Gradient #

A gradient is used to transition between two or more colours. It is defined by a list of parameter, colour pairs $(p_i, c_i)$ that distributes the colours on a scale and can be sampled using any number. If the number is outside the bounds of the parameters then then colours at the bounds will be the final colour. If the number is between two parameters then an interpolation of the two colours at those parameters will decide the final colour. $$ (p_i, c_i) \mid i = 0, 1, 2, \ldots, n \text{ and } p_0 \leq p_1 \leq \ldots \leq p_n\ $$

Colours#

The colours used to create the gradient. These colours will be interpolated in order to get the final colours.

Parameters#

The parameters used to distribute the colours on the scale. Each parameter has to be paired with a colour.

Interpolation#

The interpolation type defines how the gradient is interpolated i.e. how the gradient moves from one colour to the next. When sampling a parameter $P$ the interpolation combined with the defined colour $c$, parameter $p$ pairs are used to calculate a colour $C$.

Linear#

The linear interpolation is default in most softwares. It normalizes the parameter $P$ relative to its bounding parameters $(p_{i}, p_{i+1})$ using $f(P)$ and then samples the bounding colours $(c_{i}, c_{i+1})$ into the final colour $C$. $$ f(P) = \frac{P-p_{i}} {p_{i+1}-p_{i}} $$ $$ C = c_{i} + f(P) \cdot (c_{i+1} - c_{i}) $$

Cosine#

A cosine sampled gradient creates a smoother colour sampling, avoiding the discontinuity that a linear interpolation introduces at each point. It uses $f(P)$ to normalize the parameter $P$ between $(p_{i}, p_{i+1})$ and apply a cosine transformation before sampling the bounding colours $(c_{i}, c_{i+1})$ into the final colour $C$. $$ f(P) = (1 - cos(\frac{P-p_{i}} {p_{i+1}-p_{i}} \cdot \pi)) / 2 $$ $$ C = c_{i} + f(P) \cdot (c_{i+1} - c_{i}) $$

Logarithmic#

A logarithmic interpolation spaces colour transitions according to a logarithmic scale. This produces a greater colour variation in lower ranges and is useful when visual detail is more important near the lower end of the gradient range.

Let $f(P)$ be the interpolation factor computed using natural logarithms: $$ f(P) = \frac{\log(P) - \log(p_i)}{\log(p_{i+1}) - \log(p_i)} $$ $$ C = c_i + f(P) \cdot (c_{i+1} - c_i) $$

Nearest#

Nearest interpolation snaps the sampled parameter $P$ to the closest defined parameter $p_i$ and returns the associated colour $c_i$ without blending. This creates a stepped, non-continuous gradient.

\[ C = c_k \quad \text{where} \quad k = \arg\min_i |P - p_i| \]

Immediate#

Immediate interpolation assigns colours based on the lower bounding parameter. The output colour stays constant until the next parameter is reached, similar to a "stair-step" behaviour.

\[ C = c_i \quad \text{such that} \quad p_i \leq P < p_{i+1} \]