The first question that should be considered is, ``Does the perceptual color space have a metric?'' For a space Y to have a metric, there must exist a function for that has the properties (1) if and only if ; (2) ; and (3) for all [Itô 1993, page 1014,]. The third property is called the triangle inequality.
Color researchers have often made an implicit assumption that the perceptual color space has a metric. This is one of the foundation blocks upon which all colorimetry is based. The three formal conditions for a metric space can be reformulated as the following assumptions for the existence of a perceptual color metric space.
A stronger form of a metric space is a Euclidean metric space. An n-dimensional metric space Y is considered to be Euclidean when for all points , in the space. There have been efforts to create a representation of perceptual color space such that it conforms to a local Euclidean metric [Wyszecki & Stiles 1967], a so--called uniform color scale. However, a fully Euclidean representation of perceptual color space has as yet eluded researchers.
A weaker form of a metric space is a Riemann space. The metric in a Riemann space is the full quadratic form. Thus in an 3--dimensional Riemann space R, the metric is defined as follows:
where for . When the cross--product terms are zero, a Riemann space reduces to a Euclidean space.