From 6f6f9ab5cc64acb27045de465cde6527a5e92310 Mon Sep 17 00:00:00 2001 From: leios Date: Wed, 10 Jul 2019 19:39:46 +0900 Subject: [PATCH 1/7] adding simple text for domain coloring chapter. --- contents/domain_coloring/domain_coloring.md | 77 +++++++++++++++++++++ 1 file changed, 77 insertions(+) create mode 100644 contents/domain_coloring/domain_coloring.md diff --git a/contents/domain_coloring/domain_coloring.md b/contents/domain_coloring/domain_coloring.md new file mode 100644 index 000000000..ef6fc6a8e --- /dev/null +++ b/contents/domain_coloring/domain_coloring.md @@ -0,0 +1,77 @@ +# Domain coloring + +Domain coloring is a much more complicated plotting technique than those outlined above and is used to plot complex functions where both the input and output have imaginary and real components. +Imagine the following function: + +$$f(z) = z^2$$ + +In this case, we could create a plot that looks like this: + +ADD IMAGE + +This indicated that for various input values along $$z$$, we have different function outputs from $$f(z)$$. + +Unfortunately, complex variables and functions cannot be easily plotted in this way because the input variable and the output of $$f(z)$$ both need a two-dimensional space for plotting -- one axis for their real component and another for their imaginary component. + +Now let's imagine another function: + +$$f(z) = |z|$$ + +In this case, each complex input has a real output. +This can be plotted as a two-dimensional dataset like so: + +Here, the $$x$$-axis and $$y$$-axis represent the real and imaginary components of the input variable respectively. + +Here we see the problem. +If the output also requires plotting of the real and imaginary components, we need four dimensions to appropriately represent complex functions! +Unfortunately, feeble human minds are incapable of understanding four spatial dimensions, so we need to imoprovise. + +We do this by assuming the complex output can be represented in the following form: + +$$z = re^{i \theta} = r(\cos(\theta) + i\sin(\theta)$$ + +where, $$r$$ is a complex magnitude and $$\theta$$ is a complex phase. +This is the formula for a circle in the complex plane and we can easily find $$r$$ and $$\theta$$ like so: + +$$ +\begin{align} + r &= \sqrt{\text{real}(z)^2 + \text{imag}(z)^2} \\ + \theta &= +\end{align} +$$ + +Where $$\text{real}(z)$$ and $$\text{image}(z)$$ represent the real and imaginary components of $$z$$. + +Once we have our complex function output in this form, we then color the output domain according to a number space with more than 2 dimensions -- Often HSV or HSL. + +THESE WORK LIKE SO + +So if we are using HSV, then we set the color values for each pixel in our image like so: + +$$ +\begin{align} + H &= \\ + S &= \\ + V &= 100\% +\end{align} +$$ + +### Algorithms using this method: +- Split operator method + +##### Code Examples + +The code examples are licensed under the MIT license (found in [LICENSE.md](https://github.com/algorithm-archivists/algorithm-archive/blob/master/LICENSE.md)). + +##### Text + +The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + +[

