Three phase-offset cosine waves generate smooth procedural color palettes

To map a scalar t to color smoothly, evaluate color = a + b·cos(2π·(c·t + d)), where a, b, c, d are vec3 parameters controlling the R, G, B channels independently. Parameter a sets the center/offset of the color range; b sets the amplitude, hence saturation and range of variation; c sets the frequency, i.e. how many color cycles occur per unit of t; and d sets the per-channel phase offset that shifts which hues appear at which t. When the three channel phases are equal the waves are in sync and produce greyscale; offsetting them by 1/3 and 2/3 of a cycle produces a full color-wheel sweep. Smaller offsets yield analogous, harmonious hues; larger offsets yield complementary contrasts. Because it is a single continuous, periodic formula, the palette is differentiable and GPU-efficient, replacing lookup tables — ideal for procedural variation in shaders. Interactive palette generators let you design the four vec3 values visually.

Examples

vec3 palette(float t){
    vec3 a=vec3(0.5), b=vec3(0.5), c=vec3(1.0), d=vec3(0.0,0.33,0.67);
    return a + b*cos(6.28318*(c*t + d));
}
vec3 col = palette(length(uv));

Assessment

For a=vec3(0.5), b=vec3(0.5), c=vec3(1.0), d=vec3(0.0), describe the colors at t=0, 0.25, 0.5, 0.75. Set all three phase offsets in d equal — what does the palette look like? Then change d.x to 0.5 (or phase-shift so red appears at t=0.5) and describe the color shift.