On fields inspired with the polar HSV -- RGB theory of Colour

TL;DR

This paper introduces a new algebraic structure for the HSV-RGB color space, forming a field through factorization, and explores its mathematical properties and implications for color theory.

Contribution

It develops a novel field structure for the HSV-RGB color space by factorizing with respect to achromatic hues, extending the algebraic framework of color representation.

Findings

The HSV-RGB space forms a semi-field of parabolic-complex functions.

Factorization yields a new field structure for color triplets.

The algebraic properties of this field are analyzed in detail.

Abstract

A three-polar, cf. T. Gregor, J. Halu\v{s}ka, Lexicographical ordering and field operations in the complex plane. Stud. Mat. 41(2014), 123--133., $HSV-RGB$ Colour space $\triangle$ was introduced and studied. It was equipped with operations of addition, subtraction, multiplication, and (partially) division. Achromatic Grey Hues form an ideal $\mathfrak{S}$. Factorizing $\triangle$ by the ideal $\mathfrak{S}$, we obtain a field $\triangle | \mathfrak{S}$. An element (i.e an individual Colour) in $\triangle | \mathfrak{S}$ is a triplet of three triangular coefficients. The set of all triangular coefficients is a subset of a semi-field of parabolic-complex functions. For the parabolic-complex number set, cf.~A. A. Harkin--J. B. Harkin, Geometry of general complex numbers. Mathematics magazine, 77(2004), 118--129.