I recently decided that I wanted to make a series of solid colour backgrounds for my laptop. The reasons are nominally grounded in productivity^{1}, but given that the whole exercise sent me down a rabbithole of writing code and then this blog post, let’s just call it a little experiment to see what we can do, shall we?
Step 1: Making a solid colour PNG
Our first step will be to create a solid colour PNG of the right size. I’m going to use the ruby gem ChunkyPNG to do all the heavy lifting. Let’s get started!
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  #!/usr/bin/env ruby require 'chunky_png' HEIGHT = 800 WIDTH = 800 HUE = rand(360) puts "Making a background with hue #{HUE}" LUMINESCENCE = 0.5 image = ChunkyPNG::Image.new(WIDTH, HEIGHT) 1.upto(HEIGHT) do y 1.upto(WIDTH) do x image[x  1, y  1] = ChunkyPNG::Color.from_hsl(HUE, 1, LUMINESCENCE) end end image.save("background.png") 
This script:
 Sets the height and width of our image
 Picks an appropriate hue (randomly set in this instance) and luminence
 Generates the image and sets every pixel to the specified colour
 Saves it to
background.png
Here’s an example, generated using the script above:
Step 2: Let’s make it prettier
OK so we can make solid colour backgrounds automatically, big deal. We can do that in MS paint! What if we want to make them look a bit cooler? And what’s cooler than an automatically generated linear gradient?
This is going to take a few steps, however, and we’re going to have to go back to high school trigonometry. Ready for some maths? Let’s get stuck in.
So: if we apply a linear gradient with an angle θ, all points situated on a line at 90° to this gradient will have the same colour. We can show this visually:
So obviously when we’re just looking at this diagram, we can trace a given point back onto that gradient line. How do we do that mathematically? We use a geometric technique called projection to see how far down the gradient line a given point falls. We can do this by imagining a given point p(x,y) as a point on a rightangled triangle, as follows:
OK, so what’s that distance d, the projection of the point onto the gradient? Well, we know from trigonometry that:
OK, but what does that mean? After all, right now all we know is the angle of our gradient (we get to decide this), and the x and y coordinates of our point P (we’re going to iterate over all points, so this is going to change as we go).
So how do we calculate any of these values?
Let’s start with h. This is the distance between the topleft pixel and our point P. Pythagorus has a theorem for this:
Easy! Now, what about ɸ? This is the angle of the rightangle triangle, and it’s just the difference between our gradient angle θ and, well, whatever angle our point P makes with the top of the screen. Trigonometry to the rescue:
(We mark the whole thing as absolute as we always want to have a positive value here.)
So that’s the maths. What does it look like in code? Here’s a ruby function that, given a point with a given x and y, and a given gradient, calculates the projection of the point on the gradient line:
1 2 3 4 5 6 7 8 9 10 11 12 13 14  # Calculate a gradient projection, given a point (x,y) and a gradient # angle of `gradient` in radians. def gradient_projection(x, y, gradient) # Calculate angles point_angle = Math::atan(y / x) triangle_angle = (gradient  point_angle).abs # Calculate distance from (0, 0) to (x, y) point_distance = Math::sqrt(x**2 + y**2) # Calculate projection return point_distance * Math::cos(triangle_angle) end 
So how do we use this in constructing a gradient? There’s all kinds of ways you can do this  in the example below I’m going to transition between two luminescence values  one at the top left and one at the bottom right:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51  #!/usr/bin/env ruby require 'chunky_png' # Constants HEIGHT = 800 WIDTH = 800 # We use this for interpolating luminescence DIAGONAL = Math::sqrt(HEIGHT**2 + WIDTH**2) HUE = rand(360) LUMINESCENCE_START = 0.5 LUMINESCENCE_END = 0.3 # We need to convert this from degrees to radians GRADIENT_ANGLE_DEGREES = 45 GRADIENT_ANGLE = GRADIENT_ANGLE_DEGREES * Math::PI / 180 # Calculate a gradient projection, given a point (x,y) and a gradient # angle of `gradient` in radians. def gradient_projection(x, y, gradient) # Calculate angles point_angle = Math::atan(y / x) triangle_angle = (gradient  point_angle).abs # Calculate distance from (0, 0) to (x, y) point_distance = Math::sqrt(x**2 + y**2) # Calculate projection return point_distance * Math::cos(triangle_angle) end image = ChunkyPNG::Image.new(WIDTH, HEIGHT) 1.upto(HEIGHT) do y 1.upto(WIDTH) do x # Calculate projection as a fraction of DIAGONAL proj = gradient_projection(x, y, GRADIENT_ANGLE) proj_fxn = proj / DIAGONAL lum = LUMINESCENCE_START + (LUMINESCENCE_END  LUMINESCENCE_START) * proj_fxn image[x  1, y  1] = ChunkyPNG::Color.from_hsl(HUE, 1, lum) end end image.save("background2.png") 
Looks good? It works (almost!) perfectly!
