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Triangular Dissimiliarity Diagrams and Triangular Gradient Shading in PostScript

Visualizing data can be challenging when it involves more than two dimensions. One of my favourite three-dimensional diagrams is the bubble chart, where the two axes represent two dimensions and the size of the bublles represents the third dimension. But sometimes other types of three-dimensional diagrams squezeed into two dimensions can be helpful.

Recently, I needed to find a way to characterize urban areas in three distinct but related dimensions that can be described by three ratios \(\{\alpha\equiv a/z,\beta\equiv b/z,\gamma\equiv c/z\}\) where \(\max\{a,b,c\}\le z\) and thus the ratios are all between zero and one. Specifically, I was looking to find a diagram that characterizes how dissimilar these regions are. Consider displaying these three ratios in an equilateral triangle with axes A (pointing upward), B (pointing southwest), and C (pointing southeast), as shown below, with side length \(w\) and therefore height \(h\equiv w\cdot\sqrt{3/4}=w\cdot\sin(60^\circ)\). Starting from the center of the equilateral triangle with coordinates \((x_c,y_c)\) and noting that the maximum distance from the centre to each corner is \(l\equiv w/\sqrt{3}\), \[ x = x_c + \frac{w}{2} (\gamma-\beta) \] \[ y = y_c + \frac{w}{\sqrt{3}}\left[\alpha-\frac{1}{2}(\gamma+\beta)\right] \] Another way to describe this is to move \(\alpha\cdot l\) in the direction of A, turn 120° counterclockwise, move \(\beta\cdot l\) in the direction of B, turn another 120° counterclockwise, and move \(\gamma\cdot l\) in the direction of C.

Triangular Dissimilarity Diagram: Step 1

In the example above, there are three arrow movements into direction A, six arrow movements in direction B, and ten arrow movements in direction C. This means that the destination point should be mostly in the (green) area towards C and a little in the direction of (blue) area B.

The distance \(r\) from the center point is given by \[ r=l\cdot\sqrt{\alpha^2+\beta^2+\gamma^2-\alpha\beta-\alpha\gamma-\beta\gamma} \] When dissimilarity is smallest and \(\alpha=\beta=\gamma\), the distance is zero. The more dissimilar the values of the three parameters, the greater the distance. When one of the three measures is equal to one and the other two equal to zero, the distance is largest at length \(l\). It is easily shown that the ratio \(r/l\) is proportional to the standard deviation of the three measures so that \(\sigma\equiv(r/l)\sqrt{2}/3\). These caculations prove that the diagram is indeed a dissimilarity diagram.

Dissimilarity alone could be displayed in a one-dimensional diagram, but the second dimension is crucial as well. It describes the source of the dissimilarity by pointing in the direction of the largest contribution to the dissimilarity.

Displaying the target point in the triangular diagram looks a bit dull. It would be nice to give a visual sense of the distance from the center (as a measure of the intensity of dissimilarity) as well as the directionality of the dissimilarity using colours and shades. It makes sense to use primary colours to shade each corner of the equilateral diagram, with white in the middle. Equilateral triangles can be divided neatly into four smaller identical equilateral triangles, as shown in the next step.

Triangular Dissimilarity Diagram: Step 2

It is possible to recursively divide each equilateral triangle into four identical smaller triangles. The next step shows how this works out. One problem to take into account is that the middle triangle points downward instead of upwards.

Triangular Dissimilarity Diagram: Step 3

Repeating the recursion twice more and shading each triangle according to its direction and position relative to the centre provides for increasingly smooth shading.

Triangular Dissimilarity Diagram: Step 4

In the last step, the smooth triangular gradient emerges when the recursion level is set to 7. At that point, the triangle is filled with 16,384 small triangles, and that is sufficient to generate the impression of smoothness. For larger triangles or higher resolution, another one or two recursion steps may be necessary.

The last version also displays the points of interest. For example, the V6T district is mostly of type "B", while V5T and V6R are mostly mixtures of types "A" and "C". Region V6H is an A-B mixture, while V6Z is a B-C mixture.

Triangular Dissimilarity Diagram: Step 5

Below is the PostScript code that implements the diagram with a base length of 500 points and outputs the result into an image of 540 by 480 points. There are separate code elements for upward-pointing and downward-pointing triangles. The color shading is based on a formula that calculates the angle and distance from the center using "sethsbcolor". The hue is defined by the angle, which is calculated from coordinates using the "atan" operator. This creates a small problem when both coordinates are zero. I fix that by adding a tiny amount "epsilon" to the y-coordinate, but it can also be fixed with a simple if-then statement checking for both operators being zero. In that case, the hue can be any number because the saturation will be zero (implying white). The saturation level is the relative distance from the centre. If it is more desirable to have more whiteness in the middle, a simple modification is to remove the square root operator in the "fillUp" and "fillDown" functions.

In the PostScript code below I paint all triangles recursively, which means that the larger triangles are painted over by the small triangles. This is a bit inefficient, and you can fix that by activating the "i depth ge { } if" code that I have currently commented out. The advantage of painting smaller over larger triangles is that it removes linear artificats that may appear when the Postscript code is rendered as a raster image.

Recursion is not entirely easy to implement in PostScript, and thus the code involves tail recursion where temporary coordinates are moved onto the stack before the "trifill" routine is invoked for each of the four smaller triangles. Understand the nature of the recursion before you try to modify the code for other purposes.

