Calculate distance between colors in HSV space

Given hsv values, normalized to be in the ranges [0, 2pi), [0, 1], [0, 1], this formula will project the colors into the HSV cone and give you the squared (cartesian) distance in that cone:

   ( sin(h1)*s1*v1 - sin(h2)*s2*v2 )^2
 + ( cos(h1)*s1*v1 - cos(h2)*s2*v2 )^2
 + ( v1 - v2 )^2

_{First a short warning: Computing the distance of colors does not make sense (in most cases). Without considering the results of 50 years of research in Colorimetry, things like the CIECAM02 Color Space or perceptual linearity of distance measures, the result of such a distance measure will be counterintuitive. Colors that are "similar" according to your distance measure will appear "very different" to a viewer, and other colors, that have a large "distance" will be undistinguishable by viewers. However...}

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The actual question seems to aim mainly at the "Hue" part, which is a value between 0 and 360. And in fact, the values of 0 and 360 are the same - they both represent "red", as shown in this image:

Now, computing the difference of two of these values boils down to computing the distance of two points on a circle with a circumference of 360. You already know that the values are strictly in the range [0,360). If you did not know that, you would have to use the Floating-Point Modulo Operation to bring them into this range.

Then, you can compute the distance between these hue values, h0 and h1, as

hueDistance = min(abs(h1-h0), 360-abs(h1-h0));

Imagine this as painting both points on a circle, and picking the smaller "piece of the cake" that they describe - that is, the distance between them either in clockwise or in counterclockwise order.

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EDIT Extended for the comment:

• The "Hue" elements are in the range [0,360]. With the above formula, you can compute a distance between two hues. This distance is in the range [0,180]. Dividing the distance by 180.0 will result in a value in [0,1]

• The "Saturation" elements are in the range [0,1]. The (absolute) difference between two saturations will also be in the range [0,1].

• The "Value" elements are in the range [0,255]. The absolute difference between two values will thus be in the range [0,255] as well. Dividing this difference by 255.0 will result in a value in [0,1].

So imagine you have two HSV tuples. Call them (h0,s0,v0) and (h1,s1,v1). Then you can compute the distances as follows:

dh = min(abs(h1-h0), 360-abs(h1-h0)) / 180.0
ds = abs(s1-s0)
dv = abs(v1-v0) / 255.0

Each of these values will be in the range [0,1]. You can compute the length of this tuple:

distance = sqrt(dh*dh+ds*ds+dv*dv)

and this distance will be a metric for the HSV space.