diff options
Diffstat (limited to 'FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.py')
| -rw-r--r-- | FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.py | 123 |
1 files changed, 123 insertions, 0 deletions
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]))) |