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Diffstat (limited to 'FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature')
9 files changed, 227 insertions, 0 deletions
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_simple_visualization.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_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/file1_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/file1_simple_visualization.py new file mode 100644 index 0000000..45058d7 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_simple_visualization.py @@ -0,0 +1,69 @@ +from manimlib.imports import * + +class randomcurve(GraphScene): + CONFIG = { + "x_min": -4, + "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 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|$') + + XTD = self.x_axis_width/(self.x_max- self.x_min) + YTD = self.y_axis_height/(self.y_max- self.y_min) + + tgt1 = Arrow((-2.2*XTD,-0.5*YTD,0),(-1*XTD,1,0)) + tgt2 = Arrow((-1.2*XTD, 1.93*YTD,0),(0*XTD,1.6,0)).scale(1.2) + tgt3 = Arrow((-0.3*XTD,3*YTD, 0), (1.5*XTD, 3*YTD,0)) + tgt4 = Arrow((1.4*XTD, 2*YTD,0),(2.4*XTD, 1*YTD,0)).scale(2.8) + tgt5 = Arrow((2.4*XTD, 0, 0), (3.8*XTD,-2*YTD, 0)).scale(1.2).shift(0.26*RIGHT) + tgt6 = Arrow((3.8*XTD,-1*YTD, 0), (4.8*XTD, -1*YTD, 0)).scale(2.8).shift(0.26*RIGHT) + tgt7 = Arrow((5.3*XTD, 0, 0),(6.3*XTD,1,0)).shift(0.35*LEFT+0.1*DOWN).scale(1.3) + + dot1 = Dot(tgt1.get_start(), color = RED) + dot2 = Dot(tgt2.get_start(), color = RED) + dot3 = Dot(tgt3.get_start(), color = RED) + dot4 = Dot(tgt4.get_start(), color = RED) + dot5 = Dot(tgt5.get_start(), color = RED) + dot6 = Dot(tgt6.get_start(), color = RED) + dot7 = Dot(tgt7.get_start(), color = RED) + + arc = ArcBetweenPoints(dot1.get_center(), dot2.get_center(), color = GREEN_SCREEN, angle = 10*DEGREES).rotate(180*DEGREES) + + dots = VGroup(*[dot1, dot2, dot3, dot4, dot5, dot6, dot7]) + + ds = CurvedArrow((-4, 2, 0), (tgt1.get_start() + tgt2.get_start()) / 2, color = YELLOW) + ds_text = TextMobject(r'$ds$').next_to(ds, UP, buff = 0.1).shift(1.3*LEFT) + + self.setup_axes(hideaxes=True) + graphobj = self.get_graph(self.curve) + 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.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(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 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/file2_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/file2_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/file2_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/file2_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/file3_curvature_interpretation.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_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/file3_curvature_interpretation.gif 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 new file mode 100644 index 0000000..d8dd0a4 --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.py @@ -0,0 +1,42 @@ +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/file4_different_curvature_single_curve.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_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/file4_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/file4_different_curvature_single_curve.py new file mode 100644 index 0000000..56b7fbb --- /dev/null +++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_different_curvature_single_curve.py @@ -0,0 +1,78 @@ +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 + (2477*x**3)/31752 + (13*x**4)/784 - (17*x**5)/5292 + (17*x**6)/63504 + + graph = FunctionGraph(curve, x_min=-3, x_max=6, stroke_width = 2, color = BLUE) + + tracker = ValueTracker(-3) + + 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) + + 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) + 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 = 0.6, 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) + + 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) + + 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, 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(FadeOut(VGroup(*[curvature1, curvature2, graph, self.axes, line, dots, text]))) + self.wait() |
