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Diffstat (limited to 'FSF-2020/calculus/series-and-transformations/Fourier Transform')
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diff --git a/FSF-2020/calculus/series-and-transformations/Fourier Transform/README.md b/FSF-2020/calculus/series-and-transformations/Fourier Transform/README.md new file mode 100644 index 0000000..2fa4e04 --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/README.md @@ -0,0 +1,20 @@ +### Dividing a tone into its constituents + + +### Colors Analogy + + +### Applying the same on Graphs + + +### Fourier series for non-periodic functions-a + + +### Fourier series for non-periodic functions-b + + +### Fourier Series of Square pulse + + +### Coins Analogy + diff --git a/FSF-2020/calculus/series-and-transformations/Fourier Transform/gifs/file1.gif b/FSF-2020/calculus/series-and-transformations/Fourier Transform/gifs/file1.gif Binary files differnew file mode 100644 index 0000000..d4dc9d7 --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/gifs/file1.gif diff --git a/FSF-2020/calculus/series-and-transformations/Fourier 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a/FSF-2020/calculus/series-and-transformations/Fourier Transform/gifs/file7.gif b/FSF-2020/calculus/series-and-transformations/Fourier Transform/gifs/file7.gif Binary files differnew file mode 100644 index 0000000..ab4eed8 --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/gifs/file7.gif diff --git a/FSF-2020/calculus/series-and-transformations/Fourier Transform/video1_DividingAToneIntoItsConstituents.py b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video1_DividingAToneIntoItsConstituents.py new file mode 100644 index 0000000..fdb8719 --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video1_DividingAToneIntoItsConstituents.py @@ -0,0 +1,83 @@ +from manimlib.imports import* +import numpy as np + +class intro(GraphScene): + CONFIG = { + "x_min": -3, + "x_max": 3, + "x_axis_width": 6, + "y_min": -5, + "y_max": 5, + "graph_origin": 10.5*LEFT, + "axes_color": BLUE, + "exclude_zero_label": True, + "x_labeled_nums": range(-2, 3, 1), + } + def func(self,t,n1,n2): + s=0 + for i in range(n1,n2+1): + s+=((-2/(i*np.pi))*((-1)**i)*np.sin(2*np.pi*i*t)) + return s + + def construct(self): + image=ImageMobject('image.png').shift(5.5*LEFT+2.5*UP).scale(1.5) + self.play(ShowCreation(image)) + + self.setup_axes(scalee=1) + + mainGraphs=[ + self.get_graph(lambda x:self.func(x,2,7),x_max=2,x_min=-2).shift(9.3*RIGHT+3*UP).set_color([ORANGE,GREEN_B,RED_E,YELLOW_E,RED_D,YELLOW_D]).scale(1.4), + self.get_graph(lambda x:self.func(x,3,7),x_max=2,x_min=-2).shift(10.8*RIGHT+3*UP).set_color([GREEN_B,ORANGE,RED_D,YELLOW_E,YELLOW_D]).scale(1.4), + self.get_graph(lambda x:self.func(x,4,7),x_max=2,x_min=-2).shift(10.8*RIGHT+3*UP).set_color([GREEN_B,YELLOW_E,ORANGE,YELLOW_D]).scale(1.4), + self.get_graph(lambda x:self.func(x,5,7),x_max=2,x_min=-2).shift(10.8*RIGHT+3*UP).set_color([YELLOW_E,GREEN_B,YELLOW_D]).scale(1.4), + self.get_graph(lambda x:self.func(x,6,7),x_max=2,x_min=-2).shift(10.8*RIGHT+3*UP).set_color([YELLOW_D,GREEN_B]).scale(1.4), + self.get_graph(lambda x:self.func(x,7,7),x_max=2,x_min=-2,color=GREEN_B).shift(10.8*RIGHT+3*UP).scale(1.4), + ] + self.play(ApplyMethod(mainGraphs[0].shift,1.5*RIGHT)) + + graph1=self.get_graph(lambda x:self.func(x,2,2),x_max=2,x_min=-2,color=RED_E).shift(10.8*RIGHT+3*UP).scale(1.5) + graph2=self.get_graph(lambda x:self.func(x,3,3),x_max=2,x_min=-2,color=RED_D).shift(10.8*RIGHT+3*UP).scale(1.5) + graph3=self.get_graph(lambda x:self.func(x,4,4),x_max=2,x_min=-2,color=ORANGE).shift(10.8*RIGHT+3*UP).scale(1.5) + graph4=self.get_graph(lambda x:self.func(x,5,5),x_max=2,x_min=-2,color=YELLOW_E).shift(10.8*RIGHT+3*UP).scale(1.5) + graph5=self.get_graph(lambda x:self.func(x,6,6),x_max=2,x_min=-2,color=YELLOW_D).shift(10.8*RIGHT+3*UP).scale(1.5) + + coeff=[ + TextMobject("$\\frac { -1 }{ \pi } sin(4\pi t)$").scale(0.5).shift(DOWN+4.6*RIGHT+3*UP).set_color(RED_E), + TextMobject("$\\frac { 2 }{ 3\pi } sin(6\pi t)$").scale(0.5).shift(2*DOWN+4.6*RIGHT+3*UP).set_color(RED_D), + TextMobject("$\\frac { -1 }{ 2\pi } sin(8\pi t)$").scale(0.5).shift(3*DOWN+4.6*RIGHT+3*UP).set_color(ORANGE), + TextMobject("$\\frac { 2 }{ 5\pi } sin(10\pi t)$").scale(0.5).shift(4*DOWN+4.6*RIGHT+3*UP).set_color(YELLOW_E), + TextMobject("$\\frac { -1 }{ 3\pi } sin(12\pi t)$").scale(0.5).shift(5*DOWN+4.6*RIGHT+3*UP).set_color(YELLOW_D), + TextMobject("$\\frac { 2 }{ 7\pi } sin(14\pi t)$").scale(0.5).shift(6*DOWN+4.6*RIGHT+3*UP).set_color(GREEN_B) + ] + + self.wait(0.6) + self.play(ApplyMethod(graph1.shift,1*DOWN),ReplacementTransform(mainGraphs[0],mainGraphs[1])) + self.play(Write(coeff[0])) + self.play(ApplyMethod(graph2.shift,2*DOWN),ReplacementTransform(mainGraphs[1],mainGraphs[2])) + self.play(Write(coeff[1])) + self.play(ApplyMethod(graph3.shift,3*DOWN),ReplacementTransform(mainGraphs[2],mainGraphs[3])) + self.play(Write(coeff[2])) + self.play(ApplyMethod(graph4.shift,4*DOWN),ReplacementTransform(mainGraphs[3],mainGraphs[4])) + self.play(Write(coeff[3])) + self.play(ApplyMethod(graph5.shift,5*DOWN),ReplacementTransform(mainGraphs[4],mainGraphs[5])) + self.play(Write(coeff[4])) + self.play(ApplyMethod(mainGraphs[5].shift,6*DOWN)) + self.play(Write(coeff[5])) + + pluses=[TextMobject("+"),TextMobject("+"),TextMobject("+"),TextMobject("+"),TextMobject("+")] + for t in pluses: + t.scale(0.5).shift((2.2-1.5*pluses.index(t))*LEFT) + + finalGraph=self.get_graph(lambda x:self.func(x,2,7),x_max=2,x_min=-2).shift(10.8*RIGHT+3*UP) + finalGraph.set_color([GREEN_B,YELLOW_D,YELLOW_E,ORANGE,RED_D,RED_E]) + finalGroup=VGroup(graph1,graph2,graph3,graph4,graph5,mainGraphs[5]) + self.play(ReplacementTransform(finalGroup,finalGraph)) + self.play(ApplyMethod(coeff[0].scale,0.7),ApplyMethod(coeff[1].scale,0.7),ApplyMethod(coeff[2].scale,0.7),ApplyMethod(coeff[3].scale,0.7),ApplyMethod(coeff[4].scale,0.7),ApplyMethod(coeff[5].scale,0.7)) + #self.play(ApplyMethod(coeff[0].shift,7*LEFT+1.6*DOWN),ApplyMethod(coeff[1].shift,5.5*LEFT+0.8*DOWN),ApplyMethod(coeff[2].shift,4*LEFT),ApplyMethod(coeff[3].shift,2.5*LEFT+0.8*UP),ApplyMethod(coeff[4].shift,LEFT+1.6*UP),ApplyMethod(coeff[5].shift,0.5*RIGHT+2.4*DOWN)) + self.play(ApplyMethod(coeff[0].shift,7.6*LEFT+2*DOWN),ApplyMethod(coeff[1].shift,6.1*LEFT+DOWN),ApplyMethod(coeff[2].shift,4.6*LEFT),ApplyMethod(coeff[3].shift,3.1*LEFT+UP),ApplyMethod(coeff[4].shift,1.6*LEFT+2*UP),ApplyMethod(coeff[5].shift,0.1*LEFT+3*UP)) + equal=TextMobject("=").scale(1.5).shift(1.5*UP) + self.play(Write(equal)) + self.play(Write(pluses[0]),Write(pluses[1]),Write(pluses[2]),Write(pluses[3]),Write(pluses[4])) + group=VGroup(pluses[0],pluses[1],pluses[2],pluses[3],pluses[4],coeff[0],coeff[1],coeff[2],coeff[3],coeff[4],coeff[5]) + self.play(ApplyMethod(group.scale,1.5)) + self.wait(2) diff --git a/FSF-2020/calculus/series-and-transformations/Fourier Transform/video2_ColorsAnalogyForFourierSeries.py b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video2_ColorsAnalogyForFourierSeries.py new file mode 100644 index 0000000..c87e58e --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video2_ColorsAnalogyForFourierSeries.py @@ -0,0 +1,146 @@ +from manimlib.imports import* +import numpy as np + +def func(t,n1,n2): + s=0 + for i in range(n1,n2+1): + s+=((-2/(i*np.pi))*((-1)**i)*np.sin(2*np.pi*i*t)) + return s + +class divideColors(GraphScene): + CONFIG = { + "x_min": -2, + "x_max": 2, + "y_min": -1, + "y_max": 1, + "graph_origin": ORIGIN, + "function_color": RED, + "axes_color": BLUE, + "x_axis_label": "$t$", + "y_axis_label": "$y$", + "x_labeled_nums": range(-1, 2, 1), + "x_axis_width": 3, + "y_axis_height": 2 + } + def construct(self): + text1a=TextMobject("Consider dividing a","mixture of