diff options
Diffstat (limited to 'FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature')
11 files changed, 291 insertions, 76 deletions
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/file1_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 differindex 3f7ecd1..3f7ecd1 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_simple_visualization.gif +++ 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/file1_simple_visualization.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.py index 45058d7..05cad80 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_simple_visualization.py +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.py @@ -1,8 +1,8 @@ from manimlib.imports import * -class randomcurve(GraphScene): +class a(GraphScene): CONFIG = { - "x_min": -4, + "x_min": -3, "x_max": 6, "y_min": -6, "y_max": 10, @@ -10,12 +10,15 @@ class randomcurve(GraphScene): } def construct(self): intro = TextMobject('Consider the following curve.') - mid = TextMobject(r'Notice how the direction of the unit tangent vectors\\changes with respect to the arc length.') - outro = TextMobject(r'The rate of change of unit tangents with \\ respect to the arc length $ds$ is called curvature.\\Mathematically, curvature $ = k = \left|{\frac{dT}{ds}}\right|$') + 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|$') XTD = self.x_axis_width/(self.x_max- self.x_min) YTD = self.y_axis_height/(self.y_max- self.y_min) + circle = Circle(radius = 0.95, color = GRAY, fill_opacity = 0.2, fill_color = RED) + circle.set_stroke(width = 0.1) + tgt1 = Arrow((-2.2*XTD,-0.5*YTD,0),(-1*XTD,1,0)) tgt2 = Arrow((-1.2*XTD, 1.93*YTD,0),(0*XTD,1.6,0)).scale(1.2) tgt3 = Arrow((-0.3*XTD,3*YTD, 0), (1.5*XTD, 3*YTD,0)) @@ -40,30 +43,49 @@ class randomcurve(GraphScene): ds_text = TextMobject(r'$ds$').next_to(ds, UP, buff = 0.1).shift(1.3*LEFT) self.setup_axes(hideaxes=True) - graphobj = self.get_graph(self.curve) + + 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(graphobj), FadeIn(dots), FadeIn(ds), FadeIn(ds_text), FadeIn(arc)) - self.wait(1) - self.play(FadeOut(self.axes), FadeOut(arc), FadeOut(graphobj),FadeIn(mid), FadeOut(dots), FadeOut(ds), FadeOut(ds_text)) + 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(graphobj), FadeIn(dots)) - - tangents = [tgt1, tgt2, tgt3, tgt4, tgt5, tgt6, tgt7] - for tangent in tangents: - self.play(ShowCreation(tangent), run_time = 0.2) - self.wait(1) - tangents = VGroup(*tangents) - self.play(FadeOut(self.axes), FadeOut(graphobj), FadeOut(tangents), FadeOut(dots)) - self.wait(1) + 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 curve(self, x): - return 3 - (3653*x**2)/5292 + (2477*x**3)/31752 + (13*x**4)/784 - (17*x**5)/5292 + (17*x**6)/63504 + + + 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/file2_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 differindex 989a3b7..989a3b7 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_circle_curvature.gif +++ 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/file2_circle_curvature.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.py index 232ac41..232ac41 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_circle_curvature.py +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.py diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.py deleted file mode 100644 index d8dd0a4..0000000 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.py +++ /dev/null @@ -1,42 +0,0 @@ -from manimlib.imports import * - -class interpretation(Scene): - def construct(self): - tgt = Vector((1, 2, 0), color = YELLOW) - tgtText = TextMobject(r'$r\prime (t)$').next_to(tgt, UP, buff = 0).scale(0.7) - tgt2 = DashedLine((0,0,0),(1, 2, 0), color = GRAY).shift(DOWN + 2*RIGHT) - - nm = Vector((2, -1, 0), color = BLUE) - nmText = TextMobject(r'$r\prime\prime (t)$').next_to(nm, DOWN+RIGHT, buff = 0).scale(0.7) - nm2 = DashedLine((0,0,0),(2, -1, 0), color = GRAY).shift(2*UP + RIGHT) - square = Square(fill_color = WHITE, fill_opacity = 0.2).rotate(63*DEGREES).shift(0.5*UP +1.5*RIGHT).scale(1.1) - square.set_stroke(width = 0.1) - arrow = CurvedArrow(square.get_center() + np.array([2,1,0]), square.get_center() + np.array([0.5,0,0])) - arrowText = TextMobject(r'$r\prime (t)\times r\prime\prime (t)$').next_to(arrow.get_start(), DOWN+1*RIGHT, buff = 0).scale(0.7) - - text1 = TextMobject(r'$\left|\frac{dT}{ds}\right| = \frac{\left|\frac{dT}{dt}\right|}{\left|\frac{ds}{dt}\right|}$').shift(UP+3*LEFT) - text2 = TextMobject(r'$\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|}$').next_to(text1, DOWN, buff = 0.1) - unit = VGroup(*[tgt, tgt2, nm, nm2]) - - # self.play(FadeIn(VGroup(*[tgt, tgt2, nm, nm2, nmText, tgtText, square, arrow, arrowText]))) - tgt2text = TextMobject(r'$\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(VGroup(*[tgt, tgtText]))) - self.wait(1) - self.play(FadeIn(VGroup(*[nm, nmText]))) - self.wait(1) - self.play(FadeIn(VGroup(*[tgt2, nm2]))) - self.wait(1) - self.play(FadeIn(VGroup(*[square, arrow, arrowText]))) - self.wait(1) - self.play(FadeIn(unit2)) - 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(2) - self.play(FadeOut(VGroup(*[tgt2text, nm2text, text1, text2, 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/file3_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 differindex 22a450a..22a450a 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.gif +++ 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/file4_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 differindex 3b78b5f..3b78b5f 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_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 diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_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 index 56b7fbb..0dc06bb 100644 --- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_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 @@ -17,17 +17,17 @@ class GR(GraphScene): self.setup_axes() 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 + 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=-3, x_max=6, stroke_width = 2, color = BLUE) + graph = FunctionGraph(curve, x_min=-2, x_max=6, stroke_width = 2, color = BLUE) - tracker = ValueTracker(-3) + 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*RIGHT+3*UP).scale(0.6) + 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((4,-1, 0), color = YELLOW) + 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]) @@ -57,22 +57,20 @@ class GR(GraphScene): curvature1 = VGroup(*[circle1, tgt1, r1, r1label]) - circle2 = Circle(radius = 0.6, color = GREY, opacity = 0.2).shift(0.4*DOWN + 4*RIGHT) - tgt2 = Line((4,-2,0), (6, -2, 0), color = GREY, opacity = 0.2).scale(0.5).shift(LEFT + UP) + 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,-0.6,0]), color=GREEN_SCREEN) - r2label = TextMobject(r'$R_{2}$', color=WHITE).next_to(r2, 0.9*RIGHT, buff = 0).scale(0.6) + 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.add(graph, line, dots, text) self.wait(1.2) - self.play(tracker.set_value, 0, rate_func=smooth, run_time=5) - self.play(FadeIn(curvature1)) - self.play(tracker.set_value, 4, rate_func=smooth, run_time=5) - self.play(FadeIn(curvature2)) - self.play(tracker.set_value, 6, rate_func=smooth, run_time=3) + 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() |
