Algorithm for a diagonal non-linear gradient

Question

I'm looking for an algorithm to generate the following image:

I'm only interested in the data shown in the upper left half (diagonal) of the square.

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I'm currently at a starting point where I have the following simple linear function:

double GetColorAt(double x, double y)
{
   return 1 - (x + y);
}

where x runs from left to right and y from top to bottom in the ranges of [0...1]. The return value is a color component for RGB channels in the range of [0...1]. It generates me this:

Answer 1

You lucked out, this is the sort of puzzle that I can't resist.

Analyzing the image, two things leap out.

• r = g = b, i.e. this is a pure grayscale image.
• r + a = 255.

This simplifies things greatly, it means you only need to analyze one channel to get the full characterization of the gradient.

Here's a plot of the R values from line 0 and line 128 of the image:

This plot greatly resembles the arc of a bullet as it's pulled down by gravity. So I plugged the basic equation into Excel and asked it to solve for the closest match to line 0 for 255 - (a*(b*x)^2) for varying vales of a and b. It came up with values of a=6.998191873 and b=0.023556823. Those values also closely tracked the ones from line 128 when I substituted x/2, so I knew I was on the right track.

Here's the algorithm in Python:

for y in range(height):
    scale = (256 - y) / 256.0
    for x in range(width):
        r = int(round(255 - (6.998191873 * (0.023556823 * x/scale) ** 2)))
        r = max(r, 0)
        ld2[x, y] = (r, r, r, 255-r)

It's not quite exact, which is most noticeable at the bottom of the image where it seems to be a pixel off. But it's visually identical.

Answer 2

If anyone likes testing their theory of how to generate the gradient by looking at hex rather than the image, the following may help. It is resized down by a factor of 10 to 25px by 25px.

0000000: ff fe fd fa f7 f3 ed e8 e1 d9 d1 c8 be b2 a7 9a 8c 7e 6e 5e 4d 3c 2a 12 18  .................~n^M<*..
0000019: ff fe fc fa f6 f1 ec e6 de d6 cd c3 b8 ac 9f 91 83 73 62 51 3e 2c 15 08 1e  .................sbQ>,...
0000032: ff fe fc f9 f5 f0 ea e3 db d2 c8 bd b1 a4 96 87 77 66 54 41 2d 17 02 06 20  ................wfTA-... 
000004b: ff fe fc f9 f4 ef e8 e1 d8 ce c3 b7 aa 9c 8c 7c 6a 58 44 2f 18 04 00 06 20  ...............|jXD/.... 
0000064: ff fe fb f8 f3 ed e6 de d4 c9 bd b0 a1 92 81 6f 5b 47 31 1a 04 00 00 06 20  ...............o[G1..... 
000007d: ff fe fb f7 f2 eb e3 da cf c4 b6 a8 98 86 74 60 4a 34 1b 04 00 01 00 06 20  ..............t`J4...... 
0000096: ff fe fb f6 f0 e9 e0 d6 ca bd ae 9e 8c 79 64 4e 37 1c 05 00 00 00 00 06 20  .............ydN7....... 
00000af: ff fd fa f5 ef e7 dd d1 c4 b5 a5 93 7f 69 52 3a 1d 03 00 00 00 00 00 06 20  .............iR:........ 
00000c8: ff fd f9 f4 ed e4 d9 cc bd ac 9a 85 6f 57 3d 20 05 00 00 00 00 00 00 06 20  ............oW= ........ 
00000e1: ff fd f9 f3 ea e0 d3 c5 b4 a1 8c 75 5c 41 22 05 00 00 00 00 00 00 00 06 20  ...........u\A"......... 
00000fa: ff fd f8 f1 e7 dc cd bd a9 94 7c 61 45 25 06 00 00 00 00 00 00 00 00 06 20  ..........|aE%.......... 
0000113: ff fc f7 ef e4 d6 c6 b2 9c 84 68 4a 28 07 00 00 00 00 00 00 00 00 00 06 20  ..........hJ(........... 
000012c: ff fc f5 ec df cf bc a6 8c 6f 4f 2b 08 00 00 00 00 00 00 00 00 00 00 06 20  .........oO+............ 
0000145: ff fb f4 e9 da c7 b0 96 77 56 2e 08 00 00 00 00 00 00 00 00 00 00 00 06 20  ........wV.............. 
000015e: ff fa f1 e4 d2 bc a1 81 5d 33 09 00 01 00 00 00 00 00 00 00 00 00 00 06 20  ........]3.............. 
0000177: fe f9 ee de c8 ad 8c 66 38 0a 00 01 00 00 00 00 00 00 00 00 00 00 00 06 20  .......f8............... 
0000190: fe f8 ea d6 bb 99 70 3f 0c 00 01 00 00 00 00 00 00 00 00 00 00 00 00 06 20  ......p?................ 
00001a9: fe f6 e5 cb a8 7d 47 0d 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 06 1f  .....}G..................
00001c2: fe f3 dc ba 8d 51 0f 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 06 20  .....Q.................. 
00001db: fd ee ce a0 5e 12 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 06 20  ....^................... 
00001f4: fc e6 b9 70 16 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 06 1f  ...p.....................
000020d: fa d9 89 1d 00 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 06 20  ........................ 
0000226: f9 b4 26 00 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 06 20  ..&..................... 
000023f: e6 40 00 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 06 20  .@...................... 
0000258: 74 00 02 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 06 20  t.......................