Centroid of a set of N-dimensional points

In analytic geometry, the centroid of a set of points is a point in the same domain as the set. The centroid point is chosen to show a property which can be calculated for that set.

Consider the centroid defined as the arithmetic mean of a set of points of arbitrary dimension.

Task

Create a function in your chosen programming language to calculate such a centroid using an arbitrary number of points of arbitrary dimension.

Test your function with the following groups of points

one-dimensional: (1), (2), (3)

two-dimensional: (8, 2), (0, 0)

three-dimensional: the set (5, 5, 0), (10, 10, 0) and the set (1, 3.1, 6.5), (-2, -5, 3.4), (-7, -4, 9), (2, 0, 3)

five-dimensional: (0, 0, 0, 0, 1), (0, 0, 0, 1, 0), (0, 0, 1, 0, 0), (0, 1, 0, 0, 0)

Stretch task

   Show a 3D plot image of the second 3-dimensional set and its centroid.

See Also

[Wikipedia page] [Wolfram Mathworld on Centroid]

Julia

using Plots

struct Point{T, N}
    v::Vector{T}
end

function centroid(points::Vector{Point{T, N}}) where N where T
    arr = zeros(T, N)
    for p in points, (i, x) in enumerate(p.v)
        arr[i] += x
    end
    return Point{T, N}(arr / length(points))
end

function centroid(arr)
    isempty(arr) && return Point{Float64, 0}(arr)
    n = length(arr[begin])
    t = typeof(arr[begin][begin])
    return centroid([Point{t, n}(v) for v in arr])
end

const testvecs = [[[1], [2], [3]],
                  [(8, 2), (0, 0)],
                  [[5, 5, 0], [10, 10, 0]],
                  [[1.0, 3.1, 6.5], [-2, -5, 3.4], [-7, -4, 9.0], [2.0, 0.0, 3.0],],
                  [[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0],],
                 ]

function test_centroids(tests)
    for t in tests
        isempty(t) && error("The empty set of points $t has no centroid")
        vvec = [Point{Float64, length(t[begin])}(collect(v)) for v in t]
        println("$t => $(centroid(vvec))")
    end
    xyz = [p[1] for p in tests[4]], [p[2] for p in tests[4]], [p[3] for p in tests[4]]
    cpoint = centroid(tests[4]).v
    for i in eachindex(cpoint)
        push!(xyz[i], cpoint[i])
    end
    scatter(xyz..., color = [:navy, :navy, :navy, :navy, :red], legend = :none)
end

test_centroids(testvecs)

Output:

[[1], [2], [3]] => Point{Float64, 1}([2.0])
[(8, 2), (0, 0)] => Point{Float64, 2}([4.0, 1.0])
[[5, 5, 0], [10, 10, 0]] => Point{Float64, 3}([7.5, 7.5, 0.0])
[[1.0, 3.1, 6.5], [-2.0, -5.0, 3.4], [-7.0, -4.0, 9.0], [2.0, 0.0, 3.0]] => Point{Float64, 3}([-1.5, -1.475, 5.475])
[[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0]] => Point{Float64, 5}([0.0, 0.25, 0.25, 0.25, 0.25])

Wren

var centroid = Fn.new { |pts|
    var n = pts.count
    if (n == 0) Fiber.abort("List must contain at least one point.")
    var d = pts[0].count
    if (pts.skip(1).any { |p| p.count != d }) {
        Fiber.abort("Points must all have the same dimension.")
    }
    var res = List.filled(d, 0)
    for (j in 0...d) {
        for (i in 0...n) {
            res[j] = res[j] + pts[i][j]
        }
        res[j] = res[j] / n
    }
    return res
}

var points = [
    [ [1], [2], [3] ],
    [ [8, 2], [0, 0] ],
    [ [5, 5, 0], [10, 10, 0] ],
    [ [1, 3.1, 6.5], [-2, -5, 3.4], [-7, -4, 9], [2, 0, 3] ],
    [ [0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0] ]
]

for (pts in points) {
    System.print("%(pts) => Centroid: %(centroid.call(pts))")
}

Output:

[[1], [2], [3]] => Centroid: [2]
[[8, 2], [0, 0]] => Centroid: [4, 1]
[[5, 5, 0], [10, 10, 0]] => Centroid: [7.5, 7.5, 0]
[[1, 3.1, 6.5], [-2, -5, 3.4], [-7, -4, 9], [2, 0, 3]] => Centroid: [-1.5, -1.475, 5.475]
[[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0]] => Centroid: [0, 0.25, 0.25, 0.25, 0.25]