From 8096534204aeb06ea8e4b7a25c5bf4f7f1d52ffb Mon Sep 17 00:00:00 2001
From: Saarth Deshpande
Date: Wed, 1 Jul 2020 18:49:14 +0530
Subject: arc length video
---
.../arc-length-and-curvature/file1_arc_length.gif | Bin 0 -> 822369 bytes
.../arc-length-and-curvature/file1_arc_length.py | 325 +++++++++++++++++++++
.../file1_simple_visualization.gif | Bin 675451 -> 0 bytes
.../file1_simple_visualization.py | 69 -----
.../file2_circle_curvature.gif | Bin 346667 -> 0 bytes
.../file2_circle_curvature.py | 27 --
.../file2_simple_visualization.gif | Bin 0 -> 675451 bytes
.../file2_simple_visualization.py | 69 +++++
.../file3_circle_curvature.gif | Bin 0 -> 346667 bytes
.../file3_circle_curvature.py | 27 ++
.../file3_curvature_interpretation.gif | Bin 700862 -> 0 bytes
.../file3_curvature_interpretation.py | 42 ---
.../file4_curvature_interpretation.gif | Bin 0 -> 700862 bytes
.../file4_curvature_interpretation.py | 42 +++
.../file4_different_curvature_single_curve.gif | Bin 243868 -> 0 bytes
.../file4_different_curvature_single_curve.py | 78 -----
.../file5_different_curvature_single_curve.gif | Bin 0 -> 243868 bytes
.../file5_different_curvature_single_curve.py | 76 +++++
18 files changed, 539 insertions(+), 216 deletions(-)
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.gif
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.py
delete mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_simple_visualization.gif
delete mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_simple_visualization.py
delete mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_circle_curvature.gif
delete mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_circle_curvature.py
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.gif
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.py
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.gif
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.py
delete mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.gif
delete mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.py
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.gif
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.py
delete mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_different_curvature_single_curve.gif
delete mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_different_curvature_single_curve.py
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.gif
create mode 100644 FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.py
(limited to 'FSF-2020/calculus-of-several-variables')
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
new file mode 100644
index 0000000..bbad112
Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.gif differ
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..75c19aa
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_arc_length.py
@@ -0,0 +1,325 @@
+# Contribution creidts: Somnath Pandit
+
+from manimlib.imports import *
+
+
+class LineIntegrationAsSum(GraphScene):
+ CONFIG = {
+ "x_min" : 0,
+ "x_max" : 10,
+ "y_min" : 0,
+ "y_max" : 6,
+ "graph_origin": ORIGIN+5*LEFT+3*DOWN,
+ "x_axis_width": 10,
+ "y_axis_height": 6 ,
+ "x_tick_frequency": 2,
+ "y_tick_frequency": 2,
+ "Func":lambda x : 1+x**1.3*np.exp(-.12*(x-2)**2)*np.sin(x/4),
+ "a": 1 ,"b": 9, "n": 15,
+ }
+
+ def construct(self):
+ X = RIGHT*self.x_axis_width/(self.x_max- self.x_min)
+ Y = UP*self.y_axis_height/(self.y_max- self.y_min)
+ self.X=X ;self.Y=Y
+
+ self.setup_axes(animate=False)
+
+
+ curve=self.get_graph(
+ self.Func,
+ x_min=self.a,
+ x_max=self.b,
+ )
+ curve.set_color([BLACK,BLUE,BLUE,BLUE,BLACK])
+ curve_label= self.get_graph_label(
+ curve,
+ label="\\text{Curve for integration:}",
+ x_val=4,
+ direction=UR,
+ buff=.6,
+ color=BLUE
+ )
+ self.curve=curve
+ self.curve_label=curve_label
+
+ self.get_vector_field()
+
+
+ self.play(ShowCreation(VGroup(curve,curve_label)))
+ self.wait(.6)
+ self.break_in_arcs()
+ self.show_the_sum()
+
+ self.wait(2)
+
+
+ def get_vector_field(self):
+ func = lambda v: np.array([
+ v[0], # x
+ -v[1], # y
+ 0 # z
+ ])
+ vector_field= VectorField(
+ func,
+ delta_x=1,
+ delta_y=1,
+ colors=[GREEN_A,GREEN_C],
+ length_func= lambda norm: .8*sigmoid(norm),
+ vector_config={
+ "stroke_width": 2
+ }
+ )
+
+ self.vector_field= vector_field
+
+
+ def break_in_arcs(self):
+
+ self.write_about_breaking()
+
+ dl=0.8
+ self.get_breakers(dl)
+ self.wait(2)
+ self.play(FadeOut(self.upto_break_text))
+ self.dl=dl
+