](https://creativecommons.org/licenses/by-sa/4.0/) + +##### Images/Graphics + +##### Pull Requests + +The following pull requests have modified the text or graphics of this chapter: +- none From 545c6f2deae84e23ea47ba72619cb0bc8eb5f4d8 Mon Sep 17 00:00:00 2001 From: James Schloss Date: Thu, 21 May 2020 19:18:15 -0400 Subject: [PATCH 2/7] adding domain coloring to summary --- SUMMARY.md | 1 + 1 file changed, 1 insertion(+) diff --git a/SUMMARY.md b/SUMMARY.md index 37202640a..3f152ba7a 100644 --- a/SUMMARY.md +++ b/SUMMARY.md @@ -5,6 +5,7 @@ * [How To Contribute](contents/how_to_contribute/how_to_contribute.md) * [Plotting](contents/plotting/plotting.md) * [Iterated Function Systems](contents/IFS/IFS.md) + * [Domain Coloring](contents/domain_coloring/domain_coloring.md) * [Data Structures](contents/data_structures/data_structures.md) * [Stacks and Queues](contents/stacks_and_queues/stacks_and_queues.md) * [Mathematical Background](contents/mathematical_background/mathematical_background.md) From bf33995b3a8334b3ce6ca1c717c62567c842f601 Mon Sep 17 00:00:00 2001 From: James Schloss Date: Fri, 29 May 2020 19:37:26 -0400 Subject: [PATCH 3/7] adding new images and finishing intro for domain coloring chapter --- SUMMARY.md | 2 +- contents/domain_coloring/code/gnuplot/absz.gp | 32 +++++++++++ contents/domain_coloring/code/gnuplot/z2.gp | 9 ++++ contents/domain_coloring/domain_coloring.md | 50 +++++++++++------- contents/domain_coloring/res/absz.png | Bin 0 -> 44329 bytes contents/domain_coloring/res/z2.png | Bin 0 -> 41619 bytes 6 files changed, 73 insertions(+), 20 deletions(-) create mode 100644 contents/domain_coloring/code/gnuplot/absz.gp create mode 100644 contents/domain_coloring/code/gnuplot/z2.gp create mode 100644 contents/domain_coloring/res/absz.png create mode 100644 contents/domain_coloring/res/z2.png diff --git a/SUMMARY.md b/SUMMARY.md index 3f152ba7a..d763f88e3 100644 --- a/SUMMARY.md +++ b/SUMMARY.md @@ -4,8 +4,8 @@ * [Introduction](contents/introduction/introduction.md) * [How To Contribute](contents/how_to_contribute/how_to_contribute.md) * [Plotting](contents/plotting/plotting.md) - * [Iterated Function Systems](contents/IFS/IFS.md) * [Domain Coloring](contents/domain_coloring/domain_coloring.md) + * [Iterated Function Systems](contents/IFS/IFS.md) * [Data Structures](contents/data_structures/data_structures.md) * [Stacks and Queues](contents/stacks_and_queues/stacks_and_queues.md) * [Mathematical Background](contents/mathematical_background/mathematical_background.md) diff --git a/contents/domain_coloring/code/gnuplot/absz.gp b/contents/domain_coloring/code/gnuplot/absz.gp new file mode 100644 index 000000000..07846ab34 --- /dev/null +++ b/contents/domain_coloring/code/gnuplot/absz.gp @@ -0,0 +1,32 @@ +set terminal epslatex standalone size 12cm,10cm color +set output "absz.tex" +#set terminal pngcairo +#set output "absz.png" + +set size square +set view map + +set samples 1000 +set isosamples 1000 + +set xrange[-1:1] +set yrange[-1:1] + +set xlabel "Re$(z)$" +set ylabel "Im$(z)$" +set cblabel "$|z|$" + +# palette +set palette defined ( 0 '#2166AC',\ + 1 '#4393C3',\ + 2 '#92C5DE',\ + 3 '#D1E5F0',\ + 4 '#FDDBC7',\ + 5 '#F4A582',\ + 6 '#D6604D',\ + 7 '#B2182B') +set palette defined (0 '#000000', 1 '#FFFFFF') + +cabs(x,y) = sqrt(x**2 + y**2) + +splot "++" u 1:2:(cabs($1, $2)) with pm3d notitle diff --git a/contents/domain_coloring/code/gnuplot/z2.gp b/contents/domain_coloring/code/gnuplot/z2.gp new file mode 100644 index 000000000..aa442d41e --- /dev/null +++ b/contents/domain_coloring/code/gnuplot/z2.gp @@ -0,0 +1,9 @@ +set terminal epslatex standalone size 10cm,10cm +set output "z2.tex" + +set size square + +set xlabel "$z$" +set ylabel "$f(z)$" + +plot x**2 w l lw 4 title "$z^2$" diff --git a/contents/domain_coloring/domain_coloring.md b/contents/domain_coloring/domain_coloring.md index ef6fc6a8e..2ff09c33d 100644 --- a/contents/domain_coloring/domain_coloring.md +++ b/contents/domain_coloring/domain_coloring.md @@ -1,30 +1,34 @@ # Domain coloring -Domain coloring is a much more complicated plotting technique than those outlined above and is used to plot complex functions where both the input and output have imaginary and real components. -Imagine the following function: - -$$f(z) = z^2$$ +Domain coloring is a much more complicated plotting technique than those outlined in the [plotting chapter](../plotting/plotting.md) and is used to plot complex functions where both the input and output have imaginary and real components. +As stated in similar chapters, when discussing plotting methods, we will focus on plotting languages, instead of languages meant for number crunching or other purposes. +That is to say that this chapter will certainly have a code implementation in gnuplot, but it will not likely have an implementation in C, Fortran, or Java because these chapters do not have plotting capabilities in-built. +Imagine the following function: $$f(z) = z^2$$. In this case, we could create a plot that looks like this: -ADD IMAGE - -This indicated that for various input values along $$z$$, we have different function outputs from $$f(z)$$. +

+ +

-Unfortunately, complex variables and functions cannot be easily plotted in this way because the input variable and the output of $$f(z)$$ both need a two-dimensional space for plotting -- one axis for their real component and another for their imaginary component. +This indicates that for various input values along $$z$$, we have different function outputs from $$f(z)$$. -Now let's imagine another function: +Now let's imagine another function with complex input $$(z \in \mathbb{C})$$, but a purely real output $$(f(z) \in \mathbb{R})$$: $$f(z) = |z|$$ In this case, each complex input has a real output. This can be plotted as a two-dimensional dataset like so: -Here, the $$x$$-axis and $$y$$-axis represent the real and imaginary components of the input variable respectively. +

