representation. These definitions amount to a geometric warping of hexagons into circles: each side of the hexagon is mapped linearly onto a 60° arc of the circle (). After such a transformation, hue is precisely the angle around the origin and chroma the distance from the origin: the angle and magnitude of the vector pointing to a color. Sometimes for image analysis applications, this hexagon-to-circle transformation is skipped, and hue and chroma (we'll denote these H2 and C2) are defined by the usual cartesian-to-polar coordinate transformations (). The easiest way to derive those is via a pair of cartesian chromaticity coordinates which we'll call α and β: (The atan2 function, a "two-argument arctangent", computes the angle from a cartesian coordinate pair.) Notice that these two definitions of hue (H and H2) nearly coincide, with a maximum difference between them for any color of about 1.12°—which occurs at twelve particular hues, for instance , —and with for every multiple of 30°. The two definitions of chroma (C and C2) differ more substantially: they are equal at the corners of our hexagon, but at points halfway between two corners, such as , we have , but , a difference of about 13.4%. Lightness While the definition of hue is relatively uncontroversial—it roughly satisfies the criterion that colors of the same perceived hue should have the same numerical hue—the definition of a lightness or value dimension is less obvious: there are several possibilities depending on the purpose and goals of the representation. Here are four of the most common (; three of these are also shown in ): The simplest definition is just the arithmetic mean, i.e. average, of the three components, in the HSI model called intensity (). This is simply the projection of a point onto the neutral axis—the vertical height of a point in our tilted cube. The advantage is that, together with Euclidean-distance calculations of hue and chroma, this representation preserves distances and angles from the geometry of the RGB cube. In the HSV "hexcone" model, value is defined as the largest component of a color, our M above (). This places all three primaries, and also all of the "secondary colors"—cyan, yellow, and magenta—into a plane with white, forming a hexagonal pyramid out of the RGB cube. In the HSL "bi-hexcone" model, lightness is defined as the average of the largest and smallest color components (), i.e. the