Visualizing the Fourier Series Using Circles : Sawtooth Wave [gnuplot]-Hiro's Soliloquy

Contents[hide]

1.YouTube
2.Simulation [gnuplot]
- 2.1.Source code (PLT file)
- 2.2.Output (PNG file → GIF file)
  - 2.2.1. $n=1,\ 3,\ 5$
  - 2.2.2. $n=10,\ 20,\ 30$
3.External links

YouTube

Simulation [gnuplot]

Source code (PLT file)

	reset
	set angle rad

	#=================== Parameters ====================
	center_x = -1.3*pi
	center_y = 0

	# Parameters in drawing
	lineWidth = 2.0
	pointSize = 1.0
	numPNG = 0
	LOOP = 4

	# Parameters in graph
	numPlotArea = 3
	incDegree = 2
	graphHeight = 0.22
	axisHeight = 0.09
	offsetHeight = 0.02
	marginHeight = (1.0-(numPlotArea*graphHeight+axisHeight+offsetHeight))/3
	leftMargin = 0.21

	# Color
	array color[8] = [6, 2, 4, 1, 7, 3, 5, 8]
	colorNo(i) = color[((i%8==0) ? 8 : (i%8))]
	graphColor = "#b3000000" # W(x)'s line color, black(transparency 70)

	#=================== Functions ====================
	# Sawtooth wave
	radius(i) = (3.(-1)(i+1)) / (pii) # Amplitude of i th sine curve
	omega(i) = i # Angle velocity
	fourier(n, x) = sum[k=1:n] radius(k)sin(omega(k)x) # Fourier series
	x(i, t) = center_x - ((i>1) ? (sum[k=1:i-1]radius(k)cos(omega(k)(t+pi))) : 0) # Position of the i th circle
	y(i, t) = center_y - ((i>1) ? (sum[k=1:i-1]radius(k)sin(omega(k)(t+pi))) : 0)
	original(i, x) = ((2i-1)pi<=x && x<(2i+1)pi) ? (3./(2pi) (x-2pii)): 0

	wave(x) = sum[i=-LOOP:0] original(i, x) # Make original() the periodic function
	F(n, x, t) = (x>=0 && x<=t) ? fourier(n, (x+pi)-t) : 1/0 # Translate fourier() and restrict the domain
	W(x, t) = (x>=0 && x<=t) ? wave((x+pi)-t) : 1/0 # Translate wave() and restrict the domain

	#=================== Plot ====================
	# Setting
	set term pngcairo truecolor enhanced dashed size 1080, 720 font 'Times, 20'
	system 'mkdir png'
	set size ratio -1
	set samples 5000

	do for [deg=0:360*LOOP:2] {
	t = deg * pi/180
	set output sprintf("png/img_%04d.png", numPNG)
	numPNG = numPNG + 1

	set multiplot
	do for [i=1:numPlotArea:1]{
	numFourier = incDegree*((1+numPlotArea)-i)-1 # n=1, 3, 5
	# numFourier = 10*((1+numPlotArea)-i) # n=10, 20, 30

	# Common setting
	bottomGraphY = axisHeight+(i-1)*(graphHeight+marginHeight) + offsetHeight
	topGraphY = (axisHeight+graphHeight) + (i-1)*(marginHeight+graphHeight) + offsetHeight

	# Set the position and size of the graph
	set lmargin screen leftMargin
	set bmargin screen bottomGraphY
	set tmargin screen topGraphY

	centerGraphY = (bottomGraphY+topGraphY)/2
	set label 1 left sprintf('{/:Italic n} = %d', numFourier) at screen 0.04, centerGraphY font ', 22'

	unset key
	set grid
	set xr[2center_x:4pi]
	set yr[-2:2]
	unset xl
	set yl '{/:Italic y}' offset screen 0, 0
	set xtics nomirror (0, pi, 2pi, 3pi, 4pi, 5pi, 6*pi) offset screen 0, 0.015 tc rgb 'white' # Draw only tic marks
	set ytics nomirror (-2, '' -1, 0, '' 1, 2) offset screen 0., 0

	if(i==1){
	set xl '{/:Italic x}' offset screen 0, 0.02
	set xtics nomirror (0, '{/:Italic π}' pi, '2{/:Italic π}' 2pi, '3{/:Italic π}' 3pi, \
	'4{/:Italic π}' 4pi, '5{/:Italic π}' 5pi, '6{/:Italic π}' 6*pi) tc rgb 'black'
	}

	# Draw circles and arrows without head representing circle's rotation
	do for [j=1:numFourier]{
	set object j circle at x(j, t), y(j, t) size abs(radius(j)) fs empty border lt colorNo(j) lw lineWidth front
	set arrow j nohead from x(j, t), y(j, t) to x(j+1, t), y(j+1, t) lw lineWidth lt colorNo(j) back
	}

	# Draw the line connected between the numFourier th circle and the graph F(x)
	set arrow numFourier+1 nohead from x(numFourier+1, t), y(numFourier+1, t) to 0, y(numFourier+1, t) lw lineWidth lt colorNo(numFourier) front #lc rgb auxColor front

	# Plot function F() and W(), and draw the points representing the center of the circle
	plotCommand = sprintf("plot F(numFourier, x, t) lw %f lt %d", lineWidth, colorNo(numFourier))
	plotCommand = plotCommand.sprintf(", \"<echo '%.2f, %.2f'\" w p pt 7 ps %d lt %d", 0, y(numFourier+1, t), pointSize, colorNo(numFourier))
	plotCommand = plotCommand.sprintf(", W(x, t) lw %f lc rgb '%s'", lineWidth, graphColor)
	end = ((numFourier>3)?3:numFourier)
	do for [j=1:end]{
	plotCommand = plotCommand.sprintf(", \"<echo '%.2f, %.2f'\" w p pt 7 ps %d lt %d", x(j+1, t), y(j+1, t), pointSize, colorNo(j))
	}

	eval plotCommand

	# Unset objects and arrows in order not to display them in the other plot area
	do for [j=1:numFourier]{
	unset object j
	unset arrow j
	}
	unset arrow numFourier+1
	unset label 1

	}
	unset multiplot

	set out # Output PNG file
	}