+ +

+ +Here, the $$x$$-axis and $$y$$-axis represent the real and imaginary components of the input variable, respectively. -Here we see the problem. -If the output also requires plotting of the real and imaginary components, we need four dimensions to appropriately represent complex functions! -Unfortunately, feeble human minds are incapable of understanding four spatial dimensions, so we need to imoprovise. +At this point, we can start to see the problem. +If the output of $$f(z)$$ also requires plotting of the real and imaginary components, then we would need four dimensions to appropriately represent complex functions, one axis for their real component and another for their imaginary component of both the input ($$z$$) and the output of $$f(z)$$! +Unfortunately, feeble human minds are incapable of understanding four spatial dimensions without projecting onto lower dimensionality, so we need to improvise. We do this by assuming the complex output can be represented in the following form: @@ -35,14 +39,19 @@ This is the formula for a circle in the complex plane and we can easily find $$r $$ \begin{align} - r &= \sqrt{\text{real}(z)^2 + \text{imag}(z)^2} \\ - \theta &= + r &= \sqrt{\text{Re}(z)^2 + \text{Im}(z)^2} \\ + \theta &= \text{atan}(\frac{\text{Im}(z)}{\text{Re}(z)}) \end{align} $$ -Where $$\text{real}(z)$$ and $$\text{image}(z)$$ represent the real and imaginary components of $$z$$. +Once we have our complex function output in this form, we then color the output domain according to a number space with 2 or more dimensions, for example black and white, RGB (Red, Green, Blue), or HSV (Hue, Saturation, Value). +The choice of color space is completely dependent on what the users feel is most visually intuitive. +In any case, one dimension of the color system will be used to represent the complex magnitude of the output and another dimension of the color system will be used to represent the complex phase. +The $xy$ grid will be representing the real and imaginary inputs to these functions. +That is to say, we plug every value in the 2D complex plane into the function and then color each pixel based on the function output. + +As an example, let's again look at the fuction $$f(z) = z^2$$, but in this case $$z \in \mathbb{C}$$. -Once we have our complex function output in this form, we then color the output domain according to a number space with more than 2 dimensions -- Often HSV or HSL. THESE WORK LIKE SO @@ -56,8 +65,11 @@ $$ \end{align} $$ -### Algorithms using this method: -- Split operator method + + +## License ##### Code Examples diff --git a/contents/domain_coloring/res/absz.png b/contents/domain_coloring/res/absz.png new file mode 100644 index 0000000000000000000000000000000000000000..01d8e032de1a08462e3e20d4ad055c83b24e76de GIT binary patch literal 44329 zcmb@u2{hDy|2D2w`)a8yseDz0N~jT1DPzmNGnK4k$TG^Fv{Kd(LfMUFWEs0@`9_g# zY?-l?CCiX3GnO$tuMd6i=Xc-dKELz)&-v?gFf%^O`~6z4>w0~jYH42GzU9Cc4i1j( zXqC%4931Pa;Qzr*8{w5ng{`U_9BUrhUAm-&zH~|Gwu_Usor4tz$FVSmd0XY-Cx>PZ-4lEo zv)~{a`Cw?S>~RS3&Z&e!pI?p5sRJ4t=Pvi{Pvhg6L`}A)rMmVU4bBTs*_F6AU>csD_Pr}fRyRSa^bd#?R_f_?g>UlK3zTWexhuL*ohQu4}OKW3i2PC%* z6%GEC+roc&?N+19TMpmSd+UGt#NW1UDyPEI*d>qs-c6=|{wo5z?iv4d#)XFibMKeG z-zm1fakEQfOO5~cAe`T}b#KgHYZr`9SkfF{aV*(M5a3-e@*s92?6D9B`m&PVUFul3$4joZ&e_S{dnY~~PTIw-)ae|w<%yi* zv19PduJ!BBB)xulSv~2e^7AXF`VMsMk$#e~X507V=GSYr^>*zx7JR>fm*>#Ed-;-5 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zb#cD#n=$ZIC{JMS7vTZ7m}6#wHpK!9q^rQQLO>Txf%b!gj)MW7Lig&`E8tMTdf))d zO5nCn5pIZvozvs%BG21=?y<44nU%X9xSJW+#sf9|f%!351MK1pKY^=DfM?Tw0qzt9 zo*>oO#NZ%)ko7r3gY+rjat}}e1ZwfH@H4zfve95xH}G&#UA`l9Cc_WbMlb)Tg{ZS%0zTaVTO@1q5lDqO4#*Cnq5 zBl1oZun+;I&7Pi~u;$~|@AoWr?G_8vV0gh4@e>$KYv)>*X93Fukh#EWHTsQx)t43X z>i<>N{&?69^yV&xysg{}1>T_R7J)~azT5NJ57=G-t~7qJ`~Kf``@XKt*95k>K^q!C zC#eGuT0A@7etr{pYN}!Vx`hiJfz`vWpXck>0oU<{gLXM^f%drjEx%kE18mp>mscxI zKIw81oEQ&;PY0fH4!k_a_R|UFv)k4K=Y-eq{Wc32w7{EUO?>Cu%}oNE3+(NbAHKT| zlp;w;TFB`c!XAKB{oy9>0id%(JY5_^U|9^s8JJlNIdehSDBU+m2*gzX=RXzD)m&Tc Qv=tOCp00i_>zopr03Iz|*Z=?k literal 0 HcmV?d00001 From 54d15ff21e9a2d8ddd45cb4c09a1bcc0f0579b1b Mon Sep 17 00:00:00 2001 From: James Schloss Date: Thu, 18 Jun 2020 19:56:57 -0400 Subject: [PATCH 4/7] adding a few more images and starting HSV domain coloring --- .../code/gnuplot/domain_coloring.gp | 70 ++++++++++++++++++ .../code/gnuplot/domain_coloring_rgb.gp | 66 +++++++++++++++++ .../domain_coloring/code/gnuplot/z2_hsv.gp | 9 +++ contents/domain_coloring/domain_coloring.md | 64 +++++++++++----- contents/domain_coloring/res/hsv.png | Bin 0 -> 114294 bytes contents/domain_coloring/res/rgb1.png | Bin 0 -> 108736 bytes contents/domain_coloring/res/rgb2.png | Bin 0 -> 123306 bytes 7 files changed, 190 insertions(+), 19 deletions(-) create mode 100644 contents/domain_coloring/code/gnuplot/domain_coloring.gp create mode 100644 contents/domain_coloring/code/gnuplot/domain_coloring_rgb.gp create mode 100644 contents/domain_coloring/code/gnuplot/z2_hsv.gp create mode 100644 contents/domain_coloring/res/hsv.png create mode 100644 contents/domain_coloring/res/rgb1.png create mode 100644 contents/domain_coloring/res/rgb2.png diff --git a/contents/domain_coloring/code/gnuplot/domain_coloring.gp b/contents/domain_coloring/code/gnuplot/domain_coloring.gp new file mode 100644 index 000000000..37f0d8c7d --- /dev/null +++ b/contents/domain_coloring/code/gnuplot/domain_coloring.gp @@ -0,0 +1,70 @@ +# setting output to file of size 800 x 800 +#set terminal pngcairo size 1000, 1000 +#set output 'out_2.png' +set terminal epslatex standalone size 12cm,10cm color +set output "hsv.tex" + +# removes legend +unset key + +# projects image onto 2D plane +set view map + +# sets aspect ratio of plot to be square +set size square + +# setting x and y range +set xrange[-2:2] +set yrange[-2:2] + +set xlabel "Re$(x)$" +set ylabel "Im$(x)$" + +# scaling