+ def write_about_breaking(self):
+ breaking_text=TextMobject("\\texttt{is broken}"," into small ", "subarcs")
+ breaking_text.set_color_by_tex_to_color_map({
+ "broken":RED,"subarcs": BLUE
+ })
+ breaking_text.next_to(self.curve_label,DOWN)
+ breaking_text.align_to(self.curve_label,LEFT)
+ self.play(
+ Write(breaking_text)
+ )
+
+ self.upto_break_text=VGroup(
+ self.curve_label,
+ breaking_text,
+ )
+
+ def get_breakers(self,dl):
+ point=self.a
+ points=[]
+ while point<(self.b-dl) :
+ start=point
+ end=point+dl
+ points += [end]
+ breaker=Line(
+ self.input_to_graph_point(start,self.curve),
+ self.input_to_graph_point(end,self.curve),
+ stroke_width=2,
+ color=RED,
+ )
+ breaker.rotate(PI/2).scale(.5)
+
+ point=end
+ self.play(FadeIn(breaker),run_time=.2)
+ # self.add(breaker)
+
+ del points[-1]
+ self.points=points
+
+
+ def show_the_sum(self):
+ at_any_points_text=TextMobject("At an arbitrary ","point", " in each ", "subarc")
+ at_any_points_text.set_color_by_tex_to_color_map({
+ "point":YELLOW , "subarc": BLUE
+ })
+ at_any_points_text.to_edge(TOP,buff=SMALL_BUFF)
+
+ evaluate_text=TextMobject("$\\vec F(x,y)$ ", "is evaluated").next_to(at_any_points_text,DOWN)
+ evaluate_text.set_color_by_tex("$\\vec F(x,y)$",ORANGE)
+
+ multiply_text=TextMobject("...is multiplied with ","$\\Delta s_i$")
+ multiply_text.set_color_by_tex("\\Delta s_i", BLUE)
+ multiply_text.next_to(at_any_points_text,DOWN)
+
+
+
+ self.at_any_points_text=at_any_points_text
+ self.evaluate_text=evaluate_text
+ self.multiply_text=multiply_text
+
+ dots=[]
+ for point in self.points:
+
+ dot=Dot(
+ point=self.input_to_graph_point(point,self.curve),
+ radius= .7*DEFAULT_DOT_RADIUS,
+ stroke_width= 0,
+ fill_opacity= 1.0,
+ color= YELLOW,
+ )
+ dots+=[dot]
+
+ self.play(
+ Write(at_any_points_text),
+ FadeIn(VGroup(*dots)),run_time=1.5
+ )
+ self.dots=dots
+
+ self.wait()
+ self.show_the_dot_product()
+ self.multiply_with_ds()
+ self.construct_equation()
+
+
+ def show_the_dot_product(self):
+ index=-(len(self.points)//3)
+ self.index=index
+
+ dot=self.dots[index]
+
+
+ dot_prod_text=TextMobject("Dot product of ", "$\\vec F(x_i,y_i)$", " and ","$\\vec T(x_i,y_i)$")
+ dot_prod_text.set_color_by_tex_to_color_map({
+ "\\vec F(x_i,y_i)":ORANGE ,
+ "\\vec T(x_i,y_i)": "#DC75CD" ,
+ })
+ dot_prod_text.to_edge(TOP,buff=SMALL_BUFF)
+
+
+ point_coord=TextMobject("$(x_i,y_i)$",color=YELLOW)
+ point_coord.next_to(dot,DL,buff=.01).scale(.8)
+
+ func_val=TextMobject("$\\vec F(x_i,y_i)$",color=ORANGE)
+ func_val.next_to(dot,UR).scale(.8)
+
+ self.dot_prod_text=dot_prod_text
+ self.func_val=func_val
+
+ dot.set_color(ORANGE).scale(1.2)
+
+
+ self.play(FadeIn(VGroup(point_coord,dot)))
+ self.play(Write(self.evaluate_text))
+ self.wait(1)
+ self.play(FadeOut(self.vector_field))
+ self.get_vector_and_tangent()
+ self.dot_product()
+
+
+ self.wait(2)
+ self.remove(point_coord)
+
+
+ def get_vector_and_tangent(self):
+ dot=self.dots[self.index]
+ self.show_specific_vectors(dot)
+ self.play(Write(self.func_val))
+ self.wait(1)
+ self.show_tangent(dot)
+ self.play(FadeIn(VGroup(*[
+ dot.set_color(ORANGE).scale(1.4)
+ for dot in self.dots ]
+ )))
+
+
+ def show_specific_vectors(self,dots):
+ for dot in dots:
+ vector=self.vector_field.get_vector(dot.get_center())
+ vector.set_color(ORANGE)
+
+ self.play(Write(vector),run_time=.2)
+
+
+ def show_tangent(self,dot):
+ tangent_sym=TextMobject("$\\vec T(x_i,y_i)$",color="#DC75CD").scale(.8)
+ x=dot.get_center()
+ angle=self.angle_of_tangent(
+ self.point_to_coords(x)[0],
+ self.curve,
+ dx=0.01
+ )
+ vect = Vector().rotate(angle,about_point=x)
+ vect.set_color("#DC75CD")
+ tangent=vect.next_to(x,DR,buff=0)
+ tangent_sym.next_to(tangent,DOWN,buff=.1)
+ self.play(Write(VGroup(tangent,tangent_sym)))
+
+ self.tangent_sym=tangent_sym
+
+ def dot_product(self):
+
+ dot_sym=Dot().next_to(self.func_val,RIGHT)
+
+ self.play(FadeOut(VGroup(
+ self.at_any_points_text,
+ self.evaluate_text
+ )))
+ self.play(Write(self.dot_prod_text))
+ self.play(
+ FadeIn(dot_sym),
+ ApplyMethod(
+ self.tangent_sym.next_to,
+ dot_sym, RIGHT
+ ))
+
+ self.dot_sym=dot_sym
+
+ def multiply_with_ds(self):
+ self.get_ds()
+
+ self.play(GrowFromCenter(self.ds_brace_group))
+ self.wait(2)
+ self.play(Write(self.multiply_text))
+ self.play(ApplyMethod(
+ self.ds_brace_label.next_to,
+ self.tangent_sym, RIGHT,buff=.15
+ ))
+
+
+
+ def get_ds(self):
+ p1= self.dots[self.index]
+ p2= self.dots[self.index+1]