colors") + text1b=TextMobject("into its","components") + text1a.scale(0.8) + text1b.scale(0.8) + text1a.shift(UP) + text1b.shift(0.3*UP) + text1a.set_color_by_tex_to_color_map({"mixture of colors":[GREEN,RED,BLUE,YELLOW]}) + text1b.set_color_by_tex_to_color_map({"components":GREEN}) + self.play(Write(text1a)) + self.play(FadeIn(text1b)) + self.wait(0.8) + + self.play(FadeOut(text1a),FadeOut(text1b)) + + mainCircle=Circle(radius=1.4,color=BLACK,fill_color=[PURPLE_E,PURPLE_D,RED_B,ORANGE,YELLOW_B,YELLOW_D,GREEN_A,GREEN_C],fill_opacity=0.8) + self.play(ShowCreation(mainCircle)) + self.wait(1) + mainCirclea=Circle(radius=1.4,color=BLACK,fill_color=[RED_B,ORANGE,YELLOW_B,YELLOW_D,GREEN_A,GREEN_C],fill_opacity=0.8) + mainCircleb=Circle(radius=1.4,color=BLACK,fill_color=[YELLOW_B,YELLOW_D,GREEN_A,GREEN_C],fill_opacity=0.8) + mainCirclec=Circle(radius=1.4,color=BLACK,fill_color=[GREEN_A,GREEN_C],fill_opacity=0.8) + mainCircled=Circle(radius=1.4,color=BLACK,fill_color=[],fill_opacity=0.8) + + c1=Circle(radius=0.5,color=PURPLE_E,fill_color=PURPLE_E,fill_opacity=0.8) + c2=Circle(radius=0.5,color=PURPLE_D,fill_color=PURPLE_D,fill_opacity=0.8) + c3=Circle(radius=0.5,color=RED_D,fill_color=RED_B,fill_opacity=0.8) + c4=Circle(radius=0.5,color=ORANGE,fill_color=ORANGE,fill_opacity=0.8) + c5=Circle(radius=0.5,color=YELLOW_B,fill_color=YELLOW_B,fill_opacity=0.8) + c6=Circle(radius=0.5,color=YELLOW_D,fill_color=YELLOW_D,fill_opacity=0.8) + c7=Circle(radius=0.5,color=GREEN_A,fill_color=GREEN_A,fill_opacity=0.8) + c8=Circle(radius=0.5,color=GREEN_C,fill_color=GREEN_C,fill_opacity=0.8) + + self.play(ApplyMethod(c1.shift,3*UP+LEFT),ApplyMethod(c2.shift,3*UP+RIGHT),ReplacementTransform(mainCircle,mainCirclea)) + self.wait(0.8) + + self.play(ApplyMethod(c3.shift,UP+3*LEFT),ApplyMethod(c4.shift,DOWN+3*LEFT),ReplacementTransform(mainCirclea,mainCircleb)) + self.wait(0.8) + + self.play(ApplyMethod(c5.shift,3*DOWN+LEFT),ApplyMethod(c6.shift,3*DOWN+RIGHT),ReplacementTransform(mainCircleb,mainCirclec)) + self.wait(0.8) + + self.play(ApplyMethod(c7.shift,3*RIGHT+UP),ApplyMethod(c8.shift,3*RIGHT+DOWN),ReplacementTransform(mainCirclec,mainCircled)) + self.wait(1) + + text2=TextMobject("Similarly,").scale(0.8).shift(UP).set_color(RED) + + self.play(FadeOut(c1),FadeOut(c2),FadeOut(c3),FadeOut(c4),FadeOut(c5),FadeOut(c6),FadeOut(c7),FadeOut(c8)) + self.play(Write(text2)) + self.wait(0.8) + self.play(FadeOut(text2)) + + + coeff=[ + TextMobject("$\\frac { -2 }{ \pi } \sum _{ n=1 }^{ 24 }{ \\frac { { -1 }^{ n } }{ n } sin(2\pi nt) }$").scale(0.2).shift(RIGHT+UP), + TextMobject("$\\frac { 2 }{ \pi } sin(2\pi t)$").scale(0.3).shift(RIGHT+UP+4*LEFT+UP), + TextMobject("$\\frac { -2 }{ \pi } \sum _{ n=2 }^{ 24 }{ \\frac { { -1 }^{ n } }{ n } sin(2\pi nt) }$").scale(0.2).shift(RIGHT+UP), + TextMobject("$\\frac { -1 }{ \pi } sin(4\pi t)$").scale(0.3).shift(RIGHT+UP+4*RIGHT+UP), + TextMobject("$\\frac { -2 }{ \pi } \sum _{ n=3 }^{ 24 }{ \\frac { { -1 }^{ n } }{ n } sin(2\pi nt) }$").scale(0.2).shift(RIGHT+UP), + TextMobject("$\\frac { 2 }{ 3\pi } sin(6\pi t)$").scale(0.3).shift(RIGHT+UP+4*LEFT+2*DOWN), + TextMobject("$\\frac { -2 }{ \pi } \sum _{ n=4 }^{ 24 }{ \\frac { { -1 }^{ n } }{ n } sin(2\pi nt) }$").scale(0.2).shift(RIGHT+UP), + TextMobject("$\\frac { -1 }{ 2\pi } sin(8\pi t)$").scale(0.3).shift(RIGHT+UP+4*RIGHT+2*DOWN), + TextMobject("$\\frac { -2 }{ \pi } \sum _{ n=5 }^{ 24 }{ \\frac { { -1 }^{ n } }{ n } sin(2\pi nt) }$").scale(0.2).shift(RIGHT+UP), + TextMobject("$\\frac { 2 }{ 5\pi } sin(10\pi t)$").scale(0.3).shift(RIGHT+UP+2.5*UP), + TextMobject("$\\frac { -2 }{ \pi } \sum _{ n=6 }^{ 24 }{ \\frac { { -1 }^{ n } }{ n } sin(2\pi nt) }$").scale(0.2).shift(RIGHT+UP), + TextMobject("$\\frac { -1 }{ 3\pi } sin(12\pi t)$").scale(0.3).shift(RIGHT+UP+2.5*DOWN), + TextMobject("$\\frac { -2 }{ \pi } \sum _{ n=7 }^{ 24 }{ \\frac { { -1 }^{ n } }{ n } sin(2\pi