the x, y, and colorbar tics to zero so they are not seen in the plot +set xtics border scale 0,0 +set ytics border scale 0,0 +set cbtics border scale 0,0 + +# setting tics in color bar at 0 and 2pi +set cbtics ("0" -3.14159, '2$\pi$' 3.14159) +set title "Domain coloring of output" + +set cblabel "Phase Angle" +set cbrange [ -3.14159 : 3.14159 ] + +# use hsv for colorbar and set palette to use full hsv space +set palette model HSV +set palette defined ( 0 0 1 1, 1 1 1 1 ) + +# setting isosamples for output grid and samples for input grid +set isosamples 1000, 1000 +set samples 1000, 1000 + +# setting functions necessary for domain coloring +# setting threshold for gridlines. Smaller threshold will make smaller lines +thresh = 0.1 +f(z) = z**2 +#f(z) = z**3-1 +#f(z) = z**5 + 3*z**4 + 2*z + 1 +#f(z) = z - z**3/3! + z**5/5! +#f(z) = 1-sin(z) +angle(x,y) = (pi + atan2(-y,-x)) / (2*pi) +r(x,y) = sqrt(x*x + y*y) +theta(x,y) = atan2(x,y) +z(x,y) = r(x,y)*exp(theta(x,y)*sqrt(-1)) +ip(x,y) = imag(z(x,y)) +rp(x,y) = real(z(x,y)) +if(z) = imag(f(z)) +rf(z) = real(f(z)) +val(x,y) = 0.5 + 0.5*(abs(f(z(x,y)))-floor(abs(f(z(x,y))))) +color(x,y) = hsv2rgb(angle(rf(z(x,y)), if(z(x,y))), abs(f(z(x,y)))/8, 1) +#color(x,y) = hsv2rgb(angle(rf(z(x,y)), if(z(x,y))), abs(f(z(x,y))), val(x,y)*shade(x,y)) +#color(x,y) = hsv2rgb(angle(rp(x,y), ip(x,y)), abs(z(x,y)), val(x,y)*shade(x,y)) +#shade(x,y) = (abs(sin(rp(x,y)*pi)**thresh) * abs(sin(ip(x,y)*pi))**thresh) +shade(x,y) = (abs(sin(rf(z(x,y))*pi)**thresh) * abs(sin(if(z(x,y))*pi))**thresh) +save_encoding = "utf8" + +# Last datafile plotted: "++" +# In this case, it means, "plot the data file created with the +# samples and isosamples" +splot '++' using 1:2:(color($1,$2)) with pm3d lc rgb variable nocontour diff --git a/contents/domain_coloring/code/gnuplot/domain_coloring_rgb.gp b/contents/domain_coloring/code/gnuplot/domain_coloring_rgb.gp new file mode 100644 index 000000000..2dbc660e9 --- /dev/null +++ b/contents/domain_coloring/code/gnuplot/domain_coloring_rgb.gp @@ -0,0 +1,66 @@ +# setting output to file of size 800 x 800 +#set terminal pngcairo size 1000, 1000 +#set output 'out.png' +set terminal epslatex standalone size 12cm,10cm color +set output "rgb.tex" + +# removes legend +unset key + +# projects image onto 2D plane +set view map + +# sets aspect ratio of plot to be square +set size square + +# setting x and y range +set xrange[-2:2] +set yrange[-2:2] + +set xlabel "Re$(x)$" +set ylabel "Im$(x)$" + +# scaling the x, y, and colorbar tics to zero so they are not seen in the plot +set xtics border scale 0,0 +set ytics border scale 0,0 +set cbtics border scale 0,0 + +# setting tics in color bar at 0 and 2pi +set cbtics ("0" 0, '2$\pi$' 2*3.14159) +set title "Domain coloring of output" + +#set palette defined (0 'black', 1 'blue') +#set cblabel 'Complex Magnitude $(r)$' offset 1, 0 +#set cbrange [ 0 : 8 ] +set palette defined (0 'black', 1 'red') +set cblabel 'Phase Angle $(\theta)$' +set cbrange [ 0 : 2*3.14159 ] + +# use hsv for colorbar and set palette to use full hsv space + +# setting isosamples for output grid and samples for input grid +set isosamples 1000, 1000 +set samples 1000, 1000 + +# setting functions necessary for domain coloring +# setting threshold for gridlines. Smaller threshold will make smaller lines +thresh = 0.1 +#f(z) = z +f(z) = z**2 + +rgb(r,g,b) = 65536 * int(r) + 256 * int(g) + int(b) +angle(x,y) = (pi + atan2(-y,-x)) / (2*pi) +r(x,y) = sqrt(x*x + y*y) +theta(x,y) = atan2(x,y) +z(x,y) = r(x,y)*exp(theta(x,y)*sqrt(-1)) +ip(x,y) = imag(z(x,y)) +rp(x,y) = real(z(x,y)) +if(z) = imag(f(z)) +rf(z) = real(f(z)) +color(x,y) = rgb(angle(rf(z(x,y)), if(z(x,y)))*255, 0, abs(f(z(x,y)))*255/8) +save_encoding = "utf8" + +# Last datafile plotted: "++" +# In this case, it means, "plot the data file created with the +# samples and isosamples" +splot '++' using 1:2:(color($1,$2)) with pm3d lc rgb variable diff --git a/contents/domain_coloring/code/gnuplot/z2_hsv.gp b/contents/domain_coloring/code/gnuplot/z2_hsv.gp new file mode 100644 index 000000000..aa442d41e --- /dev/null +++ b/contents/domain_coloring/code/gnuplot/z2_hsv.gp @@ -0,0 +1,9 @@ +set terminal epslatex standalone size 10cm,10cm +set output "z2.tex" + +set size square + +set xlabel "$z$" +set ylabel "$f(z)$" + +plot x**2 w l lw 4 title "$z^2$" diff --git a/contents/domain_coloring/domain_coloring.md b/contents/domain_coloring/domain_coloring.md index 2ff09c33d..e037ec21f 100644 --- a/contents/domain_coloring/domain_coloring.md +++ b/contents/domain_coloring/domain_coloring.md @@ -1,8 +1,8 @@ # Domain coloring Domain coloring is a much more complicated plotting technique than those outlined in the [plotting chapter](../plotting/plotting.md) and is used to plot complex functions where both the input and output have imaginary and real components. -As stated in similar chapters, when discussing plotting methods, we will focus on plotting languages, instead of languages meant for number crunching or other purposes. -That is to say that this chapter will certainly have a code implementation in gnuplot, but it will not likely have an implementation in C, Fortran, or Java because these chapters do not have plotting capabilities in-built. +For the code in this chapter, we will focus on languages that are easily able to plot two-dimensional images or heatmaps, instead of languages meant for number-crunching. +That is to say that this chapter will certainly have a code implementation in gnuplot, but it will not likely have an implementation in C, Fortran, or Java because these languages do not have plotting capabilities in-built. Imagine the following function: $$f(z) = z^2$$. In this case, we could create a plot that looks like this: @@ -12,10 +12,11 @@ In this case, we could create a plot that looks like this:

This indicates that for various input values along $$z$$, we have different function outputs from $$f(z)$$. - Now let's imagine another function with complex input $$(z \in \mathbb{C})$$, but a purely real output $$(f(z) \in \mathbb{R})$$: -$$f(z) = |z|$$ +$$ +f(z) = |z| +$$ In this case, each complex input has a real output. This can be plotted as a two-dimensional dataset like so: @@ -25,14 +26,17 @@ This can be plotted as a two-dimensional dataset like so:

Here, the $$x$$-axis and $$y$$-axis represent the real and imaginary components of the input variable, respectively. +The colorbar represents the output of $$f(z)$$. At this point, we can start to see the problem. -If the output of $$f(z)$$ also requires plotting of the real and imaginary components, then we would need four dimensions to appropriately represent complex functions, one axis for their real component and another for their imaginary component of both the input ($$z$$) and the output of $$f(z)$$! +If the output of $$f(z)$$ also requires plotting of the real and imaginary components, then we would need four dimensions to appropriately represent the function space, one axis for the real component and another for the imaginary component of both the input ($$z$$) and the output of $$f(z)$$! Unfortunately, feeble human minds are incapable of understanding four spatial dimensions without projecting onto lower dimensionality, so we need to improvise. We do this by assuming the complex output can be represented in the following form: -$$z = re^{i \theta} = r(\cos(\theta) + i\sin(\theta)$$ +$$ +z = re^{i \theta} = r(\cos(\theta) + i\sin(\theta)) +$$ where, $$r$$ is a complex magnitude and $$\theta$$ is a complex phase. This is the formula for a circle in the complex plane and we can easily find $$r$$ and $$\theta$$ like so: @@ -40,30 +44,52 @@ This is the formula for a circle in the complex plane and we can easily find $$r $$ \begin{align} r &= \sqrt{\text{Re}(z)^2 + \text{Im}(z)^2} \\ - \theta &= \text{atan}(\frac{\text{Im}(z)}{\text{Re}(z)}) + \theta &= \text{atan}\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right) \end{align} $$ -Once we have our complex function output in this form, we then color the output domain according to a number space with 2 or more dimensions, for example black and white, RGB (Red, Green, Blue), or HSV (Hue, Saturation, Value). +Once we have our complex function output in this form, we then color the output domain according to a color space with at least 2 independent dimensions, like RGB (Red, Green, Blue), or HSV (Hue, Saturation, Value) [CITE]. The choice of color space is completely dependent on what the users feel is most visually intuitive. In any case, one dimension of the color system will be used to represent the complex magnitude of the output and another dimension of the color system will be used to represent the complex phase. -The $xy$ grid will be representing the real and imaginary inputs to these functions. +The $$xy$$ grid will be representing the real and imaginary inputs to these functions. That is to say, we plug every value in the 2D complex plane into the function and then color each pixel based on the function output. -As an example, let's again look at the fuction $$f(z) = z^2$$, but in this case $$z \in \mathbb{C}$$. +As an example, let's look at the simplest function we can $$f(z) = z$$, but in this case $$z \in \mathbb{C}$$. +If we use an RGB color scheme, where red represents $$\theta$$ and blue represents $$r$$, we can generate the following image: +

+ +

-THESE WORK LIKE SO +As a note here, there is a clear phase discontinuity along the vertical axis. +That is to say that the complex phase wraps around the origin, ranging from 0 (clear) to $$2\pi$$ (red). +In addition, the edges of the plot are blue because the function value increases linearly as we move from the origin. -So if we are using HSV, then we set the color values for each pixel in our image like so: +If we instead look at the function $$f(z) = z^2$$, we can generate a similar plot: -$$ -\begin{align} - H &= \\ - S &= \\ - V &= 100\% -\end{align} -$$ +

+ +

+ +Here, it is clear that the complex phase wraps around the origin twice, creating two separate phase discontinuities on top of each other. +This indicates a $$4\pi$$ phase winding, and for some purposes, such as vortex tracking for inviscid fluids, this visualizaton is ideal, because a vortex is located precisely at the phase discontinuity [CITE]. +For other purposes, the discontinuity is visually distracting, and for this reason, many people use an HSV scheme for plotting complex functions. +So here is the same function $$\left(f(z)=z^2\right)$$, but using hue to represent the complex phase and saturation to represent the magnitude. + +

