Algorithm for Additive Color Mixing for RGB Values

I'm looking for an algorithm to do additive color mixing for RGB values.

Is it as simple as adding the RGB values together to a max of 256?

(r1, g1, b1) + (r2, g2, b2) =
    (min(r1+r2, 256), min(g1+g2, 256), min(b1+b2, 256))  

Solution 1:

To blend using alpha channels, you can use these formulas:

r = new Color();
r.A = 1 - (1 - fg.A) * (1 - bg.A);
if (r.A < 1.0e-6) return r; // Fully transparent -- R,G,B not important
r.R = fg.R * fg.A / r.A + bg.R * bg.A * (1 - fg.A) / r.A;
r.G = fg.G * fg.A / r.A + bg.G * bg.A * (1 - fg.A) / r.A;
r.B = fg.B * fg.A / r.A + bg.B * bg.A * (1 - fg.A) / r.A;

fg is the paint color. bg is the background. r is the resulting color. 1.0e-6 is just a really small number, to compensate for rounding errors.

NOTE: All variables used here are in the range [0.0, 1.0]. You have to divide or multiply by 255 if you want to use values in the range [0, 255].

For example, 50% red on top of 50% green:

// background, 50% green
var bg = new Color { R = 0.00, G = 1.00, B = 0.00, A = 0.50 };
// paint, 50% red
var fg = new Color { R = 1.00, G = 0.00, B = 0.00, A = 0.50 };
// The result
var r = new Color();
r.A = 1 - (1 - fg.A) * (1 - bg.A); // 0.75
r.R = fg.R * fg.A / r.A + bg.R * bg.A * (1 - fg.A) / r.A; // 0.67
r.G = fg.G * fg.A / r.A + bg.G * bg.A * (1 - fg.A) / r.A; // 0.33
r.B = fg.B * fg.A / r.A + bg.B * bg.A * (1 - fg.A) / r.A; // 0.00

Resulting color is: (0.67, 0.33, 0.00, 0.75), or 75% brown (or dark orange).

You could also reverse these formulas:

var bg = new Color();
if (1 - fg.A <= 1.0e-6) return null; // No result -- 'fg' is fully opaque
if (r.A - fg.A < -1.0e-6) return null; // No result -- 'fg' can't make the result more transparent
if (r.A - fg.A < 1.0e-6) return bg; // Fully transparent -- R,G,B not important
bg.A = 1 - (1 - r.A) / (1 - fg.A);
bg.R = (r.R * r.A - fg.R * fg.A) / (bg.A * (1 - fg.A));
bg.G = (r.G * r.A - fg.G * fg.A) / (bg.A * (1 - fg.A));
bg.B = (r.B * r.A - fg.B * fg.A) / (bg.A * (1 - fg.A));

or

var fg = new Color();
if (1 - bg.A <= 1.0e-6) return null; // No result -- 'bg' is fully opaque
if (r.A - bg.A < -1.0e-6) return null; // No result -- 'bg' can't make the result more transparent
if (r.A - bg.A < 1.0e-6) return bg; // Fully transparent -- R,G,B not important
fg.A = 1 - (1 - r.A) / (1 - bg.A);
fg.R = (r.R * r.A - bg.R * bg.A * (1 - fg.A)) / fg.A;
fg.G = (r.G * r.A - bg.G * bg.A * (1 - fg.A)) / fg.A;
fg.B = (r.B * r.A - bg.B * bg.A * (1 - fg.A)) / fg.A;

The formulas will calculate that background or paint color would have to be to produce the given resulting color.

If your background is opaque, the result would also be opaque. The foreground color could then take a range of values with different alpha values. For each channel (red, green and blue), you have to check which range of alphas results in valid values (0 - 1).

Solution 2:

It depends on what you want, and it can help to see what the results are of different methods.

If you want

Red + Black        = Red
Red + Green        = Yellow
Red + Green + Blue = White
Red + White        = White 
Black + White      = White

then adding with a clamp works (e.g. min(r1 + r2, 255)) This is more like the light model you've referred to.

If you want

Red + Black        = Dark Red
Red + Green        = Dark Yellow
Red + Green + Blue = Dark Gray
Red + White        = Pink
Black + White      = Gray

then you'll need to average the values (e.g. (r1 + r2) / 2) This works better for lightening/darkening colors and creating gradients.

Solution 3:

Fun fact: Computer RGB values are derived from the square root of photon flux. So as a general function, your math should take that into account. The general function for this for a given channel is:

blendColorValue(a, b, t)
    return sqrt((1 - t) * a^2 + t * b^2)

Where a and b are the colors to blend, and t is a number from 0-1 representing the point in the blend you want between a and b.

The alpha channel is different; it doesn't represent photon intensity, just the percent of background that should show through; so when blending alpha values, the linear average is enough:

blendAlphaValue(a, b, t)
    return (1-t)*a + t*b;

So, to handle blending two colors, using those two functions, the following pseudocode should do you good:

blendColors(c1, c2, t)
    ret
    [r, g, b].each n ->
        ret[n] = blendColorValue(c1[n], c2[n], t)
    ret.alpha = blendAlphaValue(c1.alpha, c2.alpha, t)
    return ret

Incidentally, I long for a programming language and keyboard that both permits representing math that (or more) cleanly (the combining overline unicode character doesn't work for superscripts, symbols, and a vast array of other characters) and interpreting it correctly. sqrt((1-t)*pow(a, 2) + t * pow(b, 2)) just doesn't read as clean.