nt) }$").scale(0.2).shift(RIGHT+UP), + ] + + axes=[] + self.setup_axes(scalee=1) + axes.append(self.axes) + graphs=[self.get_graph(lambda x:func(x,1,24),x_min=-1,x_max=1).set_color([DARK_BROWN,GREEN_E,GREEN_C,GOLD_E,GOLD_C,ORANGE,RED_C]), + self.get_graph(lambda x:func(x,2,24),x_min=-1,x_max=1).set_color([DARK_BROWN,GREEN_C,GOLD_E,GOLD_C,ORANGE,RED_C]), + self.get_graph(lambda x:func(x,3,24),x_min=-1,x_max=1).set_color([DARK_BROWN,GOLD_E,GOLD_C,ORANGE,RED_C]), + self.get_graph(lambda x:func(x,4,24),x_min=-1,x_max=1).set_color([DARK_BROWN,GOLD_C,ORANGE,RED_C]), + self.get_graph(lambda x:func(x,5,24),x_min=-1,x_max=1).set_color([DARK_BROWN,ORANGE,RED_C]), + self.get_graph(lambda x:func(x,6,24),x_min=-1,x_max=1).set_color([DARK_BROWN,RED_C]), + self.get_graph(lambda x:func(x,7,24),x_min=-1,x_max=1).set_color(DARK_BROWN) + ] + + self.setup_axes(scalee=1) + axes.append(self.axes) + graph1=self.get_graph(lambda x:func(x,1,1),x_min=-1,x_max=1,color=GREEN_E) + + self.setup_axes(scalee=1) + axes.append(self.axes) + graph2=self.get_graph(lambda x:func(x,2,2),x_min=-1,x_max=1,color=GREEN_C) + + self.setup_axes(scalee=1) + axes.append(self.axes) + graph3=self.get_graph(lambda x:func(x,3,3),x_min=-1,x_max=1,color=GOLD_E) + + self.setup_axes(scalee=1) + axes.append(self.axes) + graph4=self.get_graph(lambda x:func(x,4,4),x_min=-1,x_max=1,color=GOLD_C) + + self.setup_axes(scalee=1) + axes.append(self.axes) + graph5=self.get_graph(lambda x:func(x,5,5),x_min=-1,x_max=1,color=ORANGE) + + self.setup_axes(scalee=1) + axes.append(self.axes) + graph6=self.get_graph(lambda x:func(x,6,6),x_min=-1,x_max=1,color=RED_C) + + groups=[VGroup(axes[1],graph1),VGroup(axes[2],graph2),VGroup(axes[3],graph3),VGroup(axes[4],graph4), + VGroup(axes[5],graph5),VGroup(axes[6],graph6)] + + self.play(ShowCreation(graphs[0])) + self.play(Write(coeff[0])) + self.wait(1) + + self.play(ReplacementTransform(graphs[0],graphs[1]),ApplyMethod(groups[0].shift,4*LEFT+UP),ReplacementTransform(coeff[0],coeff[2]),FadeIn(coeff[1])) + self.play(ReplacementTransform(graphs[1],graphs[2]),ApplyMethod(groups[1].shift,4*RIGHT+UP),ReplacementTransform(coeff[2],coeff[4]),FadeIn(coeff[3])) + self.play(ReplacementTransform(graphs[2],graphs[3]),ApplyMethod(groups[2].shift,4*LEFT+2*DOWN),ReplacementTransform(coeff[4],coeff[6]),FadeIn(coeff[5])) + self.play(ReplacementTransform(graphs[3],graphs[4]),ApplyMethod(groups[3].shift,4*RIGHT+2*DOWN),ReplacementTransform(coeff[6],coeff[8]),FadeIn(coeff[7])) + self.play(ReplacementTransform(graphs[4],graphs[5]),ApplyMethod(groups[4].shift,2.5*UP),ReplacementTransform(coeff[8],coeff[10]),FadeIn(coeff[9])) + self.play(ReplacementTransform(graphs[5],graphs[6]),ApplyMethod(groups[5].shift,2.5*DOWN),ReplacementTransform(coeff[10],coeff[12]),FadeIn(coeff[11])) + + + + self.wait(2) + 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..d35f8bf --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video3_seriesVSTransform.py @@ -0,0 +1,129 @@ +from manimlib.imports import * +import numpy as np + +class compare(GraphScene,MovingCameraScene): + CONFIG = { + "x_min": -3, + "x_max": 3, + "x_axis_width": 6, + "y_min": -5, + "y_max": 5, + "y_axis_label":"$\\frac { { x }^{ 2 } }{ 2 } $", + "graph_origin": ORIGIN, + "axes_color": BLUE, + "exclude_zero_label": True, + "x_labeled_nums": range(-2, 3, 1), + } + def setup(self): + GraphScene.setup(self) + MovingCameraScene.setup(self) + def returnPairLines(self,left,right,y_each_unit): + lineLeft=DashedLine(start=(0,5*y_each_unit,0),end=(0,-5*y_each_unit,0)).shift(left) + lineRight=DashedLine(start=(0,5*y_each_unit,0),end=(0,-5*y_each_unit,0)).shift(right) + return lineLeft,lineRight + + def resultFunc(self,x,n,l): + s=(l**2)/6 + for n in range(1,n+1): + s+=(2*((-1)**n))*((l**2)*np.cos(n*np.pi*x/l))*(1/((np.pi**2)*(n**2))) + return s + + def returnPartFunction(self,left,right): + return self.get_graph(lambda x:(x**2)/2,x_min=left,x_max=right,color=RED) + + def