+ +

+ +Here, the value for HSV was always set to 1. +When looking at the edges of the plot, the hue changes rapidly, but each color is mirrored on the oposite edge. +This indicated the $$4\pi$$ phase winding we saw in the RGB plots. +Also, because the complex magnitude increases as we move further from the center of the plot, the saturation also increases. +Thus the very center of the plot is completely washed out! +We need to fix this in subsequent plots to make them representative of the actual data. + +## Example Code + +Here is the full script to generate a domain colored output of $$f(z)=z^2$$. @@ -101,6 +178,12 @@ MathJax.Hub.Queue(["Typeset",MathJax.Hub]); The code examples are licensed under the MIT license (found in [LICENSE.md](https://github.com/algorithm-archivists/algorithm-archive/blob/master/LICENSE.md)). +{% method %} +{% sample lang="gnuplot" %} +[import, lang:"gnuplot"](code/gnuplot/domain_coloring.gp) +{% endmethod %} + + ##### Text The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). @@ -109,6 +192,17 @@ The text of this chapter was written by [James Schloss](https://github.com/leios ##### Images/Graphics +- The image "[z2](res/z2.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[absz](res/absz.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[rgb1](res/rgb1.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[rgb2](res/rgb2.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[hsv1](res/hsv.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[hsv2](res/hsv2.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[hsv3](res/hsv3.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[hsv4](res/hsv4.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[shade](res/shade.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). +- The image "[hsv5](res/hsv5.png)" was created by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + ##### Pull Requests The following pull requests have modified the text or graphics of this chapter: diff --git a/contents/domain_coloring/domain_coloring.md.bak b/contents/domain_coloring/domain_coloring.md.bak new file mode 100644 index 000000000..8ec9129ad --- /dev/null +++ b/contents/domain_coloring/domain_coloring.md.bak @@ -0,0 +1,185 @@ +# Domain coloring + +Domain coloring is a much more complicated plotting technique than those outlined in the [plotting chapter](../plotting/plotting.md) and is used to plot complex functions where both the input and output have imaginary and real components. +For the code in this chapter, we will focus on languages that are easily able to plot two-dimensional images or heatmaps, instead of languages meant for number-crunching. +That is to say that this chapter will certainly have a code implementation in gnuplot, but it will not likely have an implementation in C, Fortran, or Java because these languages do not have plotting capabilities in-built. + +Imagine the following function: $$f(z) = z^2$$. +In this case, we could create a plot that looks like this: + +

+ +

+ +This indicates that for various input values along $$z$$, we have different function outputs from $$f(z)$$. +Now let's imagine another function with complex input $$(z \in \mathbb{C})$$, but a purely real output $$(f(z) \in \mathbb{R})$$: + +$$ +f(z) = |z| +$$ + +In this case, each complex input has a real output. +This can be plotted as a two-dimensional dataset like so: + +

+ +

+ +Here, the $$x$$-axis and $$y$$-axis represent the real and imaginary components of the input variable, respectively. +The colorbar represents the output of $$f(z)$$. + +At this point, we can start to see the problem. +If the output of $$f(z)$$ also requires plotting of the real and imaginary components, then we would need four dimensions to appropriately represent the function space, one axis for the real component and another for the imaginary component of both the input ($$z$$) and the output of $$f(z)$$! +Unfortunately, feeble human minds are incapable of understanding four spatial dimensions without projecting onto lower dimensionality, so we need to improvise. + +We do this by assuming the complex output can be represented in the following form: + +$$ +z = re^{i \theta} = r(\cos(\theta) + i\sin(\theta)) +$$ + +where, $$r$$ is a complex magnitude and $$\theta$$ is a complex phase. +This is the formula for a circle in the complex plane and we can easily find $$r$$ and $$\theta$$ like so: + +$$ +\begin{align} + r &= \sqrt{\text{Re}(z)^2 + \text{Im}(z)^2} \\ + \theta &= \text{atan}\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right) +\end{align} +$$ + +Once we have our complex function output in this form, we then color the output domain according to a color space with at least 2 independent dimensions, like RGB (Red, Green, Blue), or HSV (Hue, Saturation, Value) [CITE]. +The choice of color space is completely dependent on what the users feel is most visually intuitive. +In any case, one dimension of the color system will be used to represent the complex magnitude of the output and another dimension of the color system will be used to represent the complex phase. +The $$xy$$ grid will be representing the real and imaginary inputs to these functions. +That is to say, we plug every value in the 2D complex plane into the function and then color each pixel based on the function output. + +As an example, let's look at the simplest function we can $$f(z) = z$$, but in this case $$z \in \mathbb{C}$$. +If we use an RGB color scheme, where red represents $$\theta$$ and blue represents $$r$$, we can generate the following image: + +

+ +

+ +As a note here, there is a clear phase discontinuity along the vertical axis. +That is to say that the complex phase wraps around the origin, ranging from 0 (clear) to $$2\pi$$ (red). +In addition, the edges of the plot are blue because the function value increases linearly as we move from the origin. + +If we instead look at the function $$f(z) = z^2$$, we can generate a similar plot: + +

+ +

+ +Here, it is clear that the complex phase wraps around the origin twice, creating two separate phase discontinuities on top of each other. +This indicates a $$4\pi$$ phase winding, and for some purposes, such as vortex tracking for inviscid fluids, this visualizaton is ideal, because a vortex is located precisely at the phase discontinuity [CITE]. +For other purposes, the discontinuity is visually distracting, and for this reason, many people use an HSV scheme for plotting complex functions. +So here is the same function $$\left(f(z)=z^2\right)$$, but using hue to represent the complex phase and saturation to represent the magnitude. + +

+ +

+ +Here, the value for HSV was always set to 1. +When looking at the edges of the plot, the hue changes rapidly, but each color is mirrored on the oposite edge. +This indicates the $$4\pi$$ phase winding we saw in the RGB plots. +Also, because the complex magnitude increases as we move further from the center of the plot, the saturation also increases. +Thus the very center of the plot is completely washed out! +We need to fix this in subsequent plots to make them representative of the actual data. + +One easy way to show that the complex magnitude is increasing as we move further from the origin is with contours. +Essentially, at ever integer value of the magnitude, we want to draw some kind of line. +There are a number of ways to generate these lines, and one simple way is by using an alternative shading function like so: + +$$ +g(r) = r-\lfloor r \rfloor. +$$ + +This will create the following image: + +

+ +

+ +This function will essentially create a smooth gradient, but because of the floor operation $$\left(\lfloor \cdot \rfloor \right)$$, the saturation will go from 0 to 1 between each integer value of the magnitude. +Here, it is clear that the magnitude is increasing as $$z^2$$ from the origin; however, because the saturation is fluctuating so much, it is difficult to see the phase pattern next to each contour. +This can be fixed simply by adding an offset to the shading function such that, + +$$ +g(r) = \frac{1}{2} + \frac{1}{2}\left(r-\lfloor r \rfloor \right). +$$ + +Which will produce the following image: + +

