-- pos - observer position
-- vec - observer vector
-- target - target position

function angle (pos, vec, target)

    -- Imagine that the observer is wearing spherical helmet of raduis 1.
    -- He looks on the target, and his line of sight crosses the helmet.
    -- We'll calculate this crossing point by normalizing his looking
    -- vector.

    -- Normalization means making length of vector equal to 1, but
    -- still pointing to the same direction.

    local vec_look = vector.normalize(vec)

    -- We also calculate another vector, that is the difference
    -- between observer and target (= position relative to target).

    local vec_diff = vector.subtract(target, pos)

    -- Well, this requires normalization too.

    vec_diff = vector.normalize(vec_diff)

    -- Now both are starting from center and pointing to some point
    -- of surface of helmet. We will calculate the distance of

    -- Calculate the length between two vectors.

    local chord = vector.length(vector.subtract(vec_look, vec_diff))

    -- What we have calculated is a chord on some circle of sphere.
    -- Having this we can calculate the angle.
    -- https://en.wikipedia.org/wiki/Chord_%28geometry%29

    -- c = 2*r*sin(phi / 2)

    -- because r = 1, it degenerates to:

    -- c = 2 * sin(phi / 2)

    -- c/2 = sin(phi / 2)

    -- asin(c/2) = phi / 2

    -- 2 * asin(c/2) = phi

    local angle = 2 * math.asin(chord / 2)

    -- note the result is in radians. Convert it to degrees.

    return angle * 180 / math.pi
end