returnPartResult(self,l,n): + return self.get_graph(lambda x:self.resultFunc(x,n,l),x_min=-3,x_max=3,color=RED) + + def construct(self): + x_each_unit = self.x_axis_width / (self.x_max - self.x_min) + y_each_unit = self.y_axis_height / (self.y_max - self.y_min) + axes=[] + self.setup_axes(animate=True,scalee=1) + axes.append(self.axes) + partFunction1=self.returnPartFunction(-1,1).shift(4*LEFT) + partFunction2=self.returnPartFunction(-2,2).shift(4*LEFT) + functionText=TextMobject("$\\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(FadeOut(axes[0]),FadeOut(left),FadeOut(right),FadeOut(function)) + self.camera_frame.save_state() + self.play(self.camera_frame.set_width, 5,self.camera_frame.move_to, 3.5*RIGHT) + + + 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])) + + self.wait(0.5) + self.play(self.camera_frame.set_width, 14,self.camera_frame.move_to, 0) + + 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(FadeIn(axes[0]),FadeIn(left),FadeIn(right),FadeIn(function)) + 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(0.4).shift(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 } } } $").scale(0.4).shift(5*RIGHT+1.5*UP).set_color(YELLOW), + TextMobject("$\\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 } } } $").scale(0.4).shift(5*RIGHT+2.2*UP).set_color(YELLOW), + TextMobject("$\\frac { { 3 }^{ 2 } }{ 6 } +\sum _{ n=1 }^{ 20 }{ \\frac { 2{ (-1) }^{ n }{ 3 }^{ 2 }cos(\\frac { n\pi x }{ 2 } ) }{ { \pi }^{ 2 }{ n }^{ 2 } } } $").scale(0.4).shift(5*RIGHT+2.2*UP).set_color(YELLOW), + ] + self.play(ApplyMethod(left.shift,LEFT*x_each_unit*0.5),ApplyMethod(right.shift,RIGHT*x_each_unit*0.5),ReplacementTransform(result1g,result2),ReplacementTransform(finalCoeff,coeffResult[0])) + self.play(ApplyMethod(left.shift,LEFT*x_each_unit*0.5),ApplyMethod(right.shift,RIGHT*x_each_unit*0.5),ReplacementTransform(result2,result3),ReplacementTransform(coeffResult[0],coeffResult[1])) + self.play(ApplyMethod(left.shift,LEFT*x_each_unit*0.5),ApplyMethod(right.shift,RIGHT*x_each_unit*0.5),ReplacementTransform(result3,result4),ReplacementTransform(coeffResult[1],coeffResult[2])) + self.play(ApplyMethod(left.shift,LEFT*x_each_unit*0.5),ApplyMethod(right.shift,RIGHT*x_each_unit*0.5),ReplacementTransform(result4,result5),ReplacementTransform(coeffResult[2],coeffResult[3])) + + + + self.wait(2) diff --git a/FSF-2020/calculus/series-and-transformations/Fourier Transform/video4_FourierSeriesOfSquarePulse.py b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video4_FourierSeriesOfSquarePulse.py new file mode 100644 index 0000000..fdf4bb3 --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video4_FourierSeriesOfSquarePulse.py @@ -0,0 +1,97 @@ +from manimlib.imports import * +import numpy as np + +def returnSum(k,x): + summ=0 + for i in range(1,k+1,2): + summ+=((np.sin(2*np.pi*i*x))/i) + return summ + +def returnFunc(self,k): + graph=self.get_graph(lambda x:(4/np.pi)*returnSum(k,x),color=WHITE,x_max=1,x_min=-1) + return graph + +class fourierSeries(GraphScene,MovingCameraScene): + CONFIG = { + "x_min": -3, + "x_max": 3, + "x_axis_width": 13, + "y_min": -3, + "y_max": 3, + "graph_origin": ORIGIN, + "function_color": RED, + "axes_color": BLUE, + "x_axis_label": "$x$", + "y_axis_label": "$y$", + "exclude_zero_label": True, + "x_labeled_nums": range(-2, 3, 1), + } + def setup(self): + GraphScene.setup(self) + MovingCameraScene.setup(self) + def construct(self): + x_each_unit = self.x_axis_width / (self.x_max - self.x_min) + y_each_unit = self.y_axis_height / (self.y_max - self.y_min) + + equation=TextMobject("$f(x)=\\frac { 4 }{ \pi } \sum _{ k=1,3,5.. }^{ \infty }{ \\frac { 1 }{ k } \sin { 2\pi kx } }$").shift(5*RIGHT+3*UP).set_color(RED).scale(0.5) + self.add(equation) + self.setup_axes(animate=True,scalee=1) + line1=Line(start=(-x_each_unit,y_each_unit,0),end=(-(1/2)*x_each_unit,y_each_unit,0),color=RED) + line2=Line(start=(-(1/2)*x_each_unit,y_each_unit,0),end=(-(1/2)*x_each_unit,-y_each_unit,0),color=RED) + line3=Line(start=(-(1/2)*x_each_unit,-y_each_unit,0),end=(0,-y_each_unit,0),color=RED) + line4=Line(start=(0,-y_each_unit,0),end=(0,y_each_unit,0),color=RED) + line5=Line(start=(0,y_each_unit,0),end=((1/2)*x_each_unit,y_each_unit,0),color=RED) + line6=Line(start=((1/2)*x_each_unit,y_each_unit,0),end=((1/2)*x_each_unit,-y_each_unit,0),color=RED) + line7=Line(start=((1/2)*x_each_unit,-y_each_unit,0),end=(x_each_unit,-y_each_unit,0),color=RED) + self.play(ShowCreation(line1)) + self.play(ShowCreation(line2)) + self.play(ShowCreation(line3)) + self.play(ShowCreation(line4)) + self.play(ShowCreation(line5)) + self.play(ShowCreation(line6)) + self.play(ShowCreation(line7)) + self.wait(0.5) + + labels=[ + TextMobject("$f_{ k=1 }(x)$"), + TextMobject("$f_{ k=3 }(x)$"), + TextMobject("$f_{ k=5 }(x)$"), + TextMobject("$f_{ k=7 }(x)$"), + TextMobject("$f_{ k=9 }(x)$"), + TextMobject("$f_{ k=11 }(x)$"), + TextMobject("$f_{ k=13 }(x)$"), + TextMobject("$f_{ k=15 }(x)$"), + TextMobject("$f_{ k=17 }(x)$"), + TextMobject("$f_{ k=19 }(x)$"), + TextMobject("$f_{ k=85 }(x)$") + ] + p=0 + for i in range(1,20,2): + if(i==1): + graphInitial=returnFunc(self,1) + label=labels[p].scale(0.5).shift(y_each_unit*1.5*UP+RIGHT*x_each_unit*0.3) + self.play(ShowCreation(graphInitial),Write(labels[0])) + old=graphInitial + oldLabel=label + else: + graph=returnFunc(self,i) + graphLabel=labels[p].scale(0.5).shift(y_each_unit*1.5*UP+RIGHT*x_each_unit*0.3) + self.play(ReplacementTransform(old,graph),ReplacementTransform(oldLabel,graphLabel)) + old=graph + oldLabel=graphLabel + p+=1 + graphFinal=returnFunc(self,85) + labelFinal=labels[10].scale(0.5).shift(y_each_unit*1.5*UP+RIGHT*x_each_unit*0.3) + self.play(FadeOut(old),FadeOut(oldLabel)) + self.play(ShowCreation(graphFinal),Write(labelFinal)) + self.wait(1) + self.camera_frame.save_state() + self.play(self.camera_frame.set_width, 2.25,self.camera_frame.move_to, y_each_unit*UP+RIGHT*x_each_unit*0.3) + circleMark=Circle(radius=0.1,color=GREEN).shift(x_each_unit*RIGHT*0.47+UP*y_each_unit*1.1) + text=TextMobject("Gibbs","phenomenon").set_color_by_tex_to_color_map({"Gibbs":BLUE,"phenomenon":YELLOW}).scale(0.1).shift(RIGHT*x_each_unit*0.65+UP*y_each_unit*1.1) + self.wait(0.7) + self.play(ShowCreation(circleMark)) + self.play(Write(text)) + self.wait(0.5) + self.play(self.camera_frame.set_width,14,self.camera_frame.move_to,0,FadeOut(circleMark),FadeOut(text)) + self.wait(2) diff --git a/FSF-2020/calculus/series-and-transformations/Fourier Transform/video5_CoinsAnalogy.py b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video5_CoinsAnalogy.py new file mode 100644 index 0000000..10ee889 --- /dev/null +++ b/FSF-2020/calculus/series-and-transformations/Fourier Transform/video5_CoinsAnalogy.py @@ -0,0 +1,225 @@ +from manimlib.imports import* +import math +import numpy as np + +class coinsAnalogy(Scene): + def construct(self): + text1=TextMobject("Consider we have","Rs 39").shift(2*UP).scale(0.75).set_color_by_tex_to_color_map({"Rs 39":[YELLOW,PURPLE]}) + text2=TextMobject("and we want to represent them only in terms of","Rs 2","and","Rs 5").shift(UP).scale(0.6).set_color_by_tex_to_color_map({"Rs 2":YELLOW,"Rs 5":PURPLE}) + text3=TextMobject("How many","Rs 2 coins","and","Rs 5 coins","do","we need?").scale(0.8).set_color_by_tex_to_color_map({"Rs 2 coins":YELLOW,"Rs 5 coins":PURPLE,"we need?":RED}) + text4=TextMobject("We","perform","the following!").scale(0.75).shift(DOWN).set_color_by_tex_to_color_map({"perform":GREEN}) + + self.play(FadeIn(text1)) + self.wait(0.6) + self.play(Write(text2)) + self.wait(0.5) + self.play(Write(text3)) + self.wait(0.7) + self.play(FadeIn(text4)) + self.wait(1) + self.play(FadeOut(text1),FadeOut(text2),FadeOut(text3),FadeOut(text4)) + + g1=self.group("Rs 