+ +

+ +This means that the saturation will fluctuate from $$\frac12$$ to 1 instead of from 0 to 1, which makes it way easier to see phase information next to contours. +Again, there is a lot of ways to play with these equations, so feel free to use whatever function you want! +As long as some sort of rounding operation is used to establish some form of integer value for the magnitude, it should be possible to create contours of various types. + +Unfortunately, there the changing saturation only shows changes in the complex magnitude, and changing hue only shows changes in the complex phase. +Neither the magnitude or the phase directly show what is happening in real or imaginary space with the output. +To show this, we might want to draw gridlines that color out grid black whenever the imaginary or real components of the output function are integer values. + +For example, let's go back to a simpler function $$f(z) = z$$. +If we draw lines on this plot, corresponding to integer values in the output, we get a simple grid + +

+ +

+ +Like before, the choice of which function to use in order to create the gridlines is somewhat arbitrary. +It is important to choose a function that sharply drops to 0 or peaks at 1 for all integer values, and then you multiply this by the output of $$f(z)$$. +For these purposes, we chose the following function + +$$ +h(z) = |\sin(\pi\times\text{Re}(f(z)))^t|\times|\sin(\pi\times\text{Im}(f(z)))^t|, +$$ + +where $$t$$ is some threshold value, and was set to be 0.1 in our plot. +A plot of h(z) for $$f(z) = z$$ where $$z\in\mathbb{R}$$ is shown below: + +

+ +

+ +So, putting it altogether and returning to the function of $$f(z) = z^2$$, we find the following image. + +

+ +

+ +Here, the diagonal lines through the center represent integer values along the imaginary axis for $$f(z)$$ and the vertical and horizontal represent integer values of the real axis for $$f(z)$$. + +## Example Code + +Here is the full script to generate a domain colored output of $$f(z)=z^2$$. + +### Bibliography + +{% references %} {% endreferences %} + + + +## License + +##### Code Examples + +The code examples are licensed under the MIT license (found in [LICENSE.md](https://github.com/algorithm-archivists/algorithm-archive/blob/master/LICENSE.md)). + +##### Text + +The text of this chapter was written by [James Schloss](https://github.com/leios) and is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International License](https://creativecommons.org/licenses/by-sa/4.0/legalcode). + +[

](https://creativecommons.org/licenses/by-sa/4.0/) + +##### Images/Graphics + +##### Pull Requests + +The following pull requests have modified the text or graphics of this chapter: +- none diff --git a/contents/domain_coloring/res/absz.png b/contents/domain_coloring/res/absz.png index 01d8e032de1a08462e3e20d4ad055c83b24e76de..9c2715043ac889f7f5a3427d2b9aec224579b48a 100644 GIT binary patch literal 51450 zcmb5V1z40{*Dj6$C}kiZAfXb9BHb;54k_IrodeRHiim_r!_eK$&SkWkXl`w+?n@mNusJPVR3_=}p}%&2VtsP}PZ+4h(e;?=R-> z2tGF=p5fQ*b-F@hkoquWaxG<{Hy+Y^zmi3R{#)zh6#?6eeqXu+gHzdJyts;yK^gbM zIBl!+opjMngv-S-^CaS6dQ9C&JTLc4Z?FqIJ`m zh~GK3ricS0Bs&>6;HY|}V z6Ps`P-g(XppX+t})%atd_5P-fx2^a`Hy1N@(MbQ?!+7uQ&w+2INIPT#1Rvhu2$aC{ z>NDi%j4*ooRVz=Tf01hEU?M{DI*OR@%SSm=$~w#PsG?}j!*bQHulqT2lN`HZ3wu}( zZp*UNMH@*mAX&95Y`*^};KA^WsyeccSQQuVlWV`f;`pRj8(O?atGgtT?CIRMxkhQ& z_C=c3bSy73Ow_+$j;o&&Iy!D3L0;eo9sFc$JjQ|mATA8m{T3r zhKCd`zEmr!Uu^BLRV<2W5%Q@B_WPKGHTA7w$n~8fW4JiUBbn!Oh;WwEl0-&w??ekM 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a/literature.bib +++ b/literature.bib @@ -217,3 +217,47 @@ @misc{chaos-game-wolf url={http://mathworld.wolfram.com/ChaosGame.html}, year={2019} } + +#------------------------------------------------------------------------------# +# Domain Coloring +#------------------------------------------------------------------------------# + +@misc{hsv, + title={Wikipedia: HSL and HSV}, + url={https://en.wikipedia.org/wiki/HSL_and_HSV}, + year={2020} +} + +@article{schloss2019, + title={Massively parallel split-step Fourier techniques for simulating quantum systems on graphics processing units}, + author={Schloss, James}, + year={2019} +} + +@book{pethick2008, + title={Bose--Einstein condensation in dilute gases}, + author={Pethick, Christopher J and Smith, Henrik}, + year={2008}, + publisher={Cambridge university press} +} + +@book{wegert2012, + title={Visual complex functions: an introduction with phase portraits}, + author={Wegert, Elias}, + year={2012}, + publisher={Springer Science \& Business Media} +} + +@article{poelkedomain, + title={Domain Coloring of Complex Functions}, + author={Poelke, Konstantin and Polthier, Konrad} +} + +@article{lundmark2004, + title={Visualizing complex analytic functions using domain coloring}, + author={Lundmark, Hans}, + journal={Recuperado el}, + volume={24}, + year={2004} +} + From 6f280268e0c9d9388d3b8474078b75d88b5390c7 Mon Sep 17 00:00:00 2001 From: James Schloss Date: Wed, 24 Jun 2020 18:09:59 -0400 Subject: [PATCH 6/7] cleaning up the PR --- contents/domain_coloring/code/gnuplot/absz.gp | 32 ------- .../domain_coloring/code/gnuplot/domain.png | Bin 739841 -> 0 bytes .../code/gnuplot/domain_coloring_rgb.gp | 66 --------------- .../code/gnuplot/domain_coloring_tex.gp | 78 ------------------ .../domain_coloring/code/gnuplot/shade.gp | 47 ----------- contents/domain_coloring/code/gnuplot/z2.gp | 9 -- .../domain_coloring/code/gnuplot/z2_hsv.gp | 9 -- contents/domain_coloring/domain_coloring.md | 11 ++- 8 files changed, 5 insertions(+), 247 deletions(-) delete mode 100644 contents/domain_coloring/code/gnuplot/absz.gp delete mode 100644 contents/domain_coloring/code/gnuplot/domain.png delete mode 100644 contents/domain_coloring/code/gnuplot/domain_coloring_rgb.gp delete mode 100644 contents/domain_coloring/code/gnuplot/domain_coloring_tex.gp delete mode 100644 contents/domain_coloring/code/gnuplot/shade.gp delete mode 100644 contents/domain_coloring/code/gnuplot/z2.gp delete mode 100644 contents/domain_coloring/code/gnuplot/z2_hsv.gp diff --git a/contents/domain_coloring/code/gnuplot/absz.gp b/contents/domain_coloring/code/gnuplot/absz.gp deleted file mode 100644 index 07846ab34..000000000 --- a/contents/domain_coloring/code/gnuplot/absz.gp +++ /dev/null @@ -1,32 +0,0 @@ -set terminal epslatex standalone size 12cm,10cm color -set output "absz.tex" -#set terminal pngcairo -#set output "absz.png" - -set size square -set view map - -set samples 1000 -set isosamples 1000 - -set xrange[-1:1] -set yrange[-1:1] 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domain coloring chapter. Ready for merge! --- contents/domain_coloring/domain_coloring.md | 25 +++++++++++++-------- 1 file changed, 16 insertions(+), 9 deletions(-) diff --git a/contents/domain_coloring/domain_coloring.md b/contents/domain_coloring/domain_coloring.md index f20779bf2..31facd2cf 100644 --- a/contents/domain_coloring/domain_coloring.md +++ b/contents/domain_coloring/domain_coloring.md @@ -61,9 +61,9 @@ If we use an RGB color scheme, where red represents $$\theta$$ and blue represen

