Color wheel graphs of complex functions

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In mathematics, a complex function is a function with the complex numbers (see the imaginary numbers and the complex plane) as both its domain and codomain. The complex color wheel method assigns a color to the point of complex plane.

Methods[edit]

The assignment of color to complex point could be diverse, but usually one of the following two:

  • the origin is white, 1 is red, -1 is cyan, and a point at the infinity is black,

or

  • the origin is black, −1 is red, 1 is cyan, and a point at the infinity is white.
HL plot of z : the origin is black, -1 is red, 1 is cyan, and a point at the infinity is white
Color wheel graph of complex function \sin(1/z) (according to the first definition). Black parts inside refer to numbers having large absolute values. This function has an essential singularity at z=0.

More precisely, the argument of the complex number defines the hue value while the modulus defines the light value of the color in the HLS (hue, lightness, saturation) color model; for a given (H,L) pair we choose the maximal saturation value. In both assignments, vivid colors of the rainbow are rotating in a continuous way on the complex unit circle, so the 6th roots of unity (counted from the 0th, the 1) are: red, yellow, green, cyan, blue, and violet. In addition, it is common in the assignments that the gradiations of colors belonging to two complex numbers close to each other are close to each other, and that colors of complex numbers of the same argument are gradiations of the same color – the one which has greater absolute value is darker (according to the first definition) or lighter (according to the second definition).

However, the HSL color space is not perceptually uniform, leading to streaks of perceived brightness at yellow, cyan, and magenta, even though their absolute values are the same as red, green, and blue, and a halo around L=0.5. Use of the Lch color space corrects this, but makes the images more drab.

History[edit]

The color wheel method was probably first used in publication by Larry Crone in 1987.[citation needed]

Other references[edit]