start()).rotate(90*DEGREES).shift(0.5*(DOWN+LEFT)) + t3 = TextMobject('Graph with a vertical line.').scale(0.7).shift(2*RIGHT) + + obj3 = VGroup(*[ld4, pd4, a1, a2]).shift(3*LEFT) + + self.play(FadeOut(obj1), Transform(obj2, obj3), ReplacementTransform(t2, t3)) + self.wait(2) + self.play(FadeOut(obj2), FadeOut(t3)) diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file3_tangent_space_curve.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file3_tangent_space_curve.gif Binary files differnew file mode 100644 index 0000000..06ed70f --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file3_tangent_space_curve.gif diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file3_tangent_space_curve.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file3_tangent_space_curve.py new file mode 100644 index 0000000..c3aecc6 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/velocity-and-differentiability/file3_tangent_space_curve.py @@ -0,0 +1,33 @@ +from manimlib.imports import * + +class tangent(ThreeDScene): + def construct(self): + axes = ThreeDAxes() + text = TextMobject(r'Tangent', r' to the ', 'space curve', r' \\ at point ', r'$P_{1}$', ' is given by:').scale(0.7).shift(3*UP + 3.5*LEFT) + text.set_color_by_tex_to_color_map({ + "Tangent": YELLOW, + '$P_{1}$': RED, + 'space curve': BLUE + }) + text.bg=BackgroundRectangle(text,fill_opacity=1, color = BLACK) + text_gr =VGroup(text.bg,text) + self.set_camera_orientation(phi = 125*DEGREES, theta = 135*DEGREES) + h = ParametricFunction( + lambda t: np.array([ + 4*(t**3) + 5, + t**2 + 2*(t**4), + -2*np.log(2*t) + ]), t_min = -3, t_max = 1.18, color = BLUE + ).shift(5*LEFT) + tgtR = Line((4,3,-2*np.log(2)), (19.5, 16, -4.772588), color=YELLOW) + tgtL =Line((4,3,-2*np.log(2)), (-11.5, -10, 2), color=YELLOW) + dot = Dot((4,3,-2*np.log(2)), color=RED, radius=0.08) + dotl = TextMobject(r'$P_{1}$', color = RED).scale(0.7).shift(2*DOWN + 5*LEFT) + self.add_fixed_in_frame_mobjects(text_gr, dotl) + self.play(FadeIn(axes),FadeIn(h), FadeIn(dot), FadeIn(dotl)) + self.wait(2) + self.play(FadeIn(tgtL), FadeIn(tgtR)) + self.begin_ambient_camera_rotation(rate=0.2) + self.play(FadeOut(dotl)) + self.wait(5) + self.play(FadeOut(axes), FadeOut(h), FadeOut(text_gr), FadeOut(dot), FadeOut(tgtL), FadeOut(tgtR)) |