39") + g1.shift(3*LEFT+0.75*UP) + l1=self.line() + l1.shift(4*LEFT) + f1=self.fiveGroup() + t1=self.twoGroup() + f1.shift(3.5*LEFT+0.7*DOWN) + andT=TextMobject("and").next_to(f1,buff=-0.1).scale(0.3) + t1.next_to(andT,buff=0.2) + equal1=TextMobject("$=$") + equal1.next_to(l1,buff=0.2) + + self.play(ShowCreation(g1)) + self.play(ShowCreation(l1)) + self.play(ShowCreation(f1),Write(andT),ShowCreation(t1)) + self.play(ShowCreation(equal1)) + self.wait(0.6) + + f2=self.fiveGroup().next_to(equal1,buff=0.4) + multiple1=TextMobject("$X7$","$\quad +$").next_to(f2,buff=0.2).set_color_by_tex_to_color_map({"$X7$":PURPLE}) + l2=self.line().next_to(multiple1,buff=0.4) + g2=self.group("Rs 4").shift(2.75*RIGHT+0.75*UP) + t2=self.twoGroup().shift(2.75*RIGHT+0.7*DOWN) + + self.play(ShowCreation(f2)) + self.play(ShowCreation(multiple1)) + self.play(ShowCreation(g2)) + self.play(ShowCreation(l2)) + self.play(ShowCreation(t2)) + self.wait(1) + + tempGrup=VGroup(g2,l2,t2) + + t3=self.twoGroup().next_to(multiple1,buff=0.4) + multiple2=TextMobject("$X2$").next_to(t3,buff=0.2).set_color_by_tex_to_color_map({"$X2$":YELLOW}) + + self.play(ReplacementTransform(tempGrup,t3)) + self.play(Write(multiple2)) + self.wait(2) + + def line(self): + l=Line(start=[0,0,0],end=[2,0,0]) + return l + + def twoGroup(self): + two=Circle(radius=0.25,color=BLACK,fill_color=YELLOW,fill_opacity=0.7) + twoText=TextMobject("Rs 2").scale(0.25).set_color(BLACK) + twoGrup=VGroup(two,twoText) + return twoGrup + + def fiveGroup(self): + five=Circle(radius=0.35,color=BLACK,fill_color=PURPLE,fill_opacity=0.7) + fiveText=TextMobject("Rs 5").scale(0.3).set_color(BLACK) + fiveGrup=VGroup(five,fiveText) + return fiveGrup + + def group(self,money): + coins=[ + Circle(radius=0.35,color=GREY,fill_color=GREY,fill_opacity=0.75), + Circle(radius=0.35,color=GREY,fill_color=GREY,fill_opacity=0.8), + Circle(radius=0.35,color=GREY,fill_color=GREY,fill_opacity=0.7), + Circle(radius=0.35,color=GREY,fill_color=GREY,fill_opacity=0.75), + Circle(radius=0.35,color=GREY,fill_color=GREY,fill_opacity=0.8), + Circle(radius=0.35,color=GREY,fill_color=GREY,fill_opacity=0.7) + ] + coinsText=TextMobject(money).set_color(BLACK) + coinsText.scale(0.35) + + coins[1].shift(0.2*RIGHT+0.2*UP) + coins[2].shift(0.2*RIGHT+0.1*DOWN) + coins[3].shift(0.2*DOWN) + coins[4].shift(0.2*UP+0.2*LEFT) + coins[5].shift(0.2*LEFT+0.1*LEFT) + + coinsGrup=VGroup(coins[0],coins[1],coins[2],coins[3],coins[4],coins[5],coinsText) + return coinsGrup + +class divideFunction(GraphScene): + CONFIG = { + "x_min": -6, + "x_max": 6, + "y_min": -300, + "y_max": 300, + "x_tick_frequency": 2, + "y_tick_frequency": 300, + "graph_origin": 3*LEFT+1.5*UP+6*LEFT, + "function_color": RED, + "axes_color": BLUE, + "x_axis_label": "$t$", + "y_axis_label": "$y$", + "x_labeled_nums": [-6,0,6], + "y_labeled_nums": [-300,0,300], + "x_axis_width": 1.5, + "y_axis_height": 1 + } + def line(self): + l=Line(start=[0,0,0],end=[2,0,0]) + return l + def construct(self): + text1=TextMobject("Similarly,").scale(0.8).shift(UP).set_color(RED) + text2=TextMobject("To find the amount of","each frequency","present in","$f(x)$").scale(0.6).set_color_by_tex_to_color_map({"each frequency":[YELLOW,RED],"$f(x)$":RED}) + text3=TextMobject("We","perform","the following!").scale(0.7).shift(DOWN).set_color_by_tex_to_color_map({"perform":GREEN}) + + self.play(FadeIn(text1)) + self.wait(0.6) + self.play(Write(text2)) + self.wait(0.7) + self.play(FadeIn(text3)) + + self.wait(1) + self.play(FadeOut(text1),FadeOut(text2),FadeOut(text3)) + + boxUP=Square(side_length=1.7,fill_color=BLUE_C,fill_opacity=0.5,color=BLACK).shift(3*LEFT+UP) + boxDOWN=Square(side_length=1.7,fill_color=BLUE_C,fill_opacity=0.5,color=BLACK).shift(3*LEFT+DOWN) + + axes=[] + self.graph_origin=10*LEFT+1.5*UP + self.setup_axes(scalee=1) + axes.append(self.axes) + fx=self.get_graph(lambda