%!PS-Adobe %%BoundingBox: 0 0 540 480 %------------------------------------------------------------- % Smooth-Gradient Equilateral Triangle via Recursion Algorithm % (C) 2016 by Werner Antweiler, University of British Columbia %------------------------------------------------------------- %------| Base Length of Equilateral Triangle /base 500 def %------| Recursion Depth /depth 6 def %------| constants, center of triangle /sqrt3 3 sqrt def /sqrt32 sqrt3 2 div def /cx base 2 div def /cy base sqrt3 div 2 div def /cr base sqrt3 div def /epsilon 1 10000 div def %------| wriggle factor for center %------| calculate the shading based on distance from center /fillUp { %--------- upward-pointing triangle /xs xc b4 add cx sub cr div def /ys yc bg 2 div add cy sub cr div def xs ys epsilon add atan 360 div xs dup mul ys dup mul add sqrt 1 sethsbcolor fill } def %------| calculate the shading based on distance from center /fillDown { %-------- downward-pointing triangle /xs xc cx sub cr div def /ys yc bl 2 div add cy sub cr div def xs ys epsilon add atan 360 div xs dup mul ys dup mul add sqrt 1 sethsbcolor fill } def %------| upward-pointing triangle, left corner at (xc,yc) /triUp { /yc exch def /xc exch def newpath xc yc moveto b2 0 rlineto b4 neg bg rlineto closepath fillUp } def %------| downward-pointing triangle, bottom corner at (xc,yc) /triDown { /yc exch def /xc exch def newpath xc yc moveto b4 neg bg rlineto b2 0 rlineto closepath fillDown } def %------| recursion routine for painting triangle up to "depth" %------| invoke as: [level] [direction] [xx] [yy] trifill %------| [level] starts at 0 and increments recursively %------| [direction] is true for upward pointing triangle %------| [direction] is false for downward pointing triangle %------| [xx] [yy] are the starting positions /trifill { /yy exch def /xx exch def /o exch def /i exch def i depth le { /u 2 i exp def /bb base u div def /b2 bb 2 div def /b4 bb 4 div def /bh bb sqrt32 mul def /bg bh 2 div def /bl bb sqrt3 div def o { % i depth ge { %------| only paint smallest triangles xx yy triUp xx b4 add bg yy add triUp xx b2 add yy triUp xx b2 add yy triDown % } if i 1 add true xx yy i 1 add true xx b4 add yy bg add i 1 add true xx b2 add yy i 1 add false xx b2 add yy trifill trifill trifill trifill }{ % i depth ge { %------| only paint smallest triangles xx yy triDown xx b4 sub yy bg add triDown xx b4 add yy bg add triDown xx b4 sub yy bg add triUp % } if i 1 add false xx yy i 1 add false xx b4 sub yy bg add i 1 add false xx b4 add yy bg add i 1 add true xx b4 sub yy bg add trifill trifill trifill trifill } ifelse } if } def %------| shift origin to (20,20) 20 20 translate %------| initialize recursion 0 true 0 0 trifill %------| outline the triangle 1 setlinewidth 0 setgray newpath 0 0 moveto base 0 rlineto base 2 div neg base sqrt32 mul rlineto closepath stroke %------------------------------------------------------------- % Diagram Data and Labels %------------------------------------------------------------- /Helvetica-Bold findfont 15 scalefont setfont -16 -11 moveto (W) show base 4 add -11 moveto (A) show base 2 div 6 sub base sqrt32 mul 4 add moveto (R) show /Helvetica findfont 12 scalefont setfont /abcz { /id exch def /zz exch def /cc exch zz div def /bb exch zz div def /aa exch zz div def gsave 0 setgray cx cy moveto 0 aa cr mul rmoveto 120 rotate 0 bb cr mul rmoveto 120 rotate 0 cc cr mul rmoveto currentpoint /yy exch def /xx exch def xx yy 4 0 360 arc fill xx yy moveto 120 rotate 5 -4 rmoveto id show grestore } def %------------------------------------------------------------- 11.366 51.910 12.238 54.011 (V6T) abcz 3.473 6.784 7.317 10.754 (V6Z) abcz 27.172 25.541 16.273 31.522 (V6H) abcz 44.527 22.172 37.123 51.205 (V5T) abcz 9.631 17.738 11.732 18.566 (V7J) abcz 42.445 31.221 37.383 45.724 (V7L) abcz 33.486 20.158 31.377 39.014 (V6R) abcz

To adjust the diagram, you need to change the value of the "base" constant, currently 500 points. The diagram is painted with a (20,20) offset.

For putting points in the diagram I use the function "abcz" that has four arguments that define my three ratios \(\{\alpha\equiv a/z,\beta\equiv b/z,\gamma\equiv c/z\}\) along with a label. Rather than calculating the x and y positions exactly, I simply move along the three different directions and rotate 120° each time.

Lastly, you may be curious how I generated the JPEG images based on the above PostScript code. For that I use the ImageMagick software that is widely available freely for many different platforms.

convert -density 400 -background white \ -flatten triangle.eps triangle.jpg mogrify -geometry 500x445 triangle.jpg
Posted on Wednesday, March 23, 2016 at 08:55 — #Econometrics | #Software
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