Astute readers will notice that something weird is going on here  the bottomleft side of the image is fine, but the topright isn’t doing what it should. If you’re feeling like a challenge, see if you can identify the issue before reading on!
OK, if you just want to know what the issue is  it’s in how ruby treats integer division (namely, it gets rid of any fractions). This means that our issue exists in the following line:
1  point_angle = Math::atan(y / x) 
If y < x
, y / x
will always equal zero. We can fix this by converting y
to a floating point decimal first:
1  point_angle = Math::atan(y.to_f / x) 
In the end, I just stepped through all 360 degrees of hue at 10 degrees per step to create 36 backgrounds to cycle through, but you may want to pick out your favourite colours, or play around with the logic.
Bonus round: radial gradients!
We’re not just limited to linear gradients, especially not given we’re generating this pixelbypixel. Let’s have a go at generating a radial gradient centred on the topleft, rather than our linear gradient.
This is actually easier than the linear gradient  all we need to do is calculate the distance from top left and use that to calculate luminescence:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34  #!/usr/bin/env ruby require 'chunky_png' # Constants HEIGHT = 800 WIDTH = 800 HUE = rand(360) LUMINESCENCE_START = 0.5 LUMINESCENCE_END = 0.3 def dist(x, y) Math::sqrt(x**2 + y**2) end DIAGONAL = dist(WIDTH, HEIGHT) image = ChunkyPNG::Image.new(WIDTH, HEIGHT) 1.upto(HEIGHT) do y 1.upto(WIDTH) do x # Calculate projection as a fraction of DIAGONAL dist_fxn = dist(x, y).to_f / DIAGONAL lum = LUMINESCENCE_START + (LUMINESCENCE_END  LUMINESCENCE_START) * dist_fxn image[x  1, y  1] = ChunkyPNG::Color.from_hsl(HUE, 1, lum) end end image.save("background4.png") 
In fact, it’s so easy that we could even specify an origin point which isn’t the top left…
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41  #!/usr/bin/env ruby require 'chunky_png' # Constants HEIGHT = 800 WIDTH = 800 HUE = rand(360) LUMINESCENCE_START = 0.5 LUMINESCENCE_END = 0 ORIGIN_X = 400 ORIGIN_Y = 200 def dist(x, y) Math::sqrt(x**2 + y**2) end def dist_from_origin(x, y) dist(x  ORIGIN_X, y  ORIGIN_Y) end DIAGONAL = dist(WIDTH, HEIGHT) image = ChunkyPNG::Image.new(WIDTH, HEIGHT) 1.upto(HEIGHT) do y 1.upto(WIDTH) do x # Calculate projection as a fraction of DIAGONAL dist_fxn = dist_from_origin(x, y).to_f / DIAGONAL lum = LUMINESCENCE_START + (LUMINESCENCE_END  LUMINESCENCE_START) * dist_fxn image[x  1, y  1] = ChunkyPNG::Color.from_hsl(HUE, 1, lum) end end image.save("background5.png") 
So there you have it. We’ve not only generated a randomcolour background, we’ve applied two different gradients to those colours. And I guess I can get back to doing all the things I was supposed to be doing before this distraction came along.

Previously I’d been using Unsplash’s app for OS X to select a random image from their site, but I was wondering if having busy backgrounds actively breaks me out of focus during the day. The obvious solution was to build a series of solidcolour wallpapers  solid colour so there was nothing distracting, and a series so I didn’t have the same colour all the time. ↩
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