x:math.pow(x,3)-math.pow(x,2)+x-2,x_min=-2*math.pi,x_max=2*math.pi,color=RED).shift(7*RIGHT+0.5*DOWN) + + l=self.line().shift(4*LEFT) + + self.graph_origin=10*LEFT+1.5*DOWN + self.y_min=-2 + self.y_max=1 + self.y_tick_frequency=1 + self.y_labeled_nums=[-1,0,1] + self.setup_axes(scalee=1) + axes.append(self.axes) + sinx=self.get_graph(lambda x:np.sin(x),x_min=-2*math.pi,x_max=2*math.pi,color=PURPLE_C).shift(7*RIGHT+0.5*UP) + + equal=TextMobject("$=$").next_to(l,buff=0.3) + result1=TextMobject("Amount of").scale(0.6).next_to(equal,buff=0.3) + boxRIGHT=Square(side_length=1.7,fill_color=GOLD_B,fill_opacity=0.5,color=BLACK).next_to(result1,buff=0.2) + self.graph_origin=10*LEFT + sinxResult=self.get_graph(lambda x:np.sin(x),color=PURPLE_C).next_to(result1,buff=0.3) + axes.append(self.axes) + result2=TextMobject("in","$f(x)$").scale(0.6).next_to(sinxResult,buff=0.2).set_color_by_tex_to_color_map({"$f(x)$":RED}) + + self.play(FadeIn(boxUP)) + self.play(ShowCreation(fx)) + self.play(ShowCreation(l)) + self.play(FadeIn(boxDOWN)) + self.play(ShowCreation(sinx)) + self.wait(0.4) + self.play(Write(equal)) + self.play(Write(result1)) + self.play(FadeIn(boxRIGHT)) + self.play(ShowCreation(sinxResult)) + self.play(Write(result2)) + aText1=TextMobject("and").scale(0.65).shift(4*RIGHT+2*DOWN).set_color(GREEN) + self.play(Write(aText1)) + self.wait(0.7) + + self.graph_origin=10*LEFT + cos4x=self.get_graph(lambda x:np.cos(4*x),color=PURPLE_A).shift(7*RIGHT+0.5*UP) + axes.append(self.axes) + self.graph_origin=10*LEFT + cos4xResult=self.get_graph(lambda x:np.cos(4*x),color=PURPLE_A).next_to(result1,buff=0.3) + axes.append(self.axes) + self.play(ReplacementTransform(sinx,cos4x),ReplacementTransform(sinxResult,cos4xResult)) + self.wait(0.7) + + soText=TextMobject("And so on..!").scale(0.65).shift(4*RIGHT+2*DOWN).set_color(GREEN) + self.play(ReplacementTransform(aText1,soText)) + + self.graph_origin=10*LEFT + cosx=self.get_graph(lambda x:np.cos(x),color=GREEN_E).shift(7*RIGHT+0.5*UP) + axes.append(self.axes) + self.graph_origin=10*LEFT + cosxResult=self.get_graph(lambda x:np.cos(x),color=GREEN_E).next_to(result1,buff=0.3) + axes.append(self.axes) + self.play(ReplacementTransform(cos4x,cosx),ReplacementTransform(cos4xResult,cosxResult)) + + self.graph_origin=10*LEFT + cos3x=self.get_graph(lambda x:np.cos(3*x),color=GREEN_C).shift(7*RIGHT+0.5*UP) + axes.append(self.axes) + self.graph_origin=10*LEFT + cos3xResult=self.get_graph(lambda x:np.cos(3*x),color=GREEN_C).next_to(result1,buff=0.3) + axes.append(self.axes) + self.play(ReplacementTransform(cosx,cos3x),ReplacementTransform(cosxResult,cos3xResult)) + + self.graph_origin=10*LEFT + const=self.get_graph(lambda x:1,color=YELLOW_B).shift(7*RIGHT+0.5*UP) + axes.append(self.axes) + self.graph_origin=10*LEFT + constResult=self.get_graph(lambda x:1,color=YELLOW_B).next_to(result1,buff=0.3) + axes.append(self.axes) + self.play(ReplacementTransform(cos3x,const),ReplacementTransform(cos3xResult,constResult)) + + self.wait(1) + + self.play(FadeOut(soText),FadeOut(const),FadeOut(constResult),FadeOut(l),FadeOut(equal),FadeOut(result1),FadeOut(result2),FadeOut(fx),FadeOut(boxRIGHT),FadeOut(boxUP),FadeOut(boxDOWN)) + + finalFormula1=TexMobject(r"Therefore,",r"F(s)",r"=",r"\int _{ -\infty }^{ \infty }",r"{f(t)",r"\over",r"sines",r"\enspace and \enspace",r"cosines}",r"dt }").scale(0.7).set_color_by_tex_to_color_map({"F(s)":RED,"sines":BLUE,"cosines}":YELLOW,"{f(t)":GREEN}) + finalFormula2=TexMobject(r"F(s)",r"=",r"\int _{ -\infty }^{ \infty }",r"{f(t)",r"\over",r"{ e }^",r"{ i\theta }}",r"dt }").set_color_by_tex_to_color_map({"F(s)":RED,"{f(t)":GREEN}) + subFinalFormula=TextMobject("where","$\\theta =2\pi st$").scale(0.5).shift(DOWN+2*RIGHT).set_color_by_tex_to_color_map({"$\\theta =2\pi st$":RED}) + + self.play(Write(finalFormula1)) + self.wait(1) + self.play(ReplacementTransform(finalFormula1,finalFormula2)) + self.play(Write(subFinalFormula)) + self.wait(2) |
