diff options
Diffstat (limited to 'FSF-2020/calculus/series-and-transformations/Fourier Transform/video3_seriesVSTransform.py')
| -rw-r--r-- | FSF-2020/calculus/series-and-transformations/Fourier Transform/video3_seriesVSTransform.py | 133 |
1 files changed, 133 insertions, 0 deletions
diff --git a/FSF-2020/calculus/series-and-transformations/Fourier Transform/video3_seriesVSTransform.py b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video3_seriesVSTransform.py new file mode 100644 index 0000000..f23e54f --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video3_seriesVSTransform.py @@ -0,0 +1,133 @@ +from manimlib.imports import * +import numpy as np + +class compare(GraphScene): + CONFIG = { + "x_min": -3, + "x_max": 3, + "x_axis_width": 6, + "y_min": -5, + "y_max": 5, + "y_axis_label":"$\\frac { { x }^{ 2 } }{ 2 } $", + "graph_origin": ORIGIN, + "axes_color": BLUE, + "exclude_zero_label": True, + "x_labeled_nums": range(-2, 3, 1), + } + def returnPairLines(self,left,right,y_each_unit): + lineLeft=DashedLine(start=(0,5*y_each_unit,0),end=(0,-5*y_each_unit,0)).shift(left) + lineRight=DashedLine(start=(0,5*y_each_unit,0),end=(0,-5*y_each_unit,0)).shift(right) + return lineLeft,lineRight + + def resultFunc(self,x,n,l): + s=(l**2)/6 + for n in range(1,n+1): + s+=(2*((-1)**n))*((l**2)*np.cos(n*np.pi*x/l))*(1/((np.pi**2)*(n**2))) + return s + + def returnPartFunction(self,left,right): + return self.get_graph(lambda x:(x**2)/2,x_min=left,x_max=right,color=RED) + + def returnPartResult(self,l,n): + return self.get_graph(lambda x:self.resultFunc(x,n,l),x_min=-3,x_max=3,color=RED) + + def construct(self): + x_each_unit = self.x_axis_width / (self.x_max - self.x_min) + y_each_unit = self.y_axis_height / (self.y_max - self.y_min) + axes=[] + self.setup_axes(animate=True,scalee=1) + axes.append(self.axes) + partFunction1=self.returnPartFunction(-1,1).shift(4*LEFT) + partFunction2=self.returnPartFunction(-2,2).shift(4*LEFT) + functionText=TextMobject("$\\frac { { x }^{ 2 } }{ 2 } $") + function=self.get_graph(lambda x:(x**2)/2,x_min=-3,x_max=3,color=GREEN) + text1=TextMobject("Non-Periodic function").scale(0.5).shift(3*DOWN+3*RIGHT).set_color(RED) + self.play(ShowCreation(function)) + self.play(FadeIn(text1)) + self.wait(1) + self.play(FadeOut(text1)) + self.play(ApplyMethod(axes[0].shift,4*LEFT),ApplyMethod(function.shift,4*LEFT)) + text2=TextMobject("For a","given","interval of $x$,").scale(0.5).shift(2.5*RIGHT+UP).set_color_by_tex_to_color_map({"given":YELLOW,"interval of $x$,":BLUE}) + text3=TextMobject("We can get the","Fourier Series","of that","particular part!").scale(0.4).shift(2.5*RIGHT+0.5*UP).set_color_by_tex_to_color_map({"particular part!":YELLOW,"Fourier Series":RED}) + self.play(Write(text2)) + left,right=self.returnPairLines((4+x_each_unit)*LEFT,(4-x_each_unit)*LEFT,y_each_unit) + self.play(ShowCreation(left),ShowCreation(right)) + self.play(Write(text3)) + self.wait(0.5) + self.play(FadeOut(text2),FadeOut(text3)) + self.graph_origin=3.5*RIGHT + self.y_axis_label="$\\frac { { l }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ \infty }{ \\frac { 2{ (-1) }^{ n }{ l }^{ 2 }cos(\\frac { n\pi x }{ l } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } }$" + self.setup_axes(animate=True,scalee=1) + axes.append(self.axes) + coeffResult=[ + TextMobject("$\\frac { { 1 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 1 }{ \\frac { 2{ (-1) }^{ n }{ 1 }^{ 2 }cos(\\frac { n\pi x }{ 1 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+UP).set_color(YELLOW), + TextMobject("$\\frac { { 1 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 3 }{ \\frac { 2{ (-1) }^{ n }{ 1 }^{ 2 }cos(\\frac { n\pi x }{ 1 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+UP).set_color(YELLOW), + TextMobject("$\\frac { { 1 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 5 }{ \\frac { 2{ (-1) }^{ n }{ 1 }^{ 2 }cos(\\frac { n\pi x }{ 1 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+UP).set_color(YELLOW), + TextMobject("$\\frac { { 1 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 7 }{ \\frac { 2{ (-1) }^{ n }{ 1 }^{ 2 }cos(\\frac { n\pi x }{ 1 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+UP).set_color(YELLOW), + TextMobject("$\\frac { { 1 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 9 }{ \\frac { 2{ (-1) }^{ n }{ 1 }^{ 2 }cos(\\frac { n\pi x }{ 1 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+UP).set_color(YELLOW), + TextMobject("$\\frac { { 1 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 11 }{ \\frac { 2{ (-1) }^{ n }{ 1 }^{ 2 }cos(\\frac { n\pi x }{ 1 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } }$").scale(0.3).shift(4.5*RIGHT+UP).set_color(YELLOW), + TextMobject("$\\frac { { 1 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 13 }{ \\frac { 2{ (-1) }^{ n }{ 1 }^{ 2 }cos(\\frac { n\pi x }{ 1 