    -As a note here, there is a clear phase discontinuity along the vertical axis. -That is to say that the complex phase wraps around the origin, ranging from 0 (clear) to $$2\pi$$ (red). -In addition, the edges of the plot are blue because the function value increases linearly as we move from the origin. +As a note here, there is a clear phase discontinuity along the horizontal axis, which is a consequence of the fact that +the complex phase wraps around the origin, ranging from 0 (clear) to $$2\pi$$ (red). +In addition, the edges of the plot are blue because the function's magnitude increases linearly as we move from the origin. If we instead look at the function $$f(z) = z^2$$, we can generate a similar plot: @@ -75,20 +75,20 @@ Here, it is clear that the complex phase wraps around the origin twice, creating This indicates a $$4\pi$$ phase winding. For some purposes, such as vortex tracking for inviscid fluids, this visualization is ideal, because a vortex is located precisely at the center of the phase discontinuity {{ "schloss2019" | cite }} {{ "pethick2008" | cite }}. For other purposes, the discontinuity is visually distracting, and for this reason, many people use an HSV scheme for plotting complex functions {{ "wegert2012" | cite }} {{ "poelkedomain" | cite }} {{ "lundmark2004" | cite }}. -So here is the same function $$\left(f(z)=z^2\right)$$, but using hue to represent the complex phase and saturation to represent the magnitude. +So here is the same function $$\left(f(z)=z^2\right)$$, but using hue to represent the complex phase and saturation to represent the magnitude:

    -Here, the value for HSV was always set to 1. +In this plot, the Value for HSV was always set to 1. When looking at the edges of the plot, the hue changes rapidly, but each color is mirrored on the opposite edge. This indicates the $$4\pi$$ phase winding we saw in the RGB plot. Also, because the complex magnitude increases as we move further from the center of the plot, the saturation also increases. Thus the center of the plot is completely washed out! We need to fix this in subsequent plots to make them more representative of the actual data. -One easy way to show that the complex magnitude is increasing as we move further from the origin is with contours. +One easy way to show the increasing complex magnitude without sacrificing phase information is by using contours. Essentially, at ever integer value of the magnitude, we want to draw some kind of line. There are a number of ways to generate these lines, and one simple way is by using an alternative shading function like so: @@ -133,7 +133,7 @@ If we draw lines on this plot, corresponding to integer values in the output, we Like before, the choice of which function to use in order to create the grid lines is somewhat arbitrary. It is important to choose a function that sharply drops to 0 or peaks at 1 for all integer values, and then we simply plug values of $$f(z)$$ into this function. -For these purposes, we chose the following function +For the purposes of this chapter, we chose the following function $$ h(z) = |\sin(\pi\times\text{Re}(f(z)))^t|\times|\sin(\pi\times\text{Im}(f(z)))^t|, @@ -154,11 +154,18 @@ So, putting it all together and returning to the function of $$f(z) = z^2$$, we Here, the diagonal lines through the center represent integer values along the imaginary axis for $$f(z)$$ and the vertical and horizontal lines represent integer values of the real axis for $$f(z)$$. An easy way to determine which lines correspond to which integer values is by plugging in certain values for $$z$$ into $$f(z)$$. -For example, there is a black line at $$z = 1 + 1i$$ where $$f(z) = 2i$$, this means that all values along that contour correspond to values that are constrainged to having an imaginary component of precisely 2. +For example, there is a black line at $$z = 1 + 1i$$ where $$f(z) = 2i$$, this means that all values along that contour correspond to values that are constrained to having an imaginary component of precisely 2. Overall, there are plenty of interesting ways to plot complex functions and make really compelling and beautiful images! -We will be using domain coloring in other contexts throughtout this text when describing methods that heavily use complex space. +We will be using domain coloring in other contexts throughout this text when describing methods that heavily use complex space. +## Video Explanation + +Here is a video describing domain coloring: + +
    + +
    ## Example Code

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