+ ds_brace=Brace(VGroup(p1,p2),DL)
+ ds_brace.move_to(p1,UR)
+ ds_brace_label=ds_brace.get_text("$\Delta s_i$", buff = .05)
+ ds_brace_label.set_color(BLUE)
+ self.ds_brace=ds_brace
+ self.ds_brace_label=ds_brace_label
+ self.ds_brace_group=VGroup(ds_brace,ds_brace_label)
+
+
+ def construct_equation(self):
+ sum_up_text=TextMobject("and"," summed ", "for all `i's")
+ sum_up_text.set_color_by_tex("summed",PURPLE_A)
+ sum_up_text.next_to(self.multiply_text,DOWN,buff=MED_SMALL_BUFF)
+ sum_up_text.shift(LEFT)
+
+ sum_eqn=TextMobject("$$\\sum_i^{ } $$").set_color(PURPLE_A)
+ sum_eqn.move_to(self.graph_origin+6.5*self.X+4*self.Y)
+
+ line_integral_text=TextMobject("The"," line ","integral's value is: ").to_edge(TOP,buff=MED_SMALL_BUFF)
+ line_integral_text.set_color_by_tex("line",BLUE_C)
+ approx=TextMobject("$\\approx$",color=RED).next_to(sum_eqn,LEFT)
+ multipled=VGroup(
+ self.func_val,
+ self.dot_sym,
+ self.tangent_sym,
+ self.ds_brace_label
+ )
+
+
+ self.play(Write(sum_up_text))
+ self.show_specific_vectors(self.dots)
+ self.play(FadeIn(sum_eqn))
+ self.play(ApplyMethod(
+ multipled.next_to,sum_eqn,RIGHT
+ ))
+ self.wait()
+ self.play(FadeOut(VGroup(
+ self.dot_prod_text,
+ self.multiply_text,
+ sum_up_text
+ )))
+ self.play(Write(line_integral_text))
+ self.play(FadeIn(approx))
+
+
+
+#uploaded by Somnath Pandit.FSF2020_Line Integrals
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
deleted file mode 100644
index 3f7ecd1..0000000
Binary files a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_simple_visualization.gif and /dev/null differ
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
deleted file mode 100644
index 45058d7..0000000
--- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file1_simple_visualization.py
+++ /dev/null
@@ -1,69 +0,0 @@
-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
deleted file mode 100644
index 989a3b7..0000000
Binary files a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_circle_curvature.gif and /dev/null differ
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
deleted file mode 100644
index 232ac41..0000000
--- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_circle_curvature.py
+++ /dev/null
@@ -1,27 +0,0 @@
-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/file2_simple_visualization.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.gif
new file mode 100644
index 0000000..3f7ecd1
Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.gif differ
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..45058d7
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file2_simple_visualization.py
@@ -0,0 +1,69 @@
+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/file3_circle_curvature.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.gif
new file mode 100644
index 0000000..989a3b7
Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_circle_curvature.gif differ
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/file3_curvature_interpretation.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.gif
deleted file mode 100644
index 22a450a..0000000
Binary files a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file3_curvature_interpretation.gif and /dev/null differ
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/file4_curvature_interpretation.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.gif
new file mode 100644
index 0000000..22a450a
Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.gif differ
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..d8dd0a4
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_curvature_interpretation.py
@@ -0,0 +1,42 @@
+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
deleted file mode 100644
index 3b78b5f..0000000
Binary files a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_different_curvature_single_curve.gif and /dev/null differ
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
deleted file mode 100644
index 56b7fbb..0000000
--- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file4_different_curvature_single_curve.py
+++ /dev/null
@@ -1,78 +0,0 @@
-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()
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
new file mode 100644
index 0000000..3b78b5f
Binary files /dev/null and b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/arc-length-and-curvature/file5_different_curvature_single_curve.gif differ
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()
--
cgit