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } }$").scale(0.3).shift(4.5*RIGHT+UP).set_color(YELLOW) + ] + result1a=self.returnPartResult(1,1) + result1b=self.returnPartResult(1,3) + result1c=self.returnPartResult(1,5) + result1d=self.returnPartResult(1,7) + result1e=self.returnPartResult(1,9) + result1f=self.returnPartResult(1,11) + result1g=self.returnPartResult(1,13) + self.play(ApplyMethod(partFunction1.shift,0.2*UP)) + self.wait(0.5) + self.play(ReplacementTransform(partFunction1,result1a),Write(coeffResult[0])) + self.play(ReplacementTransform(result1a,result1b),ReplacementTransform(coeffResult[0],coeffResult[1])) + self.play(ReplacementTransform(result1b,result1c),ReplacementTransform(coeffResult[1],coeffResult[2])) + self.play(ReplacementTransform(result1c,result1d),ReplacementTransform(coeffResult[2],coeffResult[3])) + self.play(ReplacementTransform(result1d,result1e),ReplacementTransform(coeffResult[3],coeffResult[4])) + self.play(ReplacementTransform(result1e,result1f),ReplacementTransform(coeffResult[4],coeffResult[5])) + self.play(ReplacementTransform(result1f,result1g),ReplacementTransform(coeffResult[5],coeffResult[6])) + + text4=TextMobject("Here the","obtained function","will always be","periodic","with period equal to the chosen interval").scale(0.4).shift(3.3*DOWN).set_color_by_tex_to_color_map({"obtained function":YELLOW,"periodic":RED}) + self.play(Write(text4)) + + self.wait(0.8) + + self.play(FadeOut(text4)) + text5=TextMobject("As we","increase","the","interval of $x$,").scale(0.5).shift(3*DOWN).set_color_by_tex_to_color_map({"increase":RED,"interval of $x$,":YELLOW}) + text6=TextMobject("We get","approximation","for","higher intervals!").scale(0.5).shift(3.5*DOWN).set_color_by_tex_to_color_map({"approximation":GREEN,"higher intervals!":YELLOW}) + self.play(Write(text5)) + self.play(Write(text6)) + result2=self.returnPartResult(1.5,20) + result3=self.returnPartResult(2,20) + result4=self.returnPartResult(2.5,20) + result5=self.returnPartResult(3,20) + finalCoeff=coeffResult[6] + coeffResult=[ + TextMobject("$\\frac { { 1.5 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 20 }{ \\frac { 2{ (-1) }^{ n }{ 1.5 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } }$").scale(0.3).shift(4.5*RIGHT+1.5*UP).set_color(YELLOW), + TextMobject("$\\frac { { 2 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 20 }{ \\frac { 2{ (-1) }^{ n }{ 2 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+1.5*UP).set_color(YELLOW), + TextMobject("$\\frac { { 2.5 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 20 }{ \\frac { 2{ (-1) }^{ n }{ 2.5 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+1.5*UP).set_color(YELLOW), + TextMobject("$\\frac { { 3 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 20 }{ \\frac { 2{ (-1) }^{ n }{ 3 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+1.5*UP).set_color(YELLOW), + ] + self.play(ApplyMethod(left.shift,LEFT*x_each_unit*0.5),ApplyMethod(right.shift,RIGHT*x_each_unit*0.5),ReplacementTransform(result1g,result2),ReplacementTransform(finalCoeff,coeffResult[0])) + self.play(ApplyMethod(left.shift,LEFT*x_each_unit*0.5),ApplyMethod(right.shift,RIGHT*x_each_unit*0.5),ReplacementTransform(result2,result3),ReplacementTransform(coeffResult[0],coeffResult[1])) + self.play(ApplyMethod(left.shift,LEFT*x_each_unit*0.5),ApplyMethod(right.shift,RIGHT*x_each_unit*0.5),ReplacementTransform(result3,result4),ReplacementTransform(coeffResult[1],coeffResult[2])) + self.play(ApplyMethod(left.shift,LEFT*x_each_unit*0.5),ApplyMethod(right.shift,RIGHT*x_each_unit*0.5),ReplacementTransform(result4,result5),ReplacementTransform(coeffResult[2],coeffResult[3])) + + + # coeffResult=[ + # TextMobject("$\\frac { { 2 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 1 }{ \\frac { 2{ (-1) }^{ n }{ 2 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } }$").scale(0.3).shift(4.5*RIGHT+1.5*UP), + # TextMobject("$\\frac { { 2 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 4 }{ \\frac { 2{ (-1) }^{ n }{ 2 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+1.5*UP), + # TextMobject("$\\frac { { 2 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 10 }{ \\frac { 2{ (-1) }^{ n }{ 2 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+1.5*UP), + # TextMobject("$\\frac { { 2 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 20 }{ \\frac { 2{ (-1) }^{ n }{ 2 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.3).shift(4.5*RIGHT+1.5*UP), + # ] + # result2a=self.returnPartResult(2,1) + # result2b=self.returnPartResult(2,4) + # result2c=self.returnPartResult(2,10) + # result2d=self.returnPartResult(2,20) + + # self.play(ReplacementTransform(partFunction2,result2a),ReplacementTransform(coeffResult[0],coeffResult[1])) + # self.play(ReplacementTransform(result2a,result2b),ReplacementTransform(coeffResult[0],coeffResult[1])) + # self.play(ReplacementTransform(result2b,result2c),ReplacementTransform(coeffResult[0],coeffResult[1])) + # self.play(ReplacementTransform(result2c,result2d),ReplacementTransform(coeffResult[0],coeffResult[1])) + # self.wait(0.5